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# Why does dust gather in corners?

I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is?

Edit: No, really, this is a mathematical question. Air blows around the room, which constitutes a vector field. Let's say it blows through a square room with doors near bottom-left and top-right. Air blows from bottom-left to top-right.

There are dust particles in the room, let's say uniformly distributed at first. But after application of this flow they are not uniformly distributed. They pile in the corners. Maybe that's because vortices form more in the corners, maybe some other reason.

It's not because of where I start sweeping.

It is a physics question, but obviously knowledge of Newtonian mechanics won't solve it. It comes down to fluid flow and vector fields, which is math.

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It's harder for brooms to reach corners? I'm not convinced that this is a mathematical question. I suppose any air currents in a room will tend to blow dust to the boundaries of the room, but this is a basic observation. – Qiaochu Yuan Dec 11 '10 at 7:56
Re: your edited question: isn't this easily explained by the fact that a dust particle near the middle of the room will be blown from all sides (assuming the air currents are sufficiently random), but a dust particle near a corner will tend to only be blown toward the corner? I don't think there's anything particularly deep going on here. – Qiaochu Yuan Dec 11 '10 at 8:12
@Qiaochu: I think you should post that as an answer, even if it's a basic observation and not particularly deep. – ShreevatsaR Dec 11 '10 at 8:16
You'll probably need to model additional things like the deposition/erosion of dust on the boundaries, or maybe the inertia of dust particles with respect to the air flow. If you just have massless dust particles advected passively through an incompressible air flow, the distribution of dust will remain uniform. – Rahul Dec 11 '10 at 8:17
@Qiaochu: Why wouldn't a dust particle near a corner have equal probability of being blown towards or away from the corner? If what you said was true, there would tend to be a net flow of air away from the middle of the room, which isn't physically sound. I would agree that the air currents are likely to be weaker near the corner due to the presence of boundary conditions, but I don't know if you can conclude much more than that. – Rahul Dec 11 '10 at 8:20

Air flow in a room will change direction from time to time. (Just the simple act of walking through the room in random directions will shift the air flow.)

The dust particle will be carried in the direction of the air current each time.

Eventually the dust particle will hit a wall and can obviously be carried no further in that direction.

New air currents, in slightly different directions, will then carry the dust particle in the direction of other walls. Because the dust particle is next to the wall, it will be difficult to create an air current that will move the dust particle in a direction 180 degrees in the opposite direction; therefore, the air current will have a tendency to move in the direction of one of the two perpendicular walls (assuming a square room for example's sake).

The interplay of air currents will continue to push the dust particle back and forth between the two perpendicular walls into it eventually arrives in the corner. This will happen because the closer the dust particle gets to the corner (as when it is pushed directly against a wall), the surface area available to capture air currents from the general direction of the corner DECREASES, while the sufrace area available to capture air currents from the perpendicular corner INCREASES. To say another way, the ability to capture an air current that carries the dust particle 180 degrees away from the corner exponentially decreases the closer it gets closer to the corner, while the ability to capture an air current that carries the dust particle 180 degrees towards the corner exponentially increases as it gets closer to the corner.

EVERYTHING in life is a matter of mathematics. There is not an organic or inorganic process anywhere in the known universe that can not be illustrated by mathematics (of course, there are many processes we do not yet know HOW to quantify mathematically, but that is a limitation on our part; not within the inherent reach of mathematics to explain it).

Okay, that makes for an interesting Saturday morning ... cheers.

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I especially like your the statement that EVERYTHING is a matter of mathematics. Upvote just for that. I would add that quantification is not necessary to be mathematical, for instance consider ordering relations and topological relations. – isomorphismes Dec 14 '10 at 2:54
@Guess Your answer explains why dust would be at the walls but I'm not totally convinced that it explains why dust goes to the corners. If a dust particle is taking a random walk back and forth between the corner to its "left" and the corner to its "right" then it has 100% chance of returning to the center. – isomorphismes Dec 19 '10 at 20:04
In particular why is surface area related to air that can blow from / towards a corner? – isomorphismes Dec 20 '10 at 1:00

Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. A simple Markov model of this process might be as follows:

1. the probability of the particle getting kicked up in 1 second increases with the airspeed at it's location, and

2. once it's in the air, where it falls is (uniformly) random.

Then one would expect the particle to spend more time on the ground where the airspeed is low, and less where the airspeed is high, therefore accumulating in low-speed locations.

## Simplified 2-state Markov model example

To illustrate the point, imagine the following simplified example:

• There are only two discrete locations the particle can be in: 'corner' and 'not corner', and time is divided into discrete units $t_1, t_2, \dots$

• If the particle is in the corner, then the probability of it getting kicked up and redistributed at any given time is 20%, whereas if the particle is in not corner, then the probability of it getting kicked up and redistributed is 60%

• once the particle gets kicked up, the probability of it getting distributed in one location vs. the other is 50/50.

Then, accounting for all the probabilities, the transition probability from $\textbf{corner} \rightarrow \textbf{not corner}$ is 10%, whereas the transition probability from $\textbf{not corner} \rightarrow \textbf{corner}$ is 30%. This is illustrated in the following image:

Now suppose each state (corner, not corner) starts with equal probability. What will happen to the probability after a series of timesteps?

Imagine that each state (corner, not corner) is a person, and they each have a some money, where the amount of money is proportional to probability of the state associated with that person. So, for example if each state is initially equally probable you can imagine that each person starts with $100. At each timestep, the each person gives a fraction of their money to the other person, with the fraction equal to the transition probability. Then after giving/recieving money, they each have a new amount, and the process repeats with these new amounts. After redistributing money for a while in this manner, the more greedy person will end out with more money, and the more generous person will end out with less. In general, to get the vector of probabilities at the next timestep$t_{k+1}$, you multiply the vector of probabilities at the current timestep$t_k$by the transition probability matrix. $$\begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_{k+1} = \begin{bmatrix}0.9 & 0.3 \\ 0.1 & 0.7\end{bmatrix} \begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_{k}$$ Starting at time$t_0$and repeatedly multiplying by the transition matrix, we get the following formula for the probabilities at the$N'th timestep, \begin{align} \begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_{N} &= \overbrace{\begin{bmatrix}0.9 & 0.3 \\ 0.1 & 0.7\end{bmatrix} \dots \begin{bmatrix}0.9 & 0.3 \\ 0.1 & 0.7\end{bmatrix}}^{N \text{ times}}\begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_{0}\\ &= \begin{bmatrix}0.9 & 0.3 \\ 0.1 & 0.7\end{bmatrix}^N \begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_{0} \end{align} From linear algebra, one can see that for enough timestepsN$, the probability distribution will converge to a steady state distribution given by the eigenvector of the transition probability matrix associated with the largest eigenvalue, which in this case happens to be the following: $$\begin{bmatrix}P(\textbf{corner}) \\ P(\textbf{not corner})\end{bmatrix}_\text{steady state} = \begin{bmatrix}0.75 \\ 0.25\end{bmatrix}.$$ So then we would expect a single dust particle to spend 75% of its time in the corner, and 25% of the time not in the corner, for this simplified example. Or, supposing there are a lot of dust particles, after enough time we would expect 75% of them to be in the corner and 25% of them to not be in the corner. ## Why do particles get kicked up more where the airspeed is higher? When a particle of dust is lying on the floor, static electricity forces form between the dust and the floor, creating an energy barrier that the particle must overcome in order to be kicked up into the air. Since the energy density of flowing air grows with the airspeed (actually, airspeed squared), the energy that the air is likely to impart on the particle will also grow with the airspeed. Therefore, the greater the airspeed at the location of a particle of dust, the more likely it is to be kicked up in a given amount of time. ## Why is airspeed slower in corners? If the probability of a particle getting kicked up is proportional the airspeed at its location, the question now becomes "why does the air move slower in corners?" Empirically, (at least from walking around my apartment) it seems that the airflow is indeed slower in the corners, so this is a good sanity-check. To really do this justice, you'd have to consider the full Navier-stokes equations, but those are hard, so lets consider a simplified model. Suppose the flow is an incompressible, slow, pressure-driven flow, with a fixed inflow at some air vents. In this case, the pressure field should solve the Laplace equation with no-flow boundary conditions at the walls, fixed flow boundary conditions the air vents, and fixed pressure conditions at the cracks under the door to match the outside pressure, etc. I.e., $$\begin{cases} -\Delta p = 0, & \text{inside the room}, \\ \nabla p \cdot n = 0, & \text{on the walls}, \\ \nabla p \cdot n = g_0, & \text{along air vents}, \\ p = p_0, & \text{along cracks under the door}. \end{cases}$$ We have some intuition about the solution to Laplace's equation as some sort of smooth rubber surface trying to be "as flat as possible". In particular, the solution changes smoothly from point to point - if you only move a little bit, the solution and its first derivatives will only change a little bit. Now consider a corner between one wall along$x=0$, and another wall along$y=0$. If we are near enough to the corner, we expect to be within the realm of influence of both walls, so we expect both derivatives$dp/dx$and$dp/dy$to be small, since they are zero at the$y=0$and$x=0$walls respectively. Therefore the pressure gradient is small, so the velocity (which is proportional to the pressure gradient) is small in the corner, as expected. ## Summary A simple particle model predicts dust gets kicked up more often where the air is moving quickly, and less often where the air is moving more slowly. A simple Markov model predicts that dust will accumulate in places where it doesn't get kicked up as often. A simple pressure-driven flow model predicts flow to be slow near corners. Putting these pieces together, one is left with the conclusion that dust must accumulate in corners. - Your first sentence makes for the a great first line in a really, really depressing novel :) – Mariano Suárez-Alvarez Sep 23 '11 at 17:05 Also, I believe similar logic explains why, after you buy a set of pens and pencils they evaporate from your desk, accumulating in weird places where you don't use them like your car. – Nick Alger Mar 10 '12 at 10:02 First, we must assume a more or less constant airflow in the room, for as everyone knows,in a room that is closed up for a long time with no airflow, dust accumulates evenly throughout the room. The simplest air flow type is random. Like water, or anything tbat flows, air flows through the path of least resistance. For air, the path of least resistance is a straight line. Resistance to flow is proportional to the angular change of direction produced by an obstacle; in this case a corner, which does not neccessarily imply a 90 degree corner. There are two types of corners: inside corners (less than 180 degrees and outside corners (greater than 180 degrees). Outside corners do not gather dust, simply because they do not force the air flow to change its direction. Inside corners, however present a physical barrier to straight-line air flow. The smaller the angle of deflection, the greater the resistance. The air simply avoids the highly flow resistant corner by curving around it. In doing so, it loses some of its energy, slowing its flowrate or speed. Of course, some of the air particles pass closer to the corner than others. The closer particles have to turn more sharply to avoid the black hole of the corner, thus slowing down more, thus losing more of their ability to hold a dust particle aloft, thus losing more dust particles before returning to the (assumed) random air flow of the room. In a simple rectangular or square room, this action results in dusty corners. This is the simplest and least technical answer I can think of. Such things as Navier-stokes equations, Laplace equations and pressure derivatives have their place, but are superflous, irrelevant overkill for answering such a simple question. "KISS" always applies. If you want a solution to the dusty corner problem (and seeking the cause of a problem is, after all, the first step in seeking a solution), simply place a contantly running vaccumn inlet in each corner with the (filtered) outlet in the center of the room, directed in all directions, and you will have constantly self-dusting corners. Then, (not so simply) eliminate all dust generators and dust attractors, and replace all objects that have inside corners (or modify them so that they have inside curves instead), and you will have a constantly self-dusting room. However, you should not complain if, upon enterimg this room, you friends declare, "This room really sucks!". I hope this clears the air (and the corners, of course). - I should have specified that I am talking about 90 degree corners. – isomorphismes Dec 19 '10 at 19:57 How do you know that$\mathrm{resistance } \propto \Delta \mathrm{\, direction}\$? – isomorphismes Dec 19 '10 at 19:58
Is it really true that a sealed-up room accumulates a constant amount of dust? I don't have experience one way or the other. – isomorphismes Dec 20 '10 at 1:05

If there is a net airflow in any particular direction at floor-level for any reason (doors/windows/radiators etc.), then the dust will tend to accumulate in the corner that is furthest 'down-stream'.

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I've also noticed, peculiarly, that dust accumulates as well near doorways that are not in corners. – isomorphismes Dec 18 '10 at 7:44


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# 9.5: Patterns of Economic Development in East and Southeast Asia

Despite the political changes and conflicts that marked the 20th century, the 21st century has largely been marked by economic development across East and Southeast Asia. Economic geography is a branch of geography that explores the spatial aspect of economic development. Economic geographers don’t just ask “Where is economic development occurring?” but also “Why is economic development occurring in some areas and not others?” In East and Southeast Asia, economic growth has largely resulted from regional and global trade. However, development is not spread evenly across the region and economic inequalities still persist.

Global connections have been the principle driver of economic success in East and Southeast Asia. Much of the trade connections have emerged between the countries in this region and the larger Pacific Rim, referring to the countries that border the Pacific Ocean. Many of these countries are members of the Asia-Pacific Economic Cooperation (APEC) which promotes free trade across Asia and the Pacific. In Southeast Asia, the countries of the region formed the Association of Southeast Asian Nations, or ASEAN (Figure $$\PageIndex{1}$$). The organization aimed to promote political security, economic growth, and social development among member countries.

In China and Japan, histories of relative isolation gave way to an embrace of globalization and global trade. Although China’s government is communist, it has allowed more free-market oriented economics in areas known as Special Economic Zones, or SEZs. These SEZs are located in coastal China and have special incentives to attract foreign investment (Figure $$\PageIndex{2}$$). Other capitalist shifts have occurred in China even allowing for US supermarkets and restaurants to open locations in the country. These SEZs, as well as other special development areas in China, have functioned as growth poles, which are areas that have attracted economic development in the region. In 2010, China displaced the United States as the global leader in manufacturing.

Broadly, foreign direct investment (also called FDI), the control of a business in one country by a company based in another country, has been a key driver of China’s economic success. In 2017, China was the second most attractive company for foreign investors, behind the United States, with over $136 billion in foreign direct investment flowing into the country according to the 2018 World Investment Report. China has done its own foreign direct investing as well, increasing outward flows of FDI from$5.5 billion in 2004 to over $125 billion in 2017, according to the same report. Hong Kong is the primary destination for Chinese foreign direct investment, but substantial sums also flow to countries in Africa as well as Australia, where China is the largest trading partner. Within Southeast Asia, China is now the top investor in Myanmar and has increased foreign direct investment in Singapore. While China and Japan remain the largest economies in the region, other areas have experienced high rates of economic growth. The Four Asian Tigers, in particular, referring to Hong Kong, Singapore, South Korea, and Taiwan, experienced rapid industrialization and economic development led by export-driven economies, low taxes, and free trade. Some have also theorized that the Confucian work ethic present in these countries complemented the process of industrialization. Each country also developed a distinct specialty and maintains a competitive advantage in that area. South Korea, for example, is known for its manufacturing of information technology while Hong Kong is a leading financial center. Much of the economic growth of the Asian Tigers as well as Japan has come from the export of value added goods. When countries export raw materials, their economic benefit is limited since those raw materials are often not inherently valuable. When these raw materials are made into something useful, however, value is added and the product can be sold for a higher profit. Lumber is quite cheap, for example. When the lumber is made into a dining table, it has a much higher value. Many companies located in these countries have become household names in electronics, computing, and the auto industry. China has traditionally exported relatively low value added goods, but in 2016, the Chinese government announced a shift to higher value added products like transportation technology and telecommunications. Southeast Asia has benefitted from its key geographical position as the entrepôt, a French term meaning a commercial center of trade, for the rest of Asia. The Strait of Malacca in particular is the main shipping channel between the Pacific Ocean and the Indian Ocean and a key transportation gateway (Figure $$\PageIndex{3}$$). Around one quarter of all the world’s exported goods travels through the strait each year. Malaysia’s economic success as an entrepôt is exemplified by its Petronas Towers in Kuala Lumpur, which were the tallest buildings in the world from 1998 to 2004. Indonesia has the largest economy of Southeast Asia, exporting primarily to Japan, Singapore, the United States, and China. Singapore has the highest GDP per capita in the region, again owing to its key geographical position. One lingering issue in many of these countries has been crony capitalism, the notion that the success of a business depends on its relationship to other businesses and the state. A politician might have an old friend in the manufacturing industry, for example, and give the friend a government contract with beneficial terms. In 1997, a financial crisis that started in Thailand spread throughout the Southeast Asia region and to South Korea was, in part, blamed on the business dealings of corrupt politicians. Several countries in this region rank high on an index of corruption perception, as shown in Figure $$\PageIndex{4}$$, and some have expressed concern that the communist countries of this region will continue to embrace capitalism when it is politically beneficial rather than as part of a broader and more transparent economic system. Economic development is not spread evenly throughout the region. By most estimates, North Korea remains the poorest country in the region with an estimated GDP per capita of just$1,300 in 2016. Mainland Southeast Asia, with the exception of Thailand, remains relatively poor. Cambodia, Laos, Myanmar, and Vietnam all have GDP per capitas of less than \$8,000 as of 2018 and remain far less well off than their more developed neighbors in the region.

Where economic growth has occurred, it is often confined to an urban area which can drive up population densities as people move to the city in search of work. In Indonesia, the island of Java is very densely populated while other surrounding islands are relatively sparse. The Dutch colonizers and later the Indonesian government, in a stated effort to reduce poverty and overcrowding, created a policy of transmigration seeking to relocate people to the less densely populated islands. This program has been controversial, however, and the indigenous people who inhabit the surrounding islands see the program as a threat to their way of life.

In some cases, uneven distribution of economic development has led to both interregional and intraregional migration as people move in search of economic advancement. China in particular has seen a significant amount of rural to urban migration. Around 11 percent of the entire country’s population migrated from rural to urban areas in 2009 and most of them are young adults. Many of China’s migrants are part of a floating population, which refers to members of a population who reside in an area for a period of time but do not live there permanently. Around 50 million Chinese reside overseas, mostly in Southeast Asia. Thailand has the largest population of overseas Chinese, and Chinese also represent the majority ethnic group in Singapore. Long before colonization, East and Southeast Asia was a realm of global economic influence, from the Chinese empire to the trade routes of Southeast Asia. As some countries of the realm have moved toward political stability and economic growth, others have remained in a state of political and economic turmoil. Tourism could bring some of these countries an economic boost, but the prospect of tourism in countries with government instability is limited. Still, although countries like Laos, Myanmar, and Cambodia currently have some of the world’s smallest economies, they are some of the fastest growing in the world and improvements in agricultural and natural resource development combined with a stable political infrastructure could expand the economic strength of the region.

Pacific Rim:

refers to the countries that border the Pacific Ocean

Association of Southeast Asian Nations:

an international organization that seeks to promote political security, economic growth, and social development among member countries in Southeast Asia, also known as ASEAN

Special Economic Zones:

a special region in China where more free-market oriented economics are allowed, also known as SEZ

Growth poles:

a cluster within a region that has attracted economic development

Foreign direct investment:

the control of a business in one country by a company based in another country, also known as FDI

Asian Tigers:

refers to Hong Kong, Singapore, South Korea, and Taiwan which have experienced rapid industrialization and economic development led by export-driven economies, low taxes, and free trade, sometimes also called the Four Asian Tigers

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# Gompertz growth equation

:)

Hi! I'm almost finished with a homework problem, but I cannot quite finish it. The problem is as follows:

Given the Gompertz growth equation $$\frac{dN}{dt}=K(t)N(t),\ N(0)=N_0 \\ \frac{dK}{dt}=-\alpha K(t),\ K(0)=\beta,$$ I shall first determined a closed-form expression, which will be $K(t)=\beta e^{-\alpha t}$ and $$\frac{dN}{dt}=\beta e^{-\alpha t} N(t) \Leftrightarrow \\ \int\frac{1}{N(t)}dN=\int \beta e^{-\alpha t}dt \Leftrightarrow \\ \log[N(t)]=\frac{-\beta}{\alpha} e^{-\alpha t} + C \Leftrightarrow \\ N(t)=\exp\left( \frac{-\beta}{\alpha}e^{-\alpha t}\right) \frac{N_0}{\exp\left(\frac{-\beta}{\alpha}\right)};$$ the last factor is due to the initial condition.

Then: Determine the limit $$B:=\lim_{t\rightarrow \infty}N(t),$$ it will be $B=\frac{N_0}{\exp\left(\frac{-\alpha}{\beta}\right)}$ (right?)

Then: Show that this is equivalent to $$\frac{dN}{dt}=\alpha N \log\frac{B}{N},$$ I also managed that (the LHS is $KN$, then solve the logarithm), but THEN:

For the cases $N_0>B, N_0 <B$, determine $\lim_{t\rightarrow -\infty }N(t)$. If I plug in $t=-\infty$ into $N(t)$, then I get just $0$. Or not? And with my solution for $B$, I get $\exp\left( \frac{-\beta}{\alpha}\right)<> 1$, but since $\alpha, \beta >0$, the exp cannot be smaller than one?

I either have a blonde moment, or my solutions are wrong, but magically the equivalence in the $\log$-equation works! Can someone defuse my brain? :)

-marie

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Should be $B = N_0/\exp(-\beta/\alpha) = N_0 \exp(\beta/\alpha)$. – Robert Israel May 7 '12 at 23:35
If $\beta/\alpha > 0$, there is no case $N_0 > B$. – Robert Israel May 7 '12 at 23:43
Gompertz? Gesundheit! – rschwieb May 8 '12 at 1:41
So... does that mean that the question is wrong? :/ – Marie. P. May 8 '12 at 9:22


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## arXiv Analytics

### arXiv:1906.04729 [astro-ph.CO]AbstractReferencesReviewsResources

#### B-mode Targets

Published 2019-06-11Version 1

The CMB-S4 Science Book, Astro2020 Science White Paper on gravitational waves, and PICO report contain an extensive discussion of many ongoing and planned efforts in the search for gravitational waved produced by inflation. Here we give a short executive summary of the results obtained in our recent papers, which specify the simplest available inflationary models providing physically motivated targets for these searches. Our conclusions are specific for the $10^{-3} \lesssim r \lesssim 10^{-2}$ range, where we present the B-mode benchmarks of the U-duality symmetric class of $\alpha$-attractors, and for $r \lesssim 10^{-3}$, where we present B-mode targets, for which the future precision measurements of $n_s$ will be decisive.


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# Orthogonal matrices only defined for standard inner product?

Let orthogonal matrices be defined as follows.

A matrix $$Q\in\mathcal{M}_{m\times n}(\mathbb{K})$$, where $$\mathbb{K}$$ is a field, is said to be orthogonal if $$Q^\mathrm{T}Q = \mathrm{Id}_n$$

I'm not fully sure if I'm understanding the following fact correctly:

A matrix $$Q\in\mathcal{M}_{m\times n}(\K)$$ is orthogonal iff the columns of $$Q$$ form an orthonormal set in $$\K^m$$.

Proof
Let $$q_i$$ denote the $$i$$-th column of $$Q$$ for all $$i\in\{1,\ldots,n\}$$, and let $$\ijth[A]{ij}$$ denote the $$(i,j)$$-th element of $$A$$ for any matrix $$A$$. Then, $$Q$$ being an orthogonal matrix is equivalent to $$\tp{Q}Q = \Id_n \siff \ijth{ij} = \delta_{ij}\,,$$ where $$\delta_{ij}$$ is the Kronecker delta. On the other hand, by the definition of matrix multiplication, $$\ijth{ij} = \sum_{k=1}^{m} \ijth[\tp{Q}]{ik}\ijth[{Q}]{kj} = \sum_{k=1}^{m} \ijth[Q]{ki}\ijth[{Q}]{kj} \stackrel{\color{red}*}{=} \langle q_i, q_j\rangle\,.$$ Thus $$Q$$ is orthogonal iff $$\langle q_i, q_j\rangle = \delta_{ij} \qquad\forall (i,j)\in\n\times\n\,,$$ which is true iff $$(q_i)_{i\in\n}$$ form an orthonormal set.

Particularly, I'm suspicious of the equality marked with the red asterisk. Isn't that true only for the standard inner product (i.e. the dot product), defined as $$\langle u, v \rangle = \sum_i u_iv_i\$$? So, are orthogonal matrices only treated in the context of the standard inner product? If so, is there a "generalization" of orthogonal matrices for general inner product spaces?

It might be instructive here to start with the corresponding invariant (i.e., basis-free) description of orthogonality:

On a finite-dimensional inner product space $$(\Bbb V, \langle\,\cdot\,,\,\cdot\,\rangle)$$, a linear transformation $$T : \Bbb V \to \Bbb V$$ is said to be orthogonal if it preserves the inner product, that is, if $$\langle T({\bf x}), T({\bf y}) \rangle = \langle {\bf x}, {\bf y}\rangle$$.

Fixing a basis $$({\bf e}_a)$$ of $$\Bbb V$$ determines matrix representations $$[T]$$ of $$T$$ and $$[\Sigma]$$ of the inner product: These are characterized by

$$[T({\bf e}_a)] = \sum_b [T]_{ba} [{\bf e}_a], \qquad [\Sigma]_{ab} = \langle {\bf e}_a, {\bf e}_b \rangle .$$ Unwinding all of this, we see that $$T$$ is orthogonal if $$[T]^{\top} [\Sigma] [T] = [\Sigma] .$$

In the special case that the basis $$({\bf e}_a)$$ is orthonormal, it follows immediately from the definitions that $$[\Sigma] = I$$, in which case the condition simplifies to the familiar definition of orthogonal matrix: $$[T]^{\top} [T] = I .$$ Over a real inner product space, we can always choose an orthonormal basis, so the more general construction might seem like an unnecessary formalism. But such bases are not always the most convenient in applications, and if we extend our attention to nondegenerate, symmetric bilinear forms (so, drop the condition of positive-definiteness from the definition of inner product), orthogonal bases don't exist, but we still care about the notion of orthogonality.

Yep, just as you expected, however, all of this heavily depends on your choice of basis. Since every basis $$B$$ of a vector space $$V$$ is actually a choice of isomorphism $$\varphi_B : k^{\dim(V)} \to V$$, we may interpret any homomorphism $$f: V \to W$$ as a matrix (heavily dependent on the choice of basis). Picking an orthonormal basis w.r.t an arbitrary inner product $$\langle \_ ,\_ \rangle _V$$ this isomorphism even becomes compatible with the inner product, i.e. $$\langle \varphi_B (x) , \varphi_B (y)\rangle _V =\langle x , y \rangle _{\textrm{eucl}}$$. So you can carry over all constructions. However in general you can define an orthogonal matrix as commuting with the inner product. This does, using the above identification become equivalent to the definition you know.

However, I prefer doing the general case first (i.e. commuting with inner product) and then specializing using basis.

I hope this is a satisfactory answer.


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# What method to use for interpolation and extrapolation?

Below are discrete samples { t1, f(t1) }, { t2, f(t2) }, ... ,{ tn, f(tn) } using Mathematica syntax.

{{7.0,0.354887404925574},{7.3,0.4003399403324751}, {7.6,0.5849632195845474},{7.9,0.8785638270289906}, {8.2,1.235637591701261},{8.5,1.6025351028031707}, {8.8,1.925797159944803},{9.1,2.160385431197419}, {9.4,2.276553177467603},{9.7,2.2643104784151262}, {10.0,2.1348061618705962},{10.3,1.9184189440030162}, {10.6,1.6598520161702173},{10.9,1.4109826017641867}, {11.2,1.2225616685298717},{11.5,1.1360369427321668}}

Now I plot the samples.

The samples are from a bandlimited function f(t) that I defined. However, suppose they are discrete samples of an unkown function, but we know the Fourier spectrum of f(t) is band-limited to frequencies 0.25 Hz and below. What would be a good method to use for interpolating between the samples and extrapolate beyond the samples? Here I mentioned that polynomial interpolation has less error than cubic spline interpolation in a case like this? What method would have even less approximation error in a case like this?

• for band-limited functions, sinc-interpolation yields the exact function. However, it does not work for extrapolation. Note, that sinc-interpolation also does not work nicely at the edges of the sampled interval, due to its non-causality and slow decay. Further note, that you might need much more samples (a larger sampled interval), since a max. frequency of 0.25Hz is already 4seconds period (your interval is just 5 seconds), so smaller frequencies will not even have a full period of sampled signal. This might make it more difficult. – Maximilian Matthé Apr 10 '17 at 6:53

@Maximilian Matthé, The discrete samples above were made using the function

$$\mathrm{Truth}(t) = 1.4 + \frac{\sin(13-t)}{13-t} - \sin\big((1.3)(13-t)\big)$$ In this case, the sample rate is several times the Nyquist frequncy of the signal. When that is the case, we can get a very good interpolation over the samples by finding a linear combination of eight sin(f t) terms and eight cos(f t) terms that interpolate the sixteen samples. Results using eight frequencies equally spaced from ($\tfrac{\pi}{16}$ = 0.19635 radians/second) to ($\tfrac{\pi}{2}$ = 1.5708 radians/second) are shown below.

This method is far better for extrapolation (i.e. beyond the samples) than any other method I know of, and the interpolation error (i.e. between samples) is less than $4 \times 10^{-12}$.

This example was not selected to make sure the results look very good, but it is possible to find examples where the approach doesn't excel. If the same frequencies are used and the function is nearly linear over the same sample times, then cubic spline interpolation is better than an interpolating trigonometric series. However, that issue can be overcome by selecting the range of frequencies based on how much oscillation there is between the samples. If the samples are nearly linear, then a set of frequencies closer to zero could be used. Then an interpolating trigonometric series would still work very well if the range of frequencies is appropriate. Also, the samples used in this example are equally spaced, but that is not necessary for an interpolating trigonometric series to work well.


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• Corpus ID: 119640835

# Peano's Existence Theorem revisited

@article{Pouso2012PeanosET,
title={Peano's Existence Theorem revisited},
author={Rodrigo L'opez Pouso},
journal={arXiv: Classical Analysis and ODEs},
year={2012}
}
We present new proofs to four versions of Peano's Existence Theorem for ordinary differential equations and systems. We hope to have gained readability with respect to other usual proofs. We also intend to highlight some ideas due to Peano which are still being used today but in specialized contexts: it appears that the lower and upper solutions method has one of its oldest roots in Peano's paper of 1886.
4 Citations
On the existence and uniqueness of solutions to dynamic equations
In this paper, we prove the well-known Cauchy–Peano theorem for existence of solutions to dynamic equations on time scales. Some simple examples are given to show that there may exist more than a
Peano’s 1886 existence theorem on first-order scalar differential equations: a review
• Mathematics
• 2016
In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem $$y^\prime =f(x,y)$$y′=f(x,y), $$y(a)=b$$y(a)=b, under the assumption of the continuity of
Extremal solutions of systems of measure differential equations and applications in the study of Stieltjes differential problems
• Mathematics
• 2018
We use lower and upper solutions to investigate the existence of the greatest and the least solutions for quasimonotone systems of measure differential equations. The established results are then

## References

SHOWING 1-10 OF 18 REFERENCES
Elementary proofs of Peano's existence theorem
• Mathematics
Journal of the Australian Mathematical Society
• 1973
An “elementary” proof of Peano's existence theorem is given that, in addition to avoiding the Ascoli lemma, relies neither on Dini's theorem, nor on uniform continuity of the right hand side of φ' =
Necessary conditions for solving initial value problems with infima of superfunctions
Goodman proved that the pointwise infimum of all superfunctions is the minimal absolutely continuous solution of x′ = f (t, x), t ∈ [0, 1], x(0) = 0, in case f is a L1 -bounded Caratheodory function.
Differential and Integral Inequalities
In 1964 the author's mono graph "Differential- und Integral-Un gleichungen," with the subtitle "und ihre Anwendung bei Abschatzungs und Eindeutigkeitsproblemen" was published. The present volume grew
First-Order Singular and Discontinuous Differential Equations
• Mathematics
• 2009
We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and
Minimal and maximal solutions to second‐order boundary value problems with state‐dependent deviating arguments
• Mathematics
• 2011
We prove the existence of solutions between given lower and upper solutions for second‐order boundary value problems with deviating arguments, and we show that the set of those solutions has minimal
An introduction to delay differential equations with applications to the life sciences / Hal Smith
The simplicity of the Delay Equation and its application to linear systems and linearization suggest the existence of solutions to the persistent problem of infectious disease.
Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations
• Mathematics
• 1994
Methods and Prerequisites First Order Differential Equations Second Order Differential Equations Second Order Partial Differential Equations Differential Equations in Banach Spaces.
Sull’integrabilità delle equazioni differenziali di primo ordine
• Atti. Accad. Sci. Torino 21
• 1886


text:
# zbMATH — the first resource for mathematics

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A class of two-step Steffensen type methods with fourth-order convergence. (English) Zbl 1166.65338
Summary: Based on Steffensen’s method, we derive a one-parameter class of fourth-order methods for solving nonlinear equations. In the proposed methods, an interpolating polynomial is used to get a better approximation to the derivative of the given function. Each member of the class requires three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index which equals 1.587. H. T. Kung and J. F. Traub conjectured an iteration using $n$ evaluations of $f$ or its derivatives without memory is of convergence order at most $2n-1$ [J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)]. The new class of fourth-order methods agrees with the conjecture of Kung-Traub for the case $n=3$. Numerical comparisons are made to show the performance of the presented methods.
##### MSC:
 65H05 Single nonlinear equations (numerical methods)
##### References:
 [1] Ortega, J. M.; Rheinbolt, W. C.: Iterative solution of nonlinear equations in several variables, (1970) · Zbl 0241.65046 [2] Argyros, I. K.: Convergence and application of Newton-type iterations, (2008) [3] Johnson, L. W.; Riess, R. D.: Numerical analysis, (1977) · Zbl 0412.65001 [4] Dehghan, M.; Hajarian, M.: A variant of Steffensen’s method with a better approximation to the derivative, Appl. math. Comput. (2008) [5] Jain, P.: Steffensen type methods for solving non-linear equations, Appl. math. Comput. 194, 527-533 (2007) · Zbl 1193.65063 · doi:10.1016/j.amc.2007.04.087 [6] Sharma, J. R.: A composite third order Newton – Steffensen method for solving nonlinear equations, Appl. math. Comput. 169, 242-246 (2005) · Zbl 1084.65054 · doi:10.1016/j.amc.2004.10.040 [7] Amat, S.; Busquier, S.: Convergence and numerical analysis of a family of two-step Steffensen’s methods, Comput. math. Appl. 49, 13-22 (2005) · Zbl 1075.65080 · doi:10.1016/j.camwa.2005.01.002 [8] Amat, S.; Busquier, S.: A two-step Steffensen’s method under modified convergence conditions, J. math. Anal. appl. 324, 1084-1092 (2006) · Zbl 1103.65060 · doi:10.1016/j.jmaa.2005.12.078 [9] Amat, S.; Busquier, S.: On a Steffensen’s type method and its behavior for semismooth equations, Appl. math. Comput. 177, 819-823 (2006) · Zbl 1096.65047 · doi:10.1016/j.amc.2005.11.032 [10] Alarcn, V.; Amat, S.; Busquier, S.; Lpez, D. J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. comput. Appl. math. 216, 243-250 (2008) · Zbl 1139.65040 · doi:10.1016/j.cam.2007.05.008 [11] Gautschi, W.: Numerical analysis: an introduction, (1997) [12] Bi, W.; Ren, H.; Wu, Q.: New family of seventh-order methods for nonlinear equations, Appl. math. Comput. 203, 408-412 (2008) · Zbl 1154.65323 · doi:10.1016/j.amc.2008.04.048 [13] Xu, L.; Wang, X.: Topics on methods and examples of mathematical analysis (in chinese), (1983) [14] Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, J. assoc. Comput. math. 21, 634-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860 [15] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third-order convergence, Appl. math. Lett. 13, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2


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# commutative property formula

By on Dec 29, 2020 in Uncategorized | 0 comments

The commutative property makes working with algebraic expressions easier. So if there is subtraction or division, correct it to addition or multiplication. Either way, the result (having both socks on), is the same. For example, the position and the linear momentum in the x-direction of a particle are represented by the operators Origin: The word commutative is derived from the word “commute” which means “to move around”.In commutative property the numbers are moved around for computation.. " is a metalogical symbol representing "can be replaced in a proof with.". For example, let Example 1: Commutative property with addition of the Commutative Property . Which is that you can add or multiply in any order, regardless of how the numbers are grouped. What property is illustrated by … The commutative property of multiplication states that you can multiply numbers in any order. Right here’s an instance of • Each of them Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. They want me to move stuff around, not simplify. x So whole numbers are commutative under multiplication. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. Thus, this property was not named until the 19th century, when mathematics started to become formalized. It is important to note that we cannot mix addition and multiplication. See how the orders of our letters are switched around on opposite sides of the equals sign? (i) Set union is commutative (A U B) = (B U A) (i) Set intersection is (A n Putting on socks resembles a commutative operation since which sock is put on first is unimportant. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch." Cloudflare Ray ID: 609650f98b7d1b05 i Let's look at how (and if) these properties work with addition, multiplication, subtraction and division. These two operators do not commute as may be seen by considering the effect of their compositions 1987. d I don't know what you exactly wanted to draw, so I reproduce one of the diagrams from your link, showing how to do it with pst-node and with tikz-cd.One of the main differences is that in pstricks you first describe the nodes, then the arrows, while with tikz-cd, nodes and arrows are described simultaneously. But few experiments doesn't constitute a proof and it feels unintuitive that the total of the formula would be still commutative even if it contains non-commutative operators. By Grouped We Mean How You Use Parenthesis. Addition and multiplication is commutative. Template:Transformation rules. More such examples may be found in commutative non-associative magmas. It is: a * b = b * a The different letters stand for different numbers. Addition. , so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary. For any two two sets, the following statements are true. The more flexible the computation method … {\displaystyle f(f(-4,0),+4)=+1} ∂ ÷ The "Associative Property" is a result that applies to both addition and multiplication. x , respectively (where − {\displaystyle 1\div 2\neq 2\div 1} Sep 25, 2013 - Explore Dawn Catlett (Kessler)'s board "Teaching Commutative Property", followed by 106 people on Pinterest. 4 Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. ∂ {\displaystyle f(-4,f(0,+4))=-1} Rule of replacement Properties and Operations. The commutative property changes the order of some numbers in an operation to make the work tidier or more convenient — all without affecting the result. b f + Example Charles and George learned how to calculate the area of a rectangle in math class by using the base by height formula. Statement: First Law : First law states that the union of two sets is the same no matter what the order is in the equation. Regardless of the order the bills are handed over in, they always give the same total. For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are, Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2.. = 1 Property allowing changing the order of the operands of an operation, Mathematical structures and commutativity, Non-commuting operators in quantum mechanics, Transactions of the Royal Society of Edinburgh, "Compatible Numbers to Simplify Percent Problems", "On the real nature of symbolical algebra", https://web.archive.org/web/20070713072942/http://www.ethnomath.org/resources/lumpkin1997.pdf, Earliest Known Uses Of Mathematical Terms, https://en.wikipedia.org/w/index.php?title=Commutative_property&oldid=992295657, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License. ψ d TERM TO KNOW Commutative Property A property of addition that allows terms to be added in any order; a property … This is what it lets us do: 3 lots of (2+4) is the same as 3 lots of 2 plus 3 lots of 4. In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. This is a well known number property that is used very often in math. 0 Distributive Property Basics All the numbers that are used in Mathematical calculations and have a specific value is called the real numbers. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The variable could be taken as x, y, a, b, c or any other alphabet that represents a number unknown yet. x − ( Performance & security by Cloudflare, Please complete the security check to access. 0 They use letters in place of numbers to let us know that the formula applies to all numbers. {\displaystyle f(x)=2x+1} ) What a mouthful of words! Standards: 4AF2.1 Know and understand that equals added to equals are equal. Records of the implicit use of the commutative property go back to ancient times. x − ) Students will solve 4/5 problems using commutative property. The commutative property states that regardless of the order of the addends in an addition equation, the sum remains the same. The rules are: where " but = The commutative property (or commutative law) is a property associated with binary operations and functions. . d If you are talking about the commutativity property of multiplication of natural numbers, then this is Theorem 29 of Edmund Landau’s Foundations of Analysis: The issue with this proof is that this is Theorem 29, and its proof uses f , Introduction to the commutative rule of multiplication with formula and an example with proof to derive the commutative law of multiplication in mathematics. Algebra Formulas A basic formula in Algebra represents the relationship between different variables. The commutative property of addition states that numbers may be added in any order without affecting the sum. In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and Algebra Commutative Property of Set Theory Proof. Shuffling a deck of cards is non-commutative. Commutative property of linear convolution This property states that linear convolution is a commutative operation. In truth-functional propositional logic, commutation,[13][14] or commutativity[15] refer to two valid rules of replacement. 1 f Commutative property of addition is nothing but the rule which says that, when we are doing addition, it doesn't matter, in which order the numbers are. Note that it is easy to correct subtraction, but with division, you must change it to a fraction. The Side Angle Side Formula more gifs Definition: The Commutative property states that order does not matter. Today the commutative property is a well-known and basic property used in most branches of mathematics. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commu… {\displaystyle x} {\displaystyle {\frac {d}{dx}}} Another way to prevent getting this page in the future is to use Privacy Pass. In short, in commutative property, the numbers can be added or multiplied to each other in any order without changing the answer. Commutative Property Calculator . 4AF2.2 (i) Set union is commutative (A U B) = (B U A) (i) Set intersection is commutative (A n B) = (B n A) Let us look into … Formula for the Commutative Property In math, we have a formula that says the same thing. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The generic formula for the Commutative Property of Multiplication is: ab = ba a b = b a. which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, 4 If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation. Apart from commutative, there are two more major properties of addition and multiplication of integers, and they are associative and distributive. ) Use the Commutative Property to restate " 3×4×x " in at least two ways. . : According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. 1 {\displaystyle {\frac {d}{dx}}x} Start studying Algebra 2 - Unit Test Review. For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then ( Then. mc026-1.jpg Which expression could be used to find So whole numbers are commutative under multiplication. − [4][5], Two well-known examples of commutative binary operations:[4], Some noncommutative binary operations:[7]. ⇔ In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.[16][17][18]. The commutative property of addition tells us that we can add things in any order and still get the same sum. ). Commutative Property. Answer = Given whole numbers = 23, 43 and their two orders are as follows :- Order 1 = 23 - 43 = (-20) Order 2 = 43 - 23 = 20 As, in both the orders the result is different. Commutative Property . It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The first recorded use of the term commutative was in a memoir by François Servois in 1814,[1][11] which used the word commutatives when describing functions that have what is now called the commutative property. = In English to commute means to travel or to change location. 1 how to teach properties of multiplication, Addition and multiplication both use the associative property, while subtraction and division do not. The formula for this property is: The formula for this property is: a * b = b * a Subtraction is noncommutative, since In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. R 0 The commutative property or commutative law means you can change the order you add or multiply the numbers and get the same result. The commutative property, therefore, concerns itself with the ordering of operations, including the addition and multiplication of real numbers, integers, and rational numbers. R Each of them + Commutative property vs Associative property. x Example It is a basic but important property in most branches of mathematics. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. = 0 The commutative property is one of the building blocks for the rules of algebra. Therefore, convolution is. 1 Directions: Click on each answer button to see what property goes with the statement on the left. A counterexample is the function. However it is classified more precisely as anti-commutative, since The commutative property is one of the building blocks for the rules of algebra. The term then appeared in English in 1838[2] in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.[12]. A general example to help you recognize patterns and spot the information you're looking for. The term "commutative" is used in several related senses. The commutative property of addition tells us that we can add things in any order and still get the same sum. The act of dressing is either commutative or non-commutative, depending on the items. Math Associative Property Commutative, Distributive Property. The commutative property of addition informs us we can include things in any order and still obtain the same sum. In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number.In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. true or false true 20. − ⇔ This page was last edited on 4 December 2020, at 15:19. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. = {\displaystyle \hbar } Proof of Commutative Property of Convolution The definition of convolution 1D is: First, let Then, substitute K into the equation: By definition, is the convolution of two signals h[n] and x[n], which is . This means that we can add in any order we wish, and we can multiply in any order we wish. Many mathematical proofs are based on this law and it is a basic property of many binary operations. ( In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. The commutative property of multiplication tells us that it doesn't matter in what order you multiply numbers. Commutative, Associative and Distributive Laws Wow! Remembering the formula for commutative property of addition is a + b = b + a and you are good to go! ℏ So, we can say that Subtraction is not Commutative … The commutativity of addition is observed when paying for an item with cash. . a The commutative property of multiplication is: a × b = b × a. We’re going to to get up close with each situation to get a better idea. For example, in the commutative property of addition, if you have 2 + 4, you can change it to 4 + 2, and you will have the same answer (6). The idea of commutativity revolves around the order of an operation. In math, you know how we have formulas for everything. Commutative Property. This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation). See more ideas about commutative property, commutative… Further examples of commutative binary operations include addition and multiplication of. ( And we write it like this: 0 Please enable Cookies and reload the page. ( Commutativity is a widely used term in mathematics. In this post, we’re going to see what the commutative property is all about. . For example: 2 x … • It is a fundamental property of many binary operations, and many mathematical proofs depend on it. ) This is the same example except for the constant x Some forms of symmetry can be directly linked to commutativity. Subtraction (Not Commutative) a + b = b + a a + b = b + a We can better see this relationship when using real numbers. {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing. , Formulas help us to generalize our problems. The Commutative Property of Addition is one of the crucial assumptions made on Mathematics, which you probably take for granted and use all the time without knowing. , The commutative property states that regardless of the order of the addends in an addition equation, the sum remains the same. [8][9] Euclid is known to have assumed the commutative property of multiplication in his book Elements. x Example 2 = Explain Commutative Property for Subtraction of Whole numbers 23 & 43 ? 19. a 4 Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property. 2 All the real numbers obey certain laws or have a few properties. Putting on left and right socks is commutative. {\displaystyle aRb\Leftrightarrow bRa} Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. 1 ≠ When you add 2 and 3 together, it doesn’t really matter in which order you add them. {\displaystyle \psi (x)} Commutative law is used to change the order of the operands without changing the end result. 2 For more math videos and exercises, go to HCCMathHelp.com. [1][2] A corresponding property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.[3]. ) ℏ Learn vocabulary, terms, and more with flashcards, games, and other study tools. You can use the commutative property with addition and multiplication operations, but not subtraction or division (with a few exceptions): […] d If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Plan your lesson in Math and Algebra with helpful tips from teachers like you. Commutative Property The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. x In this article, the student will learn about the commutative property with examples. This property is applicable only for addition and multiplication process, such that a + b = b + a and a × b = b × a. Matrix multiplication of square matrices is almost always noncommutative, for example: The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., b × a = −(a × b). Commutative Property under Multiplication of Integers: If we multiply two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we multiply (2×3) = (3×2) = 6. and 0 − Commutative Property Calculator When the change in the order of the operands does not change the outcome of the operation then that is called commutative property. Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. + But the ideas are simple. Similarly if we apply this to integers, (-5×3) = (3x (-5))= … The following are truth-functional tautologies. Distributive Law. and When the change in the order of the operands does not change the outcome of the operation then that is called commutative property. It refers to the ability to change the order of something without changing the final result. The commutative property changes the order of some numbers in an operation to make the work tidier or more convenient — all without affecting the result. − Let … . + ÷ ) 1 The associative property is closely related to the commutative property. Commutative Property Of Addition | The Associative Property States That You Can Add Or Multiply Regardless Of How The Numbers Are Grouped. d − Commutative Laws The "Commutative Laws" say we can swap numbers over and still get the same answer ..... when we add: [1] In physics and applied mathematics, the wedge notation a ∧ b is often used (in conjunction with the name vector product), [5] [6] [7] although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to n dimensions. Commutative property of set : Here we are going to see the commutative property used in sets. ≠ The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. ( The commutative property makes working with algebraic expressions easier. So, the 3× can be "distributed" across the 2+4, into 3×2 and 3×4. In mathematical computation, commutative property or commutative law explains that order of terms doesn’t matters while performing an operation. Your IP: 68.66.224.40 Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give In the point-slope formula, x1 represents the x coordinate of any point on the graph of a linear equation. x . x − {\displaystyle 0-1=-(1-0)} Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation. Property Example with Addition; Distributive Property: Associative: Commutative: Summary: All 3 of these properties apply to addition. a + b = b + a. Commutative Property of Multiplication. − We also have a formula for the commutative property of addition. Propositional logic. These are separate properties, but they behave the same with both operations. [10] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. However, commutativity does not imply associativity. For any two two sets, the following statements are true. Definition: According to the commutative property, order does not matter during computation.The Commutative property can only be applied in addition and multiplication. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. Most commutative operations encountered in practice are also associative. This video is provided by the Learning Assistance Center of Howard Community College. Algebra Commutative Property. The associative and commutative properties are two elements of mathematics that help determine the importance of ordering and grouping elements. ( Here’s an example of the property in use: 2 + 4 = 4 + 2. ℏ This property was first given it's name by a Frenchman named Francois Servois in 1814. Simply put, the commutative property states that the factors in an equation can be rearranged freely without affecting the outcome of the equation. 2 The commutative property is among the foundation for the rules of the algebra. 3 The commutative property of addition informs us we can include things in any order and still obtain the same sum. 7 ) g d Essentially those operations that fall under the commutative property are multiplication and addition. , The commutative property is among the foundation for the rules of the algebra. Commutative property of addition worksheet is much required to the kids who would like practice addition of numbers. 4 Commutativity is a property of some logical connectives of truth functional propositional logic. x The commutative property (or commutative law) is a property generally associated with binary operations and functions. In contrast, putting on underwear and trousers is not commutative. Remembering the formula for commutative property of addition is a + b = b + a and you are good to The commutative property is an ancient idea in mathematics that still has numerous uses today. The commutative property is one of several properties in math that allow us to evaluate expressions or compute mental math in a quicker, easier way. f This tells us that it doesn't matter what order we add our numbers in; the total will still be t… is the reduced Planck constant). Addition: $$2 + 6 = 8$$ $$6 +2 = 8$$ Multiplication: $$3 * 5 = 15$$ $$5 * 3 = 15$$ The commutative property and the commutative property are only valid for equations with addition or multiplication. {\displaystyle x{\frac {d}{dx}}} Commutative property lesson plans and worksheets from thousands of teacher-reviewed resources to help you inspire students learning. Commutative law is used to change the order of the operands without changing the end result. f 1 Commutative property of multiplication for two real numbers a, b is given below, a b = b a. An isosceles triangle's altitude will bisect its base. 1 In math, the commutative property of multiplication allows us to change the places of factors in a product. b + Commutative Property Of Multiplication Formula. {\displaystyle g(x)=3x+7} You can use the commutative property with addition and multiplication operations, but not subtraction or division (with a few exceptions): […] The "Distributive Law" is the BEST one of all, but needs careful attention. Commutative Property under Multiplication of Integers: If we multiply two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we multiply (2×3) = (3×2) = 6. Multiplication and addition are commutative. The following logical equivalences demonstrate that commutativity is a property of particular connectives. Any number of factors can be rearranged to yield the same product: 1 × 2 × 3 = 3 × 1 × 2 = 6 = 2 × 3 × 1 = 2 × 1 × 3 1 × 2 × 3 = 3 × 1 × 2 = 6 = 2 × 3 × 1 = 2 × 1 × 3. {\displaystyle 0-1\neq 1-0} b + a = a + b (Yes, algebraic expressions are also commutative for addition) Examples. of the Commutative Property for Multiplication . A sample equation would do a better job of explaining the commutative property than any explanation. In this post, we’re going to see what the commutative property is all about. {\displaystyle \Leftrightarrow } Explanation :-Subtraction is not Commutative for Whole Numbers, this means that when we change the order of numbers in subtraction expression, the result also changes. The Egyptians used the commutative property of multiplication to simplify computing products. (also called products of operators) on a one-dimensional wave function Commutative property of addition lesson plans and worksheets from thousands of teacher-reviewed resources to help you inspire students learning. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms doesn't change. 4 • 2 = 2 • 4; 5 • 3 • 2 = 5 • 2 • 3; a • b = b • a(Yes, algebraic expressions are also commutative for multiplication) Examples. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seen in the adjacent image. i Here’s an example of the When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. ) The commutative property of multiplication states that two numbers can be multiplied in either order. Example Charles and George learned how to calculate the area of a rectangle in math class by using the base by height formula. Given two ways, A and B, of shuffling a deck of cards, doing A first and then B is in general not the same as doing B first and then A. Robins, R. Gay, and Charles C. D. Shute. Captcha proves you are good to go web Store foundation for the functions noncommutative. A rectangle in math, the commutative property of addition informs us we can add in any.. 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