Numbers

In mathematics, numbers are the key units to measure, count, tally and compare to carry out calculations. From measuring time, calculating distance, paying our expenses to almost every activity where we need numbers.
 

The origin of numbers began through symbols, as humans evolved, various number systems were introduced. In the beginning, simple methods were used to keep track of quantities, numbers evolved from simple accounting methods to complex mathematical theories. 

• Greek philosophers like Pythagoras and Euclid used numbers more than just a tool, they studied numbers to understand the world. 

• The ancient Romans around 500 BCE started a roman numeral system using letters like I, V, X, L, C, D and M to represent numbers, without including zero.

• The invention of zero was around (500 CE) Indian mathematician Aryabhata introduced the concept of zero. 

• The decimal system, use of numbers from 0-9 and degrees of 10 were developed by scholars like Aryabhata.

• The Hindu Arabic numerals are used till date.
   

As you got an idea of what numbers are, now let’s discuss the mathematical operations like addition, multiplication, and division. To perform these operations, there are a specific rules/properties to be followed:

Commutative property :

• Even if the order of numbers is changed in addition or multiplication, the result remains the same.

• We apply commutative property in addition and multiplication.

• Example: addition = 2+3=3+2=5

Multiplication=4x5=5x4=20.

Associative Property:

• The way numbers are grouped does not affect the result.

• It is applied in addition and multiplication.

• Example: 

Addition:  (1+2)+3=1+(2+3)=6

Multiplication: (2×3)x4=2x(3×4)=24

Distributive Property:

• Multiplying a number with a sum or difference gives the same result as doing each multiplication separately.

• Example : 

• 2x(3+4)=(2×3)+(2×4)=14.

Identity property:

• Addition of 0 or multiplication of 1 does not bring any changes in the number.

• Example:

Addition: 5+(-5) = 0

Multiplication: 8x(1/8)=1

Inverse property :

• The inverse property reverses the effect of the operation.

• Additive inverse: When a given number is added to the same number of opposite signs, which gives the sum as zero. Example: 6+(-6)=0.

Multiplicative inverse: Multiplying a number by its reciprocal results in 1. Example: 5x(1/5)=1.

Closure property:

In the closure property when we perform a certain operation on two numbers from a particular set i.e., if we are adding two whole numbers together we get a whole number, it's the same with integers as well, the result will also belong to that same set. 

Example: integers : 5+2=7(an integer)

Whole numbers= 4×3=12(a whole number)

Numbers can be classified based on binary operations. Binary operations in mathematics, we combine two numbers to get one result. Numbers are classified into the following categories.

• Natural numbers: Natural numbers are counting numbers beginning with 1 and so on till infinity. These numbers are represented by the letter ’N’. N=1,2,34,5,...........

• Whole numbers : Whole numbers are the numbers that include natural numbers, and whole numbers begin from 0' and are represented as ‘W’. W=0,1,2,3,4,5,...........

• Integers: Integers are the numbers that have both positive and negative whole numbers but cannot be a fraction, -2,-1,0,1,2,.....

• Rational numbers: A number that is represented in  pq form, where q is not equal to 0. Example:12,-3,0.57. 

• Irrational number: The numbers that cannot be expressed as fractions are irrational numbers. Example: ,2.

• Real number: All the rational and irrational numbers form a set of real numbers.

• Complex numbers: The numbers that have real and imaginary parts are called complex numbers. In a+bi; i=-1.

Numbers play a vital role in the life of students, shaping the understanding of both academic and real-world challenges. Numbers play a very important role as a building block in mathematics for students.
 

1.Cardinal and ordinal numbers:

Cardinal numbers are the numbers that represent the quantity of a number. E.g., 1,2,3…

Ordinal numbers are the numbers that represent the position of a number. E.g., 1st,2nd,3rd….

2.Even and odd numbers:

Even : the numbers divisible by 2(e.g.,2,4,6)

Odd : the numbers that are not divisible by 2.(e.g.,1,3,5) 

3.Consecutive numbers:

Numbers that succeed further without any gaps. E.g., 1,2,3 or 5,6,7.

4.Prime and composite numbers:

Prime : The number that is divisible by 1 and the number alone. E.g., 2,3,5

Composite: The number which has more than two divisors. E.g., 4,6,8

5.Co-prime numbers:

Two numbers with no common factor apart from 1.(e.g., 8 and 15)

6.Perfect numbers:

A perfect number is a positive integer which is equal to the sum of its proper divisors, excluding the number itself. 

Example: 6=1+2+3. Here divisors of 6 are 1,2,3 which adds up to 6.

7.Fractions and Decimals:

Fractions are the numbers that represent a part of the whole, and decimal represent the same number in the base of 10. Example: 12=0.5.

8.Factors and multiples:

Factors are the number that divide the number without leaving any remainder.

Multiples are the numbers that are the result of multiplying a number by an integer.

Example: factors of 12 are 1,2,3,4,6,and 12

Multiples of 3 are 3,6,9,12, etc.

9.GCF and LCM:

GCF (also known as HCF) is the greatest common factor of number, meaning it is the largest factor which is common to two numbers. Example: GCF of 12 and 18 is 6 because it divides both.

LCM is the least common multiple which both the numbers share. Example: LCM of 4 and 6 is 12 as it is the first common multiple of both the numbers.

10.Prime factorization:

The parting down of numbers into their factors. Example : prime factors of 18 are 2×3×3.

11.Algebraic and transcendental numbers:

Algebraic numbers are roots of polynomial equations with coefficients. Example: 2.

Transcendental numbers are not the roots of any polynomial. Example:  
 

Learning numbers can be made a breeze with a few creative tricks. Let's think of it as we are playing a game, the more we practice, the easier it gets. Here are a few tips and tricks that can make understanding numbers easier.

• Let's analyze numbers of daily life situations, let's say your morning alarm that is set up to 6:48 am you can break this down to understand how many minutes you have until you wake up. E.g., 6:48=6:00 + 0:48. This means you have 6 hours and 48 minutes till the actual time you need to be up.

• Use Distributive property: Break the numbers into smaller parts, larger numbers may scare you, why not try dividing them into parts. Example: 48+36 ,40+8=48; 36=30+6, 40+30=70;8+6=14; 70+14=84.

• Rounding numbers for estimations to make calculations easier. For example: to add 487+293, round off to 490+290=780 and then we can refine. 

• Use the rule of divisibility, when the last digit is even,0 or 5 it is 2 and 5 respectively, the sum of digits is divisible by 3 then the number is 3.