Real Numbers in Grade VIII

In earlier sections we have discussed and practiced on rational numbers worksheet and Today we will discuss about the real numbers and  properties of real number for grade VIII of Maharashtra Board Syllabus.
So before starting, we will first try to find that what the real numbers actually are.
The real numbers can be defined as the set of numbers that consists of all the rational numbers(maximum used in rational expressions) together with all the irrational numbers. In general language we can say that all integers, small or large, whole number, decimal numbers are all real numbers
Except the imaginary numbers (the numbers which have negative terms under the roots are called imaginary numbers) all numbers are known as real numbers
For Example:

Given set A =  0, 2.9, -5, 4, -7, p , is a set which  consists of  natural numbers,  whole numbers,  integers,  rational numbers,  irrational numbers but all element of  set A represent the real number.

Now we will learn about some properties of real numbers which apply on all real number. These properties will be very helpful to solve algebraic problems. We will discuss each property in detail and will try to explain with some examples. Let's talk about  Commutative properties of real numbers
For addition:            a+b = b+a
For example: 1:     3+5 = 5+3 =8
     
                     2:      4 + 5 = 5 + 4 =9
   
For multiplication:       a*b =b*a
For example: 1:              3*7 =7*3 = 21
                    2:               5 × 3 = 3 × 5 =15
But this property does not satisfy for subtraction and division
                b-a ≠ a-b and   a/b ≠ b/a
  for example:    4 – 5 ≠ 5 – 4    and          4 ÷ 5 ≠ 5 ÷ 4
    Now let's move on other property that is Associative property of real numbers.
For addition : a+(b+c) =   (a+b)+c
For example:   (4x + 2x) + 7x = 4x + (2x + 7x)
                       (4 + 5) + 6 = 5 + (4 + 6)

For multiplication: a*(b*c) = (a*b)*c
For example:1:   (3x*4x)*3y = 3x * (4x*3y)
                   2:    (4 × 5) × 6 = 5 × (4 × 6)
The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.(want to Learn more about Real Numbers ,click here),
Associative property for real numbers does not satisfy in subtraction and division
                          (a-b)-c  ≠  a-(b-c)
For example:    (4-5)-6  ≠  4-(5-6)
                (a/b)/c ≠  a/(b/c)
For example (4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6)
Next is Distributive property of real numbers. In this we see that
a*(b+c) = a*b + a*c
(a+b)*(c+d) = a*c + a*d + b*c + b*d

for example 1:  4(x + 5) = 4x + 20
                   2: 3(4 – x) =12 -3x
                   3:  (a – 3) (b + 4) = ab + 4a – 3b – 12
Now comes to Identity property of real numbers
For addition:              a+0 = a
For example:           5y + 0 = 5y

The identity property for addition tells us that zero added to any number is the number itself.
Zero is called the "additive identity."

For multiplication:     a*1 =a
For example:  2c × 1 = 2c
The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself as a result. The number 1 is called the "multiplicative identity."

So this is all about real numbers. If you want to more details on the topics like range and Probability in Grade VIII then you can visit various websites on the internet.