Odd Numbers

Odd Numbers - Definition, List, Types and Properties

Math is filled with numbers. Numbers can be put into two basic categories, odd numbers, and even numbers. Even numbers are divisible by 2.


Now, what are odd numbers?


Odd numbers are the numbers that cannot be divided by two. They are integers and not fractions. 

An odd number is an integer which takes the form n=2k+1, where k is an integer. The odd numbers are therefore ..., -3, -1, 1, 3, 5, 7,.... They are also called gnomonic numbers. 

Odd numbers always leave a remainder of 1 when divided by two, i.e., the congruence n=1 (mod 2) holds for odd n. Parity is defined as the oddness of a number, so an odd number has a parity 1, while an even number always has a parity 0. Integers which are not odd are called even numbers. 

The generating function for the odd numbers is


 (x(1+x))/ ((x-1)^2)=x+3x^2+5x^3+7x^4+.... 

When an even number and an odd number is multiplied, the product is always even. 

 (2k)(2l+1)=2[k(2l+1)], 

this equation is divisible by 2 and hence is even.

Here are a few key factors of odd numbers:


1. When these numbers are divided by 2, the result or remainder should be a fraction or 1.
2. The first positive odd number in the number line is 1.

A list of Odd Numbers


There are 24 odd numbers from 1 to 50 while there are 50 odd numbers from 1 to 100. An example of an odd numbers list include numbers like -5, -3, -1, 1, 3, 5 , 7 , 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 etc. while even number list include -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 etc.


Types of Odd Numbers


Composite Odd Numbers 

All positive integers are called composite numbers or prime numbers. The composite odd numbers that are not divisible by two are called Composite odd numbers.

The composite odd numbers up to 100 are:

9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99.

Consecutive odd number

Consecutive numbers are those which follow each other in order. They have a standard difference of 2 between every two numbers. The odd numbers that are consecutive in nature are called Consecutive odd numbers. 

The consecutive odd numbers up to 10 are:

1, 3, 5, 7, 9

SOME PROPERTIES OF ODD NUMBERS 


What happens if we add two odd numbers? 

When two odd numbers are add up together, they produce an even number.

For example:
1+1=2
1+3=4

Here one and three are odd numbers and then the sum is two and four respectively are even numbers. 

ODD + ODD = EVEN 


Proof
Let two odd number be x and y

x and y can be represented in the form
x=2k1+1
and y=2k2+1
where k1,k2∈Z
Adding x+y we have,
2(k1 + k2) +2
which is surely divisible by 2.
Hence Proved.

What happens when we subtract two odd numbers?

When two odd numbers are subtracted from each other, they give an even number. 

For example: 

7-1=6
3-1=2

Here seven, one and three are odd numbers, and a result is an even Number.

ODD - ODD = EVEN 


What happens if we multiply two odd numbers?

Unlike the addition and subtraction of odd numbers, when we multiply two odd numbers, we get an odd number only. 

For example:
3x7=21
5x5=25

Here three, seven and five are odd numbers, and the product of the odd numbers which are 21 and 25 are also odd numbers.

ODD X ODD = ODD

Let two odd numbers be x and y.

These numbers can be given in the form where

x=2k1+1
and y=2k2+1
where k1,k2∈Z
x×y=(2k1+1)(2k2+1) x × y=4k1k2+2k1+2k2+1

The above equation can be re-written as x × y=2(2k1k2+k1+k2) + 1

Thus, the multiplication of two odd number results in an odd number.


hat happens when we divide two odd numbers? 

When the division of two odd numbers take place, an odd number is formed in the below. 

For example:
3/3=1
9/3=3
Here nine and three are both numbers then the division of the number is also an odd number.

ODD/ODD = ODD


Hence, odd numbers is an elementary concept to understand. It forms the basis of all mathematics relations, to help you understand further, here are a few questions you can try to solve:

1. Is three an odd number? If yes, give the reason.
2. What is the sum of seven and five?
3. Subtract one from nine. What do you get?
4. What is the product of twenty-one and three?
5. Divide nine by three. What is the answer?
6. Is zero an odd number? If yes, why?
7. What are the types of odd numbers? Give a brief explanation.