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Operations with Real Numbers

In mathematics, the real numbers are everywhere. They include all kinds of numbers except for imaginary numbers. The integers, whole numbers, natural numbers, real numbers, rational and irrational numbers, all come under real numbers. The real numbers are represented on real number line which has zero in the middle and positive real numbers located on the right side of zero, while negative real numbers are denoted on the left side of zero. In this lesson, we will learn about the operations on real numbers.
 
On the successful completion of this article, the students will be able to answer the following questions :

1) What are basic operations on real numbers?
2) How to perform operations on real numbers?
3) What are grouping symbols and exponentiation?
4) What are properties of operations on real numbers?

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Fundamental Operations With Real Numbers

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Among the most fundamental operations with real numbers, the following four are included.

1) Addition
2) Subtraction
3) Multiplication
4) Division

Addition and Subtraction of Real numbers

How do add or subtract real numbers? In order to do so, there arise two cases.

Case I : To add or subtract two numbers having the same sign, i.e. either both positive or both negative.

Step 1 :
 Consider absolute values of both numbers and add them.

Step 2 :
 Give the result a sign same as that of numbers. If both numbers are positive, the answer will have positive sign and if both are negative, the sign of answer will also be negative.
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Case II : To add or subtract two numbers having different signs, i,e, one positive, other negative.

Step 1 :
 Consider absolute values of both numbers and subtract smaller from the bigger.

Step 2 :
 Give the result a sign of the bigger number. If the bigger number is positive, the answer will have a positive sign and if it is negative, the answer will be negative.

Multiplication and Division of Real numbers

In order to multiply or divide two real numbers, following steps should be followed.

Step 1 :
 Multiply or divide absolute value of both numbers.

Step 2 :
 If both numbers are the same sign, place a positive sign with the answer. On the other hand, if both numbers are different signs, place a negative sign with the answer.

Grouping Symbols and Exponents

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Grouping Symbols

What are grouping symbols? While working with operations on real numbers, we come across with two types of grouping symbols :
Parenthesis

The parenthesis is denoted by ( ). They are used to group a number of variables or numbers. When a parenthesis is following by a negative sign, each term inside it will flip its sign during opening the parenthesis.

For example :

5 - (4 + 3 - 6) = 5 - 4 - 3 + 6 = 4
Bracket or Braces
Brackets or braces are denoted by { } and [ ]. The parenthesis are also a kind of brackets, they may be called as small brackets. But before working with brackets { } and [ ], the parenthesis ( ) should be solved.
For example :
-[7 + 2{5 - (-6)}] = -[7 + 2{5 +6}] = -[7 + 2{11}] = -[7 + 22] = -29


Exponents

Exponents are the powers of variables or numbers. They are calculated just after opening the parenthesis.

Order of Operations

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The operations of addition, multiplication, division, and subtraction are usually found together in arithmetic operations. Which operation to perform first? Confused? No worries, Use the following order of operations.

1) First open parentheses.
2) Then, solve e
xponents.
3) Then, multiply or divide (if any) from left to right.
4) Lastly, add or subtract (if any) from left to right.

You may tend to forget this rule while working with algebraic operations. You may remember this rule as "PEMDAS", where P stands for parenthesis, E for exponents, MD means multiply or divide and AS stand for add or subtract.

Properties of Real Numbers

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The main properties of real numbers are listed below.

Let us suppose that p, q and r are real numbers.

1) Associative Law

A:
 For Addition
(p + q) + r = p + (q + r)

B: For Multiplication
(p x q) x r = p x (q x r)

2) Commutative Law

A: For Addition

p + q = q + p

B : For M
ultiplication
p x q = q x p

3) Distributive Law

A: Distribution of multiplication over addition

p x (q + r) = (p x q) + (p x r)
and
(p + q) x r = (p x r) + (q x r)

B: Distribution of multiplication over subtraction
p x (q - r) = (p x q) - (p x r)
and
(p - q) x r = (p x r) - (q x r)

4) Closure Law

A: For Addition

The addition of two real numbers is also a real number.

B: For Multiplication
Multiplication of two real numbers is also a real number.

Examples

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The examples related to operations on real numbers are illustrated below.

Example 1 : Add, subtract, multiply and divide -6 and -8. Also, explain your answers.

Solution :

 Addition 
(-6) + (-8)

Step 1 : Open brackets

-6 - 8

Step 2 : Add (same sign)
-14
The answer is -14 because both the numbers have a negative sign.

Subtraction
(-6) - (-8)

Step 1 : Open brackets
-6 + 8

Step 2 : Subtract (different signs)
2

The answer is 2 because bigger number 8 has a positive sign.

Multiply
(-6) x (-8)

Step 1 : Open brackets
(-)(-) 6 x 8

Step 2 :
Multiply
48

The answer is +48 since both numbers are of same sign.

Divide

(-6) $\div$ (-8) = 0.75


The answer is +0.75 since both numbers are of same sign.
Example 2 : Solve 10 - 2 x 6 + 10$^{2}$ - (4 - 1) x 3

Solution : 10 - 2 x 6 + 10$^{2}$ - (4 - 1) x 3

Step 1 :
 Parenthesis

10 - 2 x 6 + 10$^{2}$ - 3 x 3

Step 2 : Exponent

10 - 2 x 6 + 100 - 
3 x 3

Step 3 : Multiply / Divide

10 - 12 + 100 - 9

Step 3 : Add / Subtract

110 - 21 = 89
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