## Aptitude Tests(number system)

Practice different types of aptitude tests for free, including numerical and verbal tests. All questions come with worked solutions to help you improve.

### Formulas to Solve Number System(Aptitude Tests)

- Number system is a writing system for presenting number on the number line. A number system is a system of writing or expressing numbers.
- There are generally two type of Number
**Whole Number****Natural Number.**

### Number System Formulas & Definitions(Aptitude Tests)

**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……….∞}**

- All positive integers are called natural numbers. All counting numbers from 1 to infinity are natural numbers.
**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,…………..∞}**

- 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.
**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.

- All numbers that do not have the decimal places in them are called integers.
**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}51 = 0.5,\frac{125}{4}4125 = 31.25

- b. Non-Terminating decimal fractions: For example:\frac{19}{6}619 = 3.1666666, \frac{21}{9}921 = 2.33333
**Irrational Numbers**- It is a number that cannot be written as a ratio \frac{x}{y}
*yx* form (or fraction). An Irrational numbers are non-terminating and non-periodic fractions. For example: \sqrt{2}2 = 1.414

- It is a number that cannot be written as a ratio \frac{x}{y}
**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 Number System and Basic Concept

- (a – b)(a + b) = (a² – b²).
- (a + b)² = (a² + b² + 2ab)
- (a – b)² = (a² + b² – 2ab)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- (a³ + b³) = (a + b)(a² – ab + b²)
- (a³ – b³) = (a – b)(a² + ab + b²)
- (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
- When a + b + c = 0, then a³ + b³ + c³ = 3abc.

**Read Also – How to solve number system problems Quickly**

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

- Squares of numbers 91-100:
- 97
^{2}

Step 1: 100-97 = 3

Step 2: 97-3 = 94

Step 3: 3^{2}= 09

Final result: From step 2 and

Step 3 => 97^{2}= 9409

- 91
^{2}

Step 1: 100-9 = 91

Step 2: 91-9 = 82

Step 3: 9^{2} = 81

Final Result: From step 2 and step 3 => 91^{2} = 8281

- Squares of numbers 100-109:
- 102
^{2}

Step 1: 102-100 = 2

Step 2: 102 +2 = 104

Step 3: 2^{2} = 04 Final result:

From step 2 and step 3 => 102^{2}=10404

- 107
^{2}

Step 1: 107-100 = 7

Step 2: 107+7 = 114

Step 3: 7^{2} = 49

Final Result: From step 2 and step 3 => 107^{2} = 11449

**Squares of numbers 51-60**- 53
^{2}

Step 1: 53-50 = 3

Step 2: 25+3 = 28

Step 3: 3^{2} = 09

Final result: From step 2 and step 3 => 53^{2} = 2809.

- 42
^{2}

Step 1: 50-42 = 8

Step 2: 25-8 = 17

Step 3: 8^{2} = 1764

Final Result From step 2 and step 3 => 42^{2} = 1764