THE USE OF BRACKET IN MATHEMATICS

TOPIC REVIEW

In this article, I will explain the use of a bracket, and suggest a better way of thinking about the topic. Please add your ideas in the comments

BRACKET

A bracket is used to clarify expressions by grouping those term affected by a common operator

TYPES OF BRACKET

1. Parenthesis                  (  )
2. Square Bracket            [  ]
3. Braces                         {  }
4. Angles Bracket           <  >

USE OF BRACKET

In order to simplify a Mathematical expression, it is frequently necessary to 'remove brackets' this means to rewrite an expression which includes a bracketed term in an equivalent form but without the bracket. for example,   8-(3+2) which is 8-(5) = 8-5 = 3 you can write 8-(3+2) in an equivalent form we will multiply the - sign with what is in the bracket -(3+2) which is -3-2  which rewrite our question to 8-3-2 = 3 either way the result is 3.

8-(3+2) is same as 8-3-2

this operation must be carried out according to certain rules which are described as

1. Multiplication of addition and subtraction sign
2. The associativity and commutativity of multiplication
3. Expressions of the forms

MULTIPLICATION OF ADDITION AND SUBTRACTION SIGN

addition and subtraction signs affect our result will multiplying, in this case, Positive integer multiplication Example \( 2 \times 3 \) is  \( +2 \times +3 \) mathematics don't indicate +sign for Positive integer  eg 1, 2, 3.., numbers like this should be noted as +1, +2, +3....   multiplying number without it signs are positive multiplication, for Negative integer -sign is been indicated to tell you a number is a negative number eg. -2, -3, -5... in this case we have rules

RULES FOR MULTIPLYING ADDITION AND SUBTRACTION SIGNS

$$ + \times + = + \\ - \times - = + $$ we have positive result $$ + \times - = - \\ - \times + = - $$ we have negative result

Examples

1. $ 2 \times 5 $         2. $ -3 \times (-5) $         3. $ 4 \times (-3) $     and    4. $ -2 \times 3 $

Solution

1. $$ 2 \times 5 = 10 $$
2. $$ -3 \times (-5) = 15 $$
3. $$ 4 \times (-3) = -12 $$
4. $$ -2 \times 3 = -6 $$

THE ASSOCIATIVITY AND COMMUTATIVITY OF MULTIPLICATION

  Multiplication is said to be a Commutative operation. this means' for example, $ 3 \times 5 $ has the same value as $ 5 \times 3 $, Either way, the result is 15. In symbols. $ xy $ is the same as $ yx $ and so we can interchange the order as we wish.

  Multiplication is also an Associative operation, this means that when we want to multiply three numbers together such as $ 4 \times 3 \times 5 $ it doesn't matter we evaluate $ 4 \times 3 $ first and then Multiply by 5, or evaluate $ 3 \times 5 $ first and then multiply by 4 . or evaluate $ 4 \times 5 $ first and then multiply by 3.

$ ( 4 \times 3 ) \times 5 $   is same as   $ 4 \times ( 3 \times 5 ) $   and   $ ( 4 \times 5 ) \times 3 $

Where we have used brackets to indicate which terms are multiplied first. anyway we multiply them, we have same 60 . in symbols, we have x, y and z to be a random number

$ ( x \times y ) \times z $   is the same as   $ x \times ( y \times z ) $   and   $ ( z \times x ) \times y $   e.t.c

and since the result is the same either way, the brackets make no difference at all and we can write simply x × y × z or simply XYZ. When mixing numbers and symbols we usually write the numbers first. So 7 × a × 2 = 7 × 2 × a through commutativity = 14a

Example Remove the brackets from

 1) 4(2x),   2) a(5b).

Solution 

1. 4(2x) means (2x)+(2x)+(2x)+(2x) Because of associativity of multiplication the brackets are unnecessary and we can write 4 × 2 × x which equals 8x. 
2. a(5b) means a×(5b). Because of commutativity, this is the same as (5b)×a, that is (5×b)×a. Because of associativity, the brackets are unnecessary and we write simply 5×b×a which equals 5ba. Note:- that this is also equal to 5ab because of commutativity.

 Expressions of the form a(b + c) and a(b − c)

Study the expression 4×(2 + 3). By working out the bracketed term first we obtain 4×5 which equal 20.

Note:- that this is the same as multiplying both the 2 and 3 separately by 4, and then

adding the results. That is

4 × (2 + 3) = 4 × 2 + 4 × 3 = 8 + 12 = 20

Note:- the way in which the ‘4’ multiplies both the bracketed numbers, ‘2’ and ‘3’. We say that the ‘4’ distributes itself over both the added terms in the brackets - multiplication is distributive over addition.

  Now study the expression 6 × (8 − 3). By working out the bracketed term first we obtain 6 × 5 which equals 30.

Note:- that this is the same as multiplying both the 8 and the 3 by 6 before

carrying out this subtraction:

6 × (8 − 3) = 6 × 8 − 6 × 3 = 48 − 18 = 30

Note:- the way in which the ‘6’ multiplies both the bracketed numbers. We say that the ‘6’ dis-

tributes itself over both the terms in the brackets - multiplication is distributive over subtraction.

Exactly the same property holds when we deal with symbols.

a(b + c) = ab + ac    and    a(b − c) = ab − ac

Examples

4(5 + x) is equivalent to 4 × 5 + 4 × x which equals 20 + 4x.

5(a − b) is equivalent to 5 × a − 5 × b which equals 5a − 5b.

7(x − 2y) is equivalent to 7 × x − 7 × 2y which equals 7x − 14y.

−4(5 + x) is equivalent to −4 × 5 + −4 × x which equals −20 − 4x.

−5(a − b) is equivalent to −5 × a − −5 × b which equals −5a + 5b.

−(a + b) is equivalent to −a − b.