Even Integer

Pronunciation: /ˈi.vən ˈnʌm.bər/ Explain

An integer is even if it is evenly divisible by 2.^[3] Expressed mathematically:

"Let n be an integer. We say that n is even if there is some integer k such that n = 2k."^[4]

In set notation, even integers are .

In the decimal numbering system, an integer can be identified as even by the fact that the last digit of the number is even. Since the digits 0, 2, 4, 6 and 8 are even, the numbers 2750, -54, 22, -888 and 1794830495907549234098546 are even. Any integer in decimal form that does not have an even digit as the last digit is odd.

Whether an integer is even or odd is called parity. One says, "The parity of 6 is even," or "The parity of -365 is odd."

Properties of Even Numbers

• The sum of two even integers is also even.^[3] See Proof: The Sum of Two Even Integers is Even.
• The product of two even integers is also even. See Proof: The Product of Two Even Integers is Even.
• The cardinality of the set of even integers is the same as the cardinality of the set of integers.

Proof: The Sum of Two Even Integers is Even

Consider two arbitrary even integers x and y.	Initial assertion
An even integer can be written as 2a where a is an integer.	Definition of an even integer.
We will show that there exists an integer c such that x + y = 2c.	Claim
x and y can be rewritten as x = 2a and y = 2b where a and b are integers.	Apply the definition of an even integer.
The sum x + y can be written as 2a + 2b.	Substitute 2a for x and 2b for y.
2a + 2b can be written as 2(a + b).	Apply the distributive property of multiplication.
There exists an integer c = a + b.	Apply the closure property of integers and addition.
2(a + b) can then be rewritten as 2(c) = 2c.	Substitute c for a + .
QED.	The proof is complete.

Proof: The Product of Two Even Integers is Even.

Consider two arbitrary even integers x and y.	Initial assertion
An even integer can be written as 2a where a is an integer.	Definition of an even integer.
We will show that there exists an integer c such that x · y = 2c.	Claim
x and y can be rewritten as x = 2 and y = 2b where a and b are integers.	Apply the definition of an even integer.
The product x · y can be written as 2a · 2b.	Substitute 2a for x and 2b for y.
2a · 2 can be written as 2 · (a · 2 · b).	Apply the associative property of multiplication.
There exists an integer c = a · 2 · b.	Apply the closure property of integers and multiplication.
2(a · 2 · ) can then be rewritten as 2(c) = 2c.	Substitute c for a · b.
QED.	The proof is complete.

References

1. McAdams, David E.. All Math Words Dictionary, even number. 2nd Classroom edition 20150108-4799968. pg 73. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. even. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/9/2018. http://www.merriam-webster.com/dictionary/even. Buy the book
3. parity. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/9/2018. http://www.merriam-webster.com/dictionary/parity. Buy the book
4. Ethan D. Bloch. Proofs and Fundamentals: A First Course in Abstract Mathematics. 1st Edition. pp 60-61. Birkhäuser Boston. April 20, 2000. Last Accessed 7/9/2018. Buy the book

Cite this article as:

McAdams, David E. Even Integer. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/e/evennumber.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/20/2019: Updated expressions and equations to match new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/1/2010: Added "References". (McAdams, David E.)
5/12/2009: Corrected several typographical errors. (McAdams, David E.)
1/4/2009: Changed equation to image. (McAdams, David E.)
7/8/2008: Initial version. (McAdams, David E.)