Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [parity]

This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

In mathematics, parity is a term we use to express if a given integer is even or odd.

  • The parity of a number depends only on its remainder after dividing by $2$.
  • An even number has parity $0$ because the remainder after dividing by $2$ is $0$, while an odd number has parity $1$ because the remainder after dividing by $2$ is $1$.
  • Parity is often useful for verifying whether an equality is true or false by using the parity rules of arithmetic to see whether both sides have the same parity.
  • In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code.
  • Parity is an important idea in quantum mechanics because the wavefunctions which represent particles can behave in different ways upon transformation of the coordinate system which describes them.

References:

https://en.wikipedia.org/wiki/Parity_(mathematics)

http://mathworld.wolfram.com/Parity.html

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The Mathematics of Tetris

I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game. Background: The Tetris…
Eric Naslund
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Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called…
Jon Ericson
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Is zero odd or even?

Some books say that even numbers start from $2$ but if you consider the number line concept, I think zero($0$) should be even because it is in between $-1$ and $+1$ (i.e in between two odd numbers). What is the real answer?
mvar2011
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Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
Kevin
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Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will also not be odd or even. But I want a rigorous math…
user210387
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Why is Euler's Totient function always even?

I want to prove why $\phi(n)$ is even for $n>3$. So far I am attempting to split this into 2 cases. Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will always be even. Case 2: $n$ is not a power of $2$. This is…
idkmybffjill
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Can decimal numbers be considered "even" or "odd"?

Is the concept of even/odd numbers applicable to decimal numbers? For e.g. - 4.222 is a even number?
Ravi Gupta
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Do odd imaginary numbers exist? [parity for Gaussian integers]

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
InterestedGuest
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$C(n,p)$: even or odd?

Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value? ($p\lt n$, $p$ and $n$ are positive integers)
Paulo Argolo
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$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?

The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$ Where: $n,k$ are natural numbers and $k\le n$. $t$ is taken over all integers between $0$ and $k$ such that $t$ is…
Gadi A
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Proving that all integers are even or odd

I know that $\mathbb{Z}$ is a group under addition with a multiplication defined. I have just the definition of even and odd integers: $n$ is even if $n = 2k$ for some integer $k$ and $n$ is odd if $n = 2k+1$ for some integer $k$. Using just this I…
John Doe
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$\sqrt 2$ is even?

Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n? Similar argument for odd…
jimjim
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Rearranging a $10 \times 10$ matrix of naturals $1\le n\le 100$ s.t. the sum of every two neighbouring numbers is composite in max. 35 steps

We are given a $10 \times 10$ matrix which contains every natural number between $1$ and $100$ in arbitrary order. We are to prove that it is always possible to rearrange the matrix by swapping any two entries of choice in at most 35 steps, such…
mathcactus
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Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from which we can easily find…
Bart Michels
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Do all odd functions have symmetric zeros?

Is there odd functions where a zero of positive x-axis doesn't have necessarily a symmetric? What I mean is, consider an odd function where its positive zeros are x=1, x=2, x=3. Since it is odd, it has a symmetry related to origin of referencial, so…
Vitor Aguiar
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