An even number is any number that gives a remainder of zero when divided by 2. For example, 12 gives a remainder of 0 when divided by 2, so 12 is even.
You can also say that 12 is even because there exist a number 6, such that 12 = 2 × 6. Having said that, we can come up with a formal definition.
Formal definition of an even number:
A number n is even if there exist a number k, such that n = 2k where k is an integer.
This is the formal way of saying that if n is divided by 2, we always get a quotient k with no remainder. Having no remainder means that n can in fact be divided by 2.
How to quickly check if a number is even?
We saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.
Therefore, any number whose last digit is 0, 2, 4, 6, or 8 is an even number. For example, 98716 is an even number because its last digit shown in red is 6.
Other examples of even numbers are 58, 44884, 998632, 98, 48, and 10000000.
The figure below shows the difference between even numbers and odd numbers.
An odd number is any number that gives a remainder of 1 when divided by 2. For example, 27 gives a remainder of 1 when divided by 2, so 27 is odd.
You can also say that 27 is odd because there exist a number 13, such that 27 = 2 × 13 + 1. Having said that, we can come up with a formal definition.
Formal definition of an odd number:
A number n is odd if there exist a number k, such that n = 2k + 1 where k is an integer.
This is the formal way of saying that if n is divided by 2, we always get a quotient k with a remainder of 1. Having a remainder of 1 means that n cannot in fact be divided by 2.
How to quickly check if a number is odd?
Again, we saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.
Therefore, any number whose last digit is not 0, 2, 4, 6, or 8 is an odd number.
Other examples of odd numbers are 53, 881, 238637, 99, 45, and 100000023
Adding two even numbers
even + even = even
4 + 2 = 6
Generally speaking, let n_{1} = 2k_{1} be an even number and let n_{2} = 2k_{2} be another even number.
n_{1} + n_{2} = 2k_{1} + 2k_{2}
n_{1} + n_{2} = 2(k_{1} + k_{2})
Notice that any number multiplied by 2 is even. This is the reason that 2(k_{1} + k_{2}) is an even number!
Adding an even number to an odd number
even + odd = odd
6 + 3 = 9
Generally speaking, let n_{1} = 2k_{1} be an even number and let n_{2} = 2k_{2} + 1 be an odd number.
n_{1} + n_{2} = 2k_{1} + 2k_{2} + 1
n_{1} + n_{2} = 2(k_{1} + k_{2}) + 1
Adding two odd numbers
odd + odd = even
13 + 13 = 26
Generally speaking, let n_{1} = 2k_{1} + 1 be an odd number and let n_{2} = 2k_{2} + 1 be another odd number.
n_{1} + n_{2} = 2k_{1} + 1 + 2k_{2} + 1
n_{1} + n_{2} = 2(k_{1} + k_{2}) + 2
Notice that 2(k_{1} + k_{2}) is even and 2 is even.
Therefore, 2(k_{1} + k_{2}) + 2 is even since adding two even numbers is equal to an even number
Multiplication
even × even = even
2 × 6 = 12
even × odd = even
8 × 3 = 24
odd × odd = odd
3 × 5 = 15
Subtraction
even − even = even
8 − 4 = 4
even − odd = odd
6 − 3 = 3
odd − odd = even
13 − 3 = 10
If you have any questions about this even and odd numbers lesson, just contact me.
Dec 03, 22 12:47 PM
Nov 18, 22 08:20 AM