A number (more specifically, an integer) is even if it is congruent to zero, modulo 2. In other words, when you divide the number by two, you will get a whole number (not a-number-and-a-half), or a number x is even if (and only if) it can be written as x = 2n for some integer n.

If x and y are even numbers, and z is an odd number, then:

  • x + y is even
  • x * y is even
  • x + z is odd
  • x * z is even

A function f is even if for every value x in its domain, -x is also in the domain and f(-x) = f(x). A graph of an even function is symmetrical about the y-axis.

If f and g are even functions, and h is an odd function, then:

  • nf(x) is even for any real number n
  • f(x) + g(x) is even
  • f(x) * g(x) is even
  • f(x) * h(x) is odd
  • f(g(x)) is even
  • f(h(x)) is even
  • h(f(x)) is even
  • If the domains of f and h overlap, then f(x) + h(x) is not even, but it is not odd either unless f(x) = 0

A polynomial of x is even if the power of x in every term is an even number. (Note that (-x)2n = ((-x)2)n = ((x)2)n = (x)2n; every term in an even polynomial is an even function. Also note that this includes the independent term; 0 is even and x0 = 1, which is independent of x.)

Even polynomials are even functions, and continuous even functions can be approximated to any desired accuracy by even polynomials. However, there are functions which are neither even nor odd, and there are polynomials which are neither even nor odd. This is different than for integers; every integer is either even or odd. (In fact, the function f(x) = 0 is both even and odd, thanks to ariels for setting this straight for me.)


BrianShader reminds me: You might like to point out that any function can be written as an even function plus an odd function.

E"ven (?) n. [OE. eve, even, efen, aefen. AS. �xd6;fen; akin to OS. aband, OFries, avend, D. avond, OHG. aband, Icel. aptan, Sw. afton, Dan. aften; of unknown origin. Cf. Eve, Evening.] Evening. See Eve, n. 1.

[Poetic.]

Shak.

 

© Webster 1913.


E"ven, a. [AS. efen. efn; akin to OS. eban, D. even, OHG. eban, G. efen, Icel. jafn, Dan. jevn, Sw. jamn, Goth. ibns. Cf. Anent, Ebb.]

1.

Level, smooth, or equal in surface; not rough; free from irregularities; hence uniform in rate of motion of action; as, even ground; an even speed; an even course of conduct.

2.

Equable; not easily ruffed or disturbed; calm; uniformly self-possessed; as, an even temper.

3.

Parallel; on a level; reaching the same limit.

And shall lay thee even with the ground. Luke xix. 44.

4.

Balanced; adjusted; fair; equitable; impartial; just to both side; owing nothing on either side; -- said of accounts, bargains, or persons indebted; as, our accounts are even; an even bargain.

To make the even truth in pleasure flow. Shak.

5.

Without an irregularity, flaw, or blemish; pure.

"I know my life so even."

Shak.

6.

Associate; fellow; of the same condition.

[Obs.] "His even servant."

Wyclif (Matt.).

7.

Not odd; capable of division by two without a remainder; -- said of numbers; as, 4 and 10 are even numbers.

Whether the number of the stars is even or odd. Jer. Taylor.

On even ground, with equal advantage. -- On even keel Naut., in a level or horizontal position.

 

© Webster 1913.


E"ven (?), v. t. [imp. & p. p. Evened (?); p. pr. & vb. n. Evening (?)]

1.

To make even or level; to level; to lay smooth.

His temple Xerxes evened with the soil. Sir. W. Raleigh.

It will even all inequalities Evelyn.

2.

To equal

[Obs.] "To even him in valor."

Fuller.

3.

To place in an equal state, as to obligation, or in a state in which nothing is due on either side; to balance, as accounts; to make quits.

Shak.

4.

To set right; to complete.

5.

To act up to; to keep pace with.

Shak.

 

© Webster 1913.


E"ven (?), v. i.

To be equal.

[Obs.]

R. Carew.

 

© Webster 1913.


E"ven, adv. [AS. efne. See Even, a., and cf. E'en.]

1.

In an equal or precisely similar manner; equally; precisely; just; likewise; as well.

"Is it even so?"

Shak.

Even so did these Gauls possess the coast. Spenser.

2.

Up to, or down to, an unusual measure or level; so much as; fully; quite.

Thou wast a soldier Even to Cato's wish. Shak.

Without . . . making us even sensible of the change. Swift.

3.

As might not be expected; -- serving to introduce what is unexpected or less expected.

I have made several discoveries, which appear new, even to those who are versed in critical learning. Addison.

4.

At the very time; in the very case.

I knew they were had enough to please, even when I wrote them. Dryden.

Even is sometimes used to emphasize a word or phrase. "I have debated even in my soul."

Shak.

By these presence, even the presence of Lord Mortimer. Shak.

 

© Webster 1913.

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