# Number System Formulas

## Important Formulas for Number System

Number system is a method of representing  a number on the number line. Number system is a system of writing or expressing numbers. This Page from here on contains Formulas and definition of Number System. ### Formulas of Number System:

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

2. (1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1)/6

3. (1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/2)²

4. Entirety of first n odd numbers = n²

5. Entirety of first n even numbers = n(n + 1)

Mathematical Formulas to solve questions

1. (a + b)(a – b) = (a² – b²)

2. (a + b)² = (a² + b² + 2ab)

3. (a – b)² = (a² + b² – 2ab)

4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

5. (a³ + b³) = (a + b)(a² – ab + b²)

6.(a³ – b³) = (a – b)(a² + ab + b²)

7. (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)

8. when a + b + c = 0, then a³ + b³ + c³ = 3abc

### Types of Number System:

• Natural Numbers
• All positive integers are called natural numbers. All counting numbers from 1 to infinity are natural numbers. N = {1, 2, 3, 4, 5, 6……….∞}
• Whole Numbers
• The set of numbers that includes all natural numbers and the number zero are called whole numbers. They are also called as Non-negative integers. W = { 0,1,2,3,4,5,6,7,8,…………..∞}
• Integers
• All numbers that do not have the decimal places in them are called integers. Z = {∞…….-3, -2, -1, 0, 1, 2, 3………∞}
• a. Positive Integers: 1, 2, 3, 4….. is the set of all positive integers.
• b. Negative Integers: −1, −2, −3….. is the set of all negative integers.
• c. Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
• Real Numbers
• All numbers that can be represented on the number line are called real numbers.
• Rational Numbers
• A rational number is defined as a number of the form a/b where ‘a’ and ‘b’ are integers and b ≠ 0. The rational numbers that are not integers will have decimal values. These values can be of two types
• a. Terminating decimal fractions: For example: $\frac{1}{5}$ = 0.5,$\frac{125}{4}$ = 31.25
• b. Non-Terminating decimal fractions: For example:$\frac{19}{6}$ = 3.1666666, $\frac{21}{9}$ = 2.33333
• Irrational Numbers
•  It is a number that cannot be written as a ratio $\frac{x}{y}$ form (or fraction). An Irrational numbers are non-terminating and non-periodic fractions. For example: $\sqrt{2}$ = 1.414
• Complex Numbers
• The complex numbers are the set {a+bi}, where, a and b are real numbers and ‘i’  is the imaginary unit.
• Imaginary Numbers
• A number does not exist on the number line is called imaginary number. For example square root of negative numbers are imaginary numbers. It is denoted by ‘i’ or ‘j.
• Even Numbers
• A number divisible by 2 is called an even number.
• For example: 2, 6, 8, 14, 18, 246, etc.
• Odd Numbers
• A number not divisible by 2 is called an odd number.
• For example: 3, 7, 9, 15, 17, 373, etc.
• Prime numbers
• A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
• For example: 2, 3, 5, 7, 11, 13, 17, etc.
• Composite numbers
• Numbers greater than 1 which are not prime, are known as composite numbers. For example: 4, 6, 8, 10, etc.

### Formulas for finding the Squares of a number.

Squares of numbers between 91-100:

• 972

Step 1: 97 can be written as (100-3)

Step 2: $(100-3)^2, using the formula of (a-b)^2$

$(100-3)^2 = 100^2 + 3^2 – 2*100*3$

= 10000 + 9 – 6000

= 10009 -600 = 9409

• 912

Step 1: 91 can be written as (100-9)

Step 2: $(100-9)^2, using the formula (a-b)^2$

$(100-9)^2 = 100^2 + 9^2 – 2*100*9$

10000 + 81 – 1800 = 8281

Final Result: From step 2 and step 3 => 912 = 8281

Squares of numbers between 100-109:

• 1022

Step 1: 102 can be written as (100+2)

Step 2: $(100+2)^2, using the formula (a+b)^2$

[/latex](100+2)^2 = 100^2 + 2^2 + 2*100*2[/latex]

10000 + 4 + 400 = 10404

• 1072

Step 1: 107 can be written as (100+7)

Step 2: $(100+7)^2, using the formula (a+b)^2$

$(100+7)^2 = 100^2 + 7^2 + 2*100*7$

10000 + 49 + 1400 = 11449

Squares of numbers between 51-60

• 532

Step 1: 53-50 = 3

Step 2: 25+3 = 28

Step 3: 32 = 09

Final result: From step 2 and step 3 => 532 = 2809.

• 422

Step 1: 50-42 = 8

Step 2: 25-8 = 17

Step 3: 82 = 1764

Final Result From step 2 and step 3 => 422 = 1764

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### 8 comments on “Number System Formulas”

• maninageswarv 0
• usha

how can i access coginitive ability games and i have paid for ibm prime mock could i access that?? 0
• Ashutosh Sinha

where can I find the squares material which was said in the number system video? 10
• HelpPrepInsta

Hello Ashutosh ,
we have resolved your concern regarding square.
we updated our page .

warm Regards
Prepinsta Team 18
• WatchToLearn

first formula is wrong a square minus b square is equal to a plus b into a -b 8
• PrepInsta

Actually its a cube minus b cube on the page, but although your formula is also correct. 4
• Ashish mark Daniel

1. (a – b) = (a² – b²) 15
• ten

superb well said 1