Algebra

Understand Factors and Coefficients of terms; Like and Unlike Terms, etc

We know that whole numbers are commutative under addition. According to this property, if the order of numbers is interchanged in addition, it does not affect the result. For example,

2 + 3 = 5 and 3 + 2 = 5

∴ 2 + 3 = 3 + 2

11 + 14 = 25 and 14 + 11 = 25

∴ 11 + 14 = 14 + 11

Is there any other way to write commutative property of whole numbers?

We can easily write this property in a general way by the use of variables. Therefore, let us take two variables p and q, where p and q are any two whole numbers. Therefore, we can write the commutative property of whole numbers as follows: p + q = q + p, where p and q are whole numbers.  

Similarly, the commutative property also holds true under multiplication. We can write this property in a general way by making use of variables, say p and q, where p and q are whole numbers. We can write the rule for the commutativity of multiplication of whole numbers as follows: p × q = q × p, where p and q can be any two whole numbers.

In a similar way, we can make use of variables in generalising the distributive and the associative property also. Let us see how.

Distributive property of multiplication over addition Consider three numbers 6, 25, and 5. We have, 6 × (25 + 5) = 6 × 30 = 180 Also, 6 × 25 + 6 × 5 = 150 + 30 = 180 ∴ 6× (25 + 5) = 6 × 25 + 6 × 5

This property is known as distributive property and is applicable for all whole numbers. Let us now write the generalised form of this property by using variables. Let us take three variables a, b, and c where a, b, and c are whole numbers. Using these variables, we can write the distributive property as follows: a × (b + c) = a × b + a × c, where a, b, and c are any three whole numbers.

Associative property under addition and multiplication Consider the whole numbers 9, 15, and 46. Then, according to the associative property, we obtain 9 + (15 + 46) = 9 + 61 = 70 Also, (9 +15) + 46 = 24 + 46=70 ∴ 9 + (15 + 46) = (9 + 15) + 46

By using the variables p, q, and r, the associative property can be written as follows: p + (q + r) = (p + q) + r, where p, q, and r are whole numbers.

Similarly, the associative property under multiplication can be written as follows: p × (q × r) = (p × q) × r, where p, q, and r are whole numbers.

Now, let us observe the results, when a whole number is multiplied with 0. 1 × 0 = 0; 5 × 0 = 0; 99 × 0 = 0; 625 × 0 = 0, etc.

It can be seen that the product of any number with 0, gives 0 as result. Thus, the rule can be written as follows: number × 0 = 0 or, x × 0 = 0, where x is a whole number. 

Example 1.

Identify the property used in the a(b + c) = ab + ac. Answer: Distributive property of multiplication over addition is used in the given expression.

Example 2.

Fill in the blanks in the following expression and mention the property used.

3a × (2b + c) = 6ab + __ a × (–c) = (–c) × __    (x × 2y) × 9z = x × (__)    2p + __ = 4r + __

Answer:

3a × (2b + c) = 6ab + 3ac             (Distributive property of multiplication over addition) a × (–c) = (–c) × a                         (Commutativity of multiplication) (x × 2y) × 9z = x × (2y × 9z)       (Associative property under multiplication) 2p + 4r = 4r + 2p                          (Commutativity of addition)

We know that whole numbers are commutative under addition. According to this property, if the order of numbers is interchanged in addition, it does not affect the result. For example,

2 + 3 = 5 and 3 + 2 = 5

∴ 2 + 3 = 3 + 2

11 + 14 = 25 and 14 + 11 = 25

∴ 11 + 14 = 14 + 11

Is there any other way to write commutative property of whole numbers?

We can easily write this property in a general way by the use of variables. Therefore, let us take two variables p and q, where p and q are any two whole numbers. Therefore, we can write the commutative property of whole numbers as follows: p + q = q + p, where p and q are whole numbers.  

Similarly, the commutative property also holds true under multiplication. We can write this property in a general way by making use of variables, say p and q, where p and q are whole numbers. We can write the rule for the commutativity of multiplication of whole numbers as follows: p × q = q × p, where p and q can be any two whole numbers.

In a similar way, we can make use of variables in generalising the distributive and the associative property also. Let us see how.

Distributive property of multiplication over addition Consider three numbers 6, 25, and 5. We have, 6 × (25 + 5) = 6 × 30 = 180 Also, 6 × 25 + 6 × 5 = 150 + 30 = 180 ∴ 6× (25 + 5) = 6 × 25 + 6 × 5

This property is known as distributive property and is applicable for all whole numbers. Let us now write the generalised form of this property by using variables. Let us take three variables a, b, and c where a, b, and c are whole numbers. Using these variables, we can write the distributive property as follows: a × (b + c) = a × b + a × c, where a, b, and c are any three whole numbers.

Associative property under addition and multiplication Consider the whole numbers 9, 15, and 46. Then, according to the associative property, we obtain 9 + (15 + 46) = 9 + 61 = 70 Also, (9 +15) + 46 = 24 + 46=70 ∴ 9 + (15 + 46) = (9 + 15) + 46

By using the variables p, q, and r, the associative property can be written as follows: p + (q + r) = (p + q) + r, where p, q, and r are whole numbers.

Similarly, the associative property under multiplication can be written as follows: p × (q × r) = (p × q) × r, where p, q, and r are whole numbers.

Now, let us observe the results, when a whole number is multiplied with 0. 1 × 0 = 0; 5 × 0 = 0; 99 × 0 = 0; 625 × 0 = 0, etc.

It can be seen that the product of any number with 0, gives 0 as result. Thus, the rule can be written as follows: number × 0 = 0 or, x × 0 = 0, where x is a whole number. 

Example 1.

Identify the property used in the a(b + c) = ab + ac. Answer: Distributive property of multiplication over addition is used in the given expression.

Example 2.

Fill in the blanks in the following expression and mention the property used.

3a × (2b + c) = 6ab + __ a × (–c) = (–c) × __    (x × 2y) × 9z = x × (__)    2p + __ = 4r + __

Answer:

3a × (2b + c) = 6ab + 3ac             (Distributive property of multiplication over addition) a × (–c) = (–c) × a                         (Commutativity of multiplication) (x × 2y) × 9z = x × (2y × 9z)       (Associative property under multiplication) 2p + 4r = 4r + 2p                          (Commutativity of addition)

We will now learn two important properties of whole numbers. Let us start by taking two numbers and adding them.

What do we obtain on adding 3 and 4?

We obtain, 3 + 4 = 7

Now, what would be the result, if we interchange the places of th…