Understanding Even Numbers

• Even numbers are integers exactly divisible by 2
• An even number can be expressed as 2n, where n is an integer
• Examples of even numbers include -4, -2, 0, 2, 4, 6, and 8
• The set of even numbers includes all integers that, when divided by 2, leave a remainder of 0
• Zero is considered an even number because it satisfies the definition (0 = 2 * 0)

Properties of Even Numbers

• The sum of two even numbers is always even
  • For example, 2n + 2m = 2(n + m), which is divisible by 2
• The difference between two even numbers is always even
  • For example, 2n - 2m = 2(n - m), which is divisible by 2
• The product of two even numbers is always even
  • For example, (2n) * (2m) = 4nm = 2(2nm), which is divisible by 2
• The product of an even number and an odd number is always even
  • For example, (2n) * (2m + 1) = 4nm + 2n = 2(2nm + n), which is divisible by 2
• The additive inverse of an even number is also even
  • The additive inverse of 2n is -2n, which is 2*(-n), and therefore even
• Even numbers alternate with odd numbers in the set of integers

Identifying Even Numbers

• A number is even if its last digit is 0, 2, 4, 6, or 8 in base 10
• Any integer can be expressed as 10k + d, where k is an integer and d is the last digit
• Since 10k is always divisible by 2, the divisibility by 2 depends only on the last digit d

Even Numbers in Arithmetic Operations

• Division of an even number by 2 results in an integer
  • By definition, an even number 2n divided by 2 equals n, which is an integer
• When dividing an odd number by 2, the result is not an integer
  • It will always have a fractional component of 0.5

Mathematical Significance

• Even numbers are fundamental in number theory and used extensively in various mathematical proofs and concepts
• They play a crucial role in modular arithmetic, particularly in understanding remainders and congruences
• Even and odd numbers are the basis for parity, which is very important in computer science