Knowing Our Numbers

In this chapter, understanding numbers and developing numeracy skills is vital for academic success, problem-solving, critical thinking and a wide range of future opportunities. Numeracy is a fundamental skill that enriches our lives in countless ways.

Number

Playing with Numbers

Natural and Whole Numbers

Operations on Whole Numbers

More About Numbers

Division Algorithm

Prime Factorization

HCF (Highest Common Factor)

Lowest or Least Common Multiple (LCM)

Solved Questions on Knowing Our Numbers

Number

A number is a numerical value used to denote and quantify a specific quantity.

In a nine-digit numerical system, the smallest number is 100,000,000 (Hundred million) and the largest is 999,999,999 (Nine hundred ninety-nine million nine hundred ninety-nine thousand nine hundred and ninety-nine).

Playing with Numbers

Playing with numbers refers to engaging in mathematical activities, games or exercises that involve numbers and mathematical concepts. It has a wide range of activities, from solving puzzles to practicing arithmetic operations, exploring number patterns and enjoying mathematical games.

Play with numbers in the Snakes and Ladders game.

Successor and Predecessor of Numbers

→ The successor of a number is the one that immediately follows it. It means 1 more than the given number.

Successor of a given number = Given Number + 1

→ The predecessor of a number is the one that comes just before it. It means 1 less than the given number.

Predecessor of a given number = Given Number − 1

Natural and Whole Numbers

Natural numbers are a set of counting numbers that begin with 1 and continue infinitely. They are a fundamental set of positive integers used for counting. They are typically represented as {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,..............}.

Whole numbers are a set of numbers that includes zero along with all the natural numbers. They are typically represented as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,..............}.

Operations on Whole Numbers

Operations on whole numbers refer to the mathematical calculations that can be performed using the set of whole numbers which includes positive integers and zero. The primary operations on whole numbers are:

Properties of Addition:

Properties of Addition for Whole Numbers are as follows:

a. Closure Property: When you add two whole numbers, the result is always a whole number.

If A and B are two whole numbers, then

A + B = C

Here, C is also a whole number.

→ Example: 15 + 12 = 27

15, 12 and 27 are whole numbers.

Here, C is also a whole number.

b. Commutative Property: The order of numbers in an operation addition does not affect the result. It is also known as commutative law.

Adding A to B is the same as adding B to A.

A + B = B + A

→ Example: 15 + 12 = 12 + 15

Both left-hand and right-hand expressions have the same result as 27.

c. Additive Property of Zero: Adding zero to any whole number doesn't change the value.

If A is a whole number, then

A + 0 = 0 + A

→ Example: 15 + 0 = 0 + 15

Both left-hand and right-hand expressions have the same result as 15.

d. Associative Property: When adding three or more whole numbers, the grouping of numbers does not affect the result. It is also known as associative law.

If A, B and C are three whole numbers, then

(A + B) + C = A + (B + C)

→ Example: (15 + 12) + 11 = 15 + (12 + 11) which simplifies to 

(15 + 12) + 11 = (27) + 11 = 38

15 + (12 + 11) = 15 + (23) = 38

Both left-hand and right-hand expressions have the same result as 38.

Properties of Subtraction:

The properties of Subtraction for Whole Numbers are as follows:

a. Closure Property: Subtraction of two whole numbers results in a whole number only if the first number is greater than or equal to the second number. Otherwise, subtraction is not a whole number.

If A and B are two whole numbers and A > B or A = B, then 

A − B = C

Here, C is also a whole number.

If A and B are two whole numbers and A < B, then

A − B = −C

Here, −C is not a whole number.

→ Example: 15 − 12 = 3

Here, the subtraction of 15 and 12 is a whole number (3) because 15 is greater than 12  (15 > 12).

→ Example: 15 − 15 = 0

Here, the subtraction of 15 and 15 is a whole number (0) because 15 is equal to 15.

→ Example: 12 − 15 = −3 (Not a Whole Number)

Here, subtraction of 12 and 15 is not a whole number because 12 is less than 15. When you subtract a larger number from a smaller number such as subtracting 15 from 12, the result is not a whole number within the set of whole numbers.

b. Commutative Property: The order of numbers in an operation subtraction affects the result. It is also known as commutative law.

Subtracting B from A is not the same as subtracting A from B.

A − B ≠ B − A

These are not equal (≠).

→ Example: (15 − 12) ≠ (12 − 15)

(15 − 12): Subtracting 12 from 15 is 3 which is a whole number. 

(12 − 15): Subtracting 15 from 12 is not a whole number. 

c. Subtracting Zero: Subtracting zero from any whole number results in the same whole number but subtracting a whole number from zero is not defined in whole numbers.

If A is a whole number, then  A − 0 ≠ 0 − A.

Therefore, A − 0 = A
0 − A ≠ A      [0 − A = −A]

→ Example: (15 − 0) ≠ (0 − 15)

15 − 0: Subtracting 0 from 15 is 15 which is a whole number. 

0 − 15: Subtracting 15 from 0 is −15 which is not a whole number.

d. Associative Property: When subtracting three or more whole numbers, the grouping of numbers does not have the same result. It is also known as associative law.

If A, B and C are three whole numbers, then 

(A − B) − C ≠ A − (B − C)

→ Example: (15 − 12) − 11 ≠ 15 − (12 − 11)

(15 − 12) − 11: The result is not a whole number which is −8. 

15 − (12 − 11): The result is a whole number which is 14.  

Properties of Multiplication:

The properties of Multiplication for Whole Numbers are as follows:

a. Closure Property: When you multiply two whole numbers, the result is also a whole number. 

If A and B are two whole numbers, then 

A × B = C

Here, C is also a whole number.   

Example: 15 × 12 = 180

15, 12 and 180 are whole numbers.

b. Commutative Property: The order of numbers in an operation multiplication does not affect the result. It is also known as commutative law.

Multiplying A to B is the same as multiplying B to A.

A × B = B × A

Example: 15 × 12 = 12 × 15

Both left-hand and right-hand expressions have the same result as 180.

c. Multiplicative Property of Zero: Any whole number multiplied by zero equals zero. 

If A is a whole number, then 

A × 0 = 0 × A

Example: 15 × 0 = 0 × 15 

Both left-hand and right-hand expressions have the same result as 0.

d. Multiplicative Property of One: When you multiply any whole number by 1, the result is the same whole number. 

If A is a whole number, then

A × 1 = 1 × A 

Example: 15 × 1 = 1 × 15 

Both left-hand and right-hand expressions have the same result as 15.

e. Associative Property: When you multiply three whole numbers together, the grouping of the numbers doesn't affect the result. It is also known as associative law.

If A, B and C are three whole numbers, then

(A × B) × C = A × (B × C)

→ Example: (15 × 12) × 11 = 15 × (12 × 11)

which simplifies to 

LHS = (15 × 12) × 11 = (180) × 11 = 1980

RHS = 15 × (12 × 11) = 15 × (132) = 1980 

Both left-hand and right-hand expressions have the same result as 1980.

f. Distributive Property of Multiplication over Addition: You can distribute the multiplication over addition. It is also known as distributive law.

If A, B and C are three whole numbers, then 

A × (B + C) = (A × B) + (A × C)

→ Example: 15 × (12 + 11) = (15 × 12) + (15 × 11)

Which simplifies to 

LHS = 15 × (12 + 11) = 15 × (23) = 345

RHS = (15 × 12) + (15 × 11) = (180) + (165)  = 345

Both left-hand and right-hand expressions have the same result as 345.

Properties of Division

The properties of Division for Whole Numbers are as follows:

a. Closure Property: When you divide two non-zero whole numbers, the result is not necessarily a whole number. 

If A and B are two whole numbers, then 

A ÷ B = C

Here, C is not a whole number. C can be a fraction, a decimal or an integer.

→ Example: 15 ÷ 12 = 1.25 which is not a whole number.

b. Division by Zero: Dividing any whole number by 0, the operation is undefined or meaningless.

If A is a whole number, then 

A ÷ 0 = Undefined

→ Example: 15 ÷ 0 is undefined or meaningless.

c. Dividing Zero by any Whole Number:  Dividing 0 by any non-zero whole number, the result is always 0.

If A is a  non-zero whole number, then

0 ÷ A = 0

→ Example: 0 ÷ 15 is undefined or meaningless.

More About Numbers

Even and Odd Numbers:

→ Even numbers are numbers where the digit at the one's place is 0, 2, 4, 6, or 8. These numbers are exactly divided by 2 without leaving a remainder. Examples of even numbers are 2, 4, 8, 16, 18, 22, 134, 1956, etc.

→ Odd numbers are numbers where the digit at the one's place is 1, 3, 5, 7 or 9. These numbers are divided by 2 and leave some remainder. Examples of odd numbers are 1, 3, 5, 17, 19, 23, 135, 1957, etc.

Prime Numbers:

→ Prime numbers are those numbers that have only two factors: 1 and themselves. Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, etc. 

Co-Primes:

→ Co-primes are pairs of numbers that have no common factors other than 1. Examples of co-primes are (3, 5), (4, 5), (7, 9), etc.

Twin Primes:

→ Twin primes are prime numbers that differ from each other by 2. Examples of twin primes are (3, 5), (5, 7), (11, 13), etc.

Prime Triplets:

→ Prime triplets are sets of three consecutive prime numbers that differ by 2. Examples of prime triplets are (3, 5, 7) and (19, 21, 23), etc.

Composite Numbers:

→ Composite numbers are numbers that have more than two factors. Examples of composite numbers are 4, 6, 8, 9, 10, 12, 14, 16, 18, etc.

Perfect Numbers:

→ A perfect number is a number whose sum of all its factors, including 1 and itself, equals twice that number. An example of a perfect number is 6 because its factors are 1, 2, 3 and 6. The sum of these factors (1 + 2 + 3 + 6) equals 12 which is 2 times 6. Other perfect numbers are 28, 496, etc.

Division Algorithm

In a division operation, there are four key quantities involved - the Dividend, Divisor, Quotient and Remainder. The relationship among them is expressed as:

Dividend = Divisor × Quotient + Remainder

Divisibility Tests:

a. Divisibility by 1:

→ Every integer is divisible by 1. In other words, any whole number can be divided by 1 without leaving a remainder. This is because when you divide any number by 1, the result is the same number itself.
→ Example: 98763 is divisible by 1.
To check divisibility by1,

   98763 ÷ 1 = 98763 (Same number)   

b. Divisibility by 2:

→ A number is divisible by 2 if its last digit is even.
→ Example: 1578 is divisible by 2.

To check divisibility by 2, 

1578 is divisible by 2 because its last digit (8) is even.

c. Divisibility by 3:

→ A number is divisible by 3 if the sum of its digits is divisible by 3.
→ Example: 3651 is divisible by 3.

To check divisibility by 3,

The sum of its digits is 3 + 6 + 5 + 1 = 15 which is divisible by 3.

d. Divisibility by 4:

→ A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
→ Example: 79428 is divisible by 4.

To check divisibility by 4,

79428 is divisible by 4 because its last two digits are divisible by 4.

e. Divisibility by 5:

→ A number is divisible by 5 if its last digit is 5 or 0.
→ Example: 565 and 560 are divisible by 5.

To check divisibility by 5,

565 is divisible by 5 because its last digit is 5 and 560 is divisible by 5 because its last digit is 0. 

f. Divisibility by 6:

→ A number is divisible by 6 if it is even and divisible by 3.
→ Example: 8052 is divisible by 6.

To check divisibility by 6,

8052 is an even number.

Sum of its digits =  (8 + 0 + 5 + 2) = 15 which is divisible by 3.

g. Divisibility by 7:

→ To check divisibility by 7, double the last digit and subtract the result from the remaining part of the number. If the resulting number is exactly divisible by 7, then the original number is divisible by 7.

→ Example: The number 378 is divisible by 7.
To check divisibility by 7,
Last digit = 8
Double the last digit = 2 × 8 = 16
Subtract the result from the remaining part of the number: 37 − 16 = 21
Hence, 21 is divisible by 7.

h. Divisibility by 8:

→ A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
→ Example: 89736 is divisible by.

To check divisibility by 8,

89736 is divisible by 8 because the number formed by the last three digits is divisible by 8

i. Divisibility by 9:

→ A number is divisible by 9 if the sum of its digits is divisible by 9.
→ Example: 61128 is divisible by 9.

To check divisibility by 9,

The sum of its digits is (6 + 1 + 1 + 2 + 8) = 18 which is divisible by 9. 

j. Divisibility by 10:

→ A number is divisible by 10 if it ends with the digit 0.
→ Example: 46890 and 67800 are divisible by 10.

To check divisibility by 9,

46890 and 67800 are divisible by 10 because they end with the digit 0.

k. Divisibility by 11:

→ To check divisibility by 11, find the difference between the sum of digits in odd places and the sum of digits in even places starting from the unit's place. If the difference is 0 or divisible by 11, then the original number is divisible by 11.

→ Example: 37246 and 41679  are divisible by 11.

To check divisibility by 9,

For 37246, the calculation is (3 + 2 + 6) − (7 + 4) = 0 which is divisible by 11.

For 41679, the calculation is (4 + 6 + 9) − (1 + 7) = 11 which is divisible by 11.

Prime Factor

A prime factor of a given number is a factor that is itself a prime number. In other words, if a number can be divided evenly by a prime number, then that prime number is considered a prime factor of the original number.

Example: The prime factors of 48 are 2 and 3. 

Prime Factorization

Prime Factorization: Prime factorization is the way of expressing a given number as the product of its prime factors. 

Example: The prime factorisation of 48 = 2 × 2 × 2 × 2 × 3 

A factor tree is used to represent the prime factorization of a number by breaking it down into its prime factors step by step.

HCF (Highest Common Factor)

The HCF of a set of whole numbers is the largest positive number that evenly divides all the given numbers. It is also called the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD).

For example, when considering the numbers 9 and 15, the HCF is 3 because it is the largest number that can evenly divide both 9 and 15.

Methods to Find HCF

Division Method for HCF:

→ Start by dividing the larger number by the smaller number.
→ If there is a remainder left after this initial division, proceed to divide the first divisor (the smaller number) by the remainder.
→ If the remainder divides the first divisor without leaving any remainder, then this result is the H.C.F. of the two given numbers.
→ If the remainder does not divide the first divisor evenly, repeat the process by treating the remainder as the new divisor and using the original first divisor as the new dividend.
→ The last divisor is the HCF of the numbers.

Prime Factorisation Method for HCF:

→ Find the factors of the given numbers separately.
→ Identify the factors that are common to both numbers.
→ Calculate the product of these common factors, which is equal to the highest common factor (HCF) of the given numbers.

These two methods provide a systematic way to find the HCF of a set of numbers, helping us determine the largest number that divides them all evenly.

Example: Find the GCF of 105 and 42.

a) 14
b) 21
c) 28
d) 42

Answer: b) 21

Explanation: To find the HCF/GCF, the following methods are used:

Division Method:

HCF = Last divisor = 21

Prime Factorisation Method:

Lowest or Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest positive number that can be evenly divided by all the given numbers.

The LCM of the co-prime numbers is the product of these two numbers.

For example, when considering the numbers 5 and 6, the LCM is 30 because it's the smallest number that can be evenly divided by both 5 and 6. 

Methods to Find LCM:

Division Method for LCM:

→ Create a table and arrange the given numbers horizontally.
→ Continuously divide the numbers by their common factors.
→ Keep dividing until there are no common factors left between the given numbers.
→ Finally, multiply both the common factors and the remaining values to determine the L.C.M.

Prime Factorisation Method for LCM:

→ Find the factors of each of the given numbers separately.
→ Identify the common prime factors shared among the numbers as well as the unique factors.
→ Calculate the lowest common multiple (L.C.M.) of multiple numbers by multiplying together all the common prime factors once and then including the unique factors as well.

These methods offer systematic approaches to finding the LCM, the smallest number that evenly divides a set of numbers.

Example: Find the LCM of 105, 42 and 56.

a) 42
b) 56
c) 168
d) 198

Answer: c) 168

Explanation: To find the LCM, the following methods are used:

Division Method:

Prime Factorisation: