Number System

The number system is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a digital computer. there are two basic ways of representing the numerical values of the various physical quantities with which we constantly deal in our day-to-day lives. The arithmetic value which is used for expressing the quantity and used in making calculations are defined as NUMBERS. A symbol like “4, 5, 6” representing a number is known as a numeral. Without numbers, counting things is not possible, date, time, money, etc. these numbers are also used for measurement and labeling. The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words

Number System Definition

A Number system is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as a Number system. The numeral system Represents a useful set of numbers, reflects the arithmetic and algebraic structure of a number and Provides standard representation. The digits from 0 to 9 can be used to form all the numbers. With these digits, anyone can create infinite numbers. For example, 156,3907, 3456, 1298, 784859 etc.

Number and Its Types

Numbers used in mathematics are mostly decimal number systems. In the decimal number system, digits used are from 0 to 9 and base 10 is used. There are many types of numbers in the decimal number system, below are some of the types of numbers mentioned,

Numbers that are represented on the right side of the zero are termed Positive Numbers. The value of these numbers increases on moving towards the right. Positive numbers are used for Addition between numbers. Example: 1, 2, 3, 4.

Numbers that are represented on the left side of the zero are termed Negative Numbers. The value of these numbers decreases on moving towards the left. Negative numbers are used for Subtraction between numbers. Example: -1, -2, -3, -4.

Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity and are also used for counting purpose. It does not include zero (0). In fact, 1,2,3,4,5,6,7,8,9…., are also called counting Numbers

Natural numbers are part of real numbers that include only the positive integers i.e. 1, 2, 3, 4,5,6, ………. excluding zero, fractions, decimals, and negative numbers.

The whole numbers are the numbers without fractions and it is a collection of positive integers and zero. It is represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}. Zero as a whole represents nothing or a null value. These numbers are positive integers including zero and do not include fractional or decimal parts (3/4, 2.2, and 5.3 are not whole numbers). Also, arithmetic operations such as addition, subtraction, multiplication, and division are possible on whole numbers.

Integers are the collection of Whole Numbers plus the negative values of the Natural Numbers. Integers do not include fraction numbers i.e. they can’t be written in a/b form. The range of Integers is from the Infinity at the Negative end and Infinity at the Positive end, including zero. Integers are represented by the symbol Z. We can perform all the arithmetic operations, like addition, subtraction, multiplication, and division, on integers. The examples of integers are, 1, 2, 5,8, -9, -12, etc. The symbol of integers is “Z“. Now, let us discuss the definition of integers, symbols, types, operations on integers, rules, and properties associated with integers, and how to represent integers on a number line with many solved examples in detail.

Rational numbers are the numbers that can be represented in the fraction form i.e. a/b. Here, a and b both are integers, and b≠0. All the fractions are rational numbers but not all rational numbers are fractions.Rational numbers are numbers that can be expressed as a fraction and also as positive numbers, negative numbers, and zero. It can be written as p/q, where q is not equal to zero.

The word “rational” is derived from the word ‘ratio’, which actually means a comparison of two or more values or integer numbers and is known as a fraction. In simple words, it is the ratio of two integers.

Example: 3/2 is a rational number. It means integer 3 is divided by another integer 2.

Irrational numbers are the numbers that can’t be represented in the form of fractions i.e. they can not be written as a/b.The numbers which are not rational numbers are called irrational numbers Now, let us elaborate, irrational numbers could be written in decimals but not in the form of fractions, which means they cannot be written as the ratio of two integers.

Irrational numbers have endless non-repeating digits after the decimal point. Below is an example of an irrational number:

Example: √8 = 2.828…

Numbers that do not have any factors other than 1 and the number itself are termed Prime Numbers. All the numbers other than Prime Numbers are termed as Composite Numbers except 0. Zero is neither prime nor a composite number.

Types of Number System

There are various types of number systems in mathematics. The four most common number system types are:

Decimal number system

The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands, and so on. This system is expressed in Decimal Numbers. Every position shows a particular power of the base (10).

For example, 10264 has place values as,

(1 × 10⁴) + (0 × 10³) + (2 × 10²) + (6 × 10¹) + (4 × 10⁰)

= 1 × 10000 + 0 × 1000 + 2 × 100 + 6 × 10 + 4 × 1

= 10000 + 0 + 200 + 60 + 4

= 10264

Binary number system

The base 2 number system is also known as the Binary number system. wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.

For example, 14 can be written as 1110, 19 can be written as 10011, and 50 can be written as 110010.

Octal number system

In the octal number system, the base is 8 and it uses numbers from 0 to 7 to represent numbers. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.

Example:

(135)¹⁰ can be written as (207)⁸

(215)¹⁰ can be written as (327)⁸

Hexadecimal number system

In the hexadecimal system, numbers are written or represented with base 16. In the hex system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabet from A to F. The below-given table shows the representation of numbers in the Hexadecimal number system

Example

(255)¹⁰ can be written as (FF)¹⁶

(1096)¹⁰ can be written as (448)¹⁶

(4090)¹⁰ can be written as (FFA)¹⁶

Binary to Octal

Example: 1100011

Octal means 8

Starting from the least significant bit, we will make a group of three bits First pair = 011

Second group = 100

Third group = 001

Now, we know that 011 = 3, 100 = 4, and 001 = 1, so the equivalent octal number system will be (143)₈

Binary to Hexadecimal

Example: 101010001

We know that hexadecimal means 16, so we will start from the least significant bit and make groups of 4 bits:

First pair = 0001 = 1

Second pair = 0101 = 5

Third pair = 0001 = 1

So, the hexadecimal equivalent will be (151)₁₆.

Decimal to Octal Conversion

Example: (425)₁₀

Divide 425 by 8 till you get the quotient as 0.

1. Dividing 425 by 8, we get the quotient is 53, and the remainder as 1
2. Divide 53 by 8, we get the quotient is 6, and the remainder as 5
3. Divide 6 by 8 by taking the quotient as 0, we get the remainder as 6

Importance of Number System in MBA Entrance Exams