Numbers, Numerals and Digits

Dhristi JEE 2022-24

What are Numbers?

A number can be defined as an arithmetic value used for representing the quantity and is used in making calculations. We count things using numbers. For example, in the image given below, this is one butterfly and these are 4 butterflies.


this


The collection of no objects in an element is symbolised as '0' and we call it zero. Digits are used to represent numbers, which include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For denoting a number we use a group of digits known as numerals. For example, 1234 2314 56111 are numerals. The method of expressing numbers in words is known as numeration. A number system is defined as a writing system for denoting numbers using digits or using symbols in a logical manner. 


The numeral system is used to:


  • Represents a useful set of different numbers.

  • Reflects the arithmetic as well as the algebraic structure of a number.

  • The numeral system provides a standard representation.


One-Digit Numbers

Examples of one-digit numbers are 1, 2, 3, 4, 5,6,7,8, and 9.


Two Digits Numbers

When we add one unit to the greatest one-digit number, we get the smallest two-digit number, which is 1+9 = 10. The greatest two-digit number is 99.


Two


Three-Digit Numbers

When we add one unit to the greatest two-digit number then we get the smallest three-digit number, which is 1+99 = 100. The greatest three-digit number is 999.


Three-Digit

 

Four-Digit Numbers

When we add one unit to the greatest three-digit number, we get the smallest four-digit number that is, 1+999 = 1000. The smallest four-digit number is 1000 and the greatest number is 9999.


 

Four


Five-Digit Numbers

When we add one unit to the greatest four-digit number, we get the smallest five-digit number, which is 1+ 9999 = 10000. The smallest five-digit number is 10000 and the greatest five-digit number is equal to 99999. 


Five

Types of Numbers

The numbers can be classified into sets, which is known as the number system. The different types of numbers in maths are:


  • Natural Numbers 

Natural numbers are also known as counting numbers that contain positive integers from 1 to infinity. The set of natural numbers is denoted as “N” and it includes N = {1, 2, 3, 4, 5, ……….} For example 0, 1, 2, 3, 4, 5, 6, 7,  8, and 9.


  • Prime Numbers

Prime numbers are natural numbers that are greater than 1 and have only 1 and themselves as factors. For example 2, 3, 5, 7, 11, 13… and so on.


  • Composite Numbers

A composite number is a natural number that is greater than 1 and has more factors including one. For example 2,4,6,8,9… and so on.


  • Whole Numbers

Whole numbers are the set of natural numbers in which zero is adjoined. They are known as non-negative integers and it does not include any fractional or decimal part. It is denoted as “W” and the set of whole numbers includes W = {0,1, 2, 3, 4, 5, ……….} For example 0, 1, 2, 3, 4, 5, 6, 7… and so on.


  • Integers

Integers are the set of all whole numbers but it includes a negative set of natural numbers also. “Z” represents integers and the set of integers are Z = { -3, -2, -1, 0, 1, 2, 3}


They are whole numbers with negative numbers adjoined. For example -9, -8, -7, -6… and so on.


  • Real Numbers 

All the positive and negative integers, fractional and decimal numbers without imaginary numbers are called real numbers. They are denoted by the symbol R.  Real numbers are the set of rational numbers with a set of irrational adjoins. For example- 3, 0, 1.5, √2, etc.


  • Rational Numbers

Any number that can be written as a ratio of one number over another number is written as a rational number. This means that any number that can be written in the form of p/q is a rational number. The symbol “Q” represents a rational number. They can also be defined as fractions with integers. For example ⅔, ⅚, etc.


  • Irrational Numbers

The number that cannot be expressed as the ratio of one over another or that cannot be written as fractions is known as irrational number, and it is represented by the symbol ”P”.  For example √3,√5,√7, etc.


  • Complex Numbers

The number that can be written in the form of a+bi, where the variables “a and b” are the real numbers and variable “i” is an imaginary number, is known as complex number and is denoted by the letter “C”. For example- 2+3i, 5+2i, etc.


  • Imaginary Numbers

The imaginary numbers are known to be complex numbers that can be written in the form of the product of a real number and such numbers are denoted by the letter “i”.


What is the Indian Numeral System?

Let us consider a number, say 225. Notice that the digit 2 is used twice in this given number, however, both of them have different values. We differentiate them by stating their place value in mathematics, which is defined as the numerical value of a digit on the basis of its position in any given number. So, the place value of the leftmost 2 is Hundreds while for the 2 in the centre is Tens.


Coming back to the Indian numeral system, the place values of the various digits go in the sequence of:


  1. Ones

  2. Tens

  3. Hundreds

  4. Thousands

  5. Ten Thousand

  6. Lakhs

  7. Ten Lakhs

  8. Crores, and so on.


In the given number 10,23,45,678 the place values of each of the digits present in the number are given below:


We will begin from the right side of the number.


  • 8 – Ones

  • 7 – Tens

  • 6 – Hundreds

  • 5 – Thousands

  • 4 – Ten Thousand

  • 3 – Lakhs

  • 2 – Ten Lakhs

  • 0 – Crores

  • 1 – Ten Crores


The relationship between them is given below:


  • 1 hundred = 10 tens

  • 1 thousand = 10 hundreds = 100 tens

  • 1 lakh = 100 thousands = 1000 hundreds

  • 1 crore = 100 lakhs = 10,000 thousands


International Numeral System

The place values of digits is in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousands, Millions, Ten Million and so on, in the international numeral system. In the given number 12,345,678 the place values of each digit is:


  • 8 – Ones

  • 7 – Tens

  • 6 – Hundreds

  • 5 – Thousands

  • 4 – Ten Thousand

  • 3 – Hundred Thousand

  • 2 – Millions

  • 1 – Ten Million


The relations between them are:


  • 1 hundred = 10 tens

  • 1 thousand = 10 hundreds = 100 tens

  • 1 million = 1000 thousand

  • 1 billion = 1000 millions

FAQs on Numbers, Numerals and Digits

1. Is 0 a real number?

Real numbers are any numbers that you can think of. These are not imaginary numbers. Real numbers can be negative or positive and yes, they do include the number zero. Real numbers can be represented over a number line and their centre point or the origin is zero. Numbers to the right on a number line are all positive numbers and those to the left are all negative.

2. How many types of number systems are there?

The number system is a representation of numbers using symbols or digits. The number system includes four main types-


  • The binary number system only uses two digits 0 and 1 and numbers have a base of 2.

  • The octal number system has a base of 8 and uses 8 digits 0, 1, 2, 3, 4, 5, 6 and 7. 

  • The decimal number system has a base of 10 and uses 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 

  • In hexadecimal numbers, the base is 16 and it uses 16 digits or alphabets.

3. How many types of numbers are there in the English language?

There are generally three common types of numbers in the English language, which include: 


  • Ordinal numbers- They represent the order, for example, first, second, third, fourth, fifth, sixth.…..and so on. 

  • Cardinal numbers- They are those which show the quantity, for example, one, two, three, four, five….and so on,

  • Nominal numbers- They show the identity for example bank account numbers or security numbers etc.

4. What is a positional number system?

The positional number system is the one in which numbers are represented in an ordered form. The position of a numeral is its place value. There are generally four types of positional number systems - binary (base 2), octal (base 8), decimal (base 10) and hexadecimal (base 16).

Comment