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.

Number System in Grade VIII

Hello friends, in today's session we are going to discuss about the Number System. The Number System is defined as a set of numbers arranged together in a such a manner so that we can perform different operations like addition, multiplication etc.

The number system is classified as follows:

1. Natural Numbers
2. Integers
3. Rational Numbers
4. Polynomials
5. Real Numbers
6. Complex numbers.

Let's just take them one by one and understand what are these:

Natural numbers is a set of all positive whole number greater than zero.

The set of natural numbers is denoted by N. There are various laws which the set of natural numbers follow:

Commutative axioms: a + b = b + a; a · b =b · a.

Associative axioms: a + (b + c) = (a + b) +c ; a · (b · c) = (a · b) · c.

Distributive axioms: a · (b +c ) = a · b +a · c; (b · c) n = b n · c n.

Identity axioms: a + 0 = a ; a · 1 = a ; a ⁰ = 1.

Now we come to Integers which are defined as for every natural number “a” there exists a Integer “-a” and the set of these numbers is represented by Z.

To perform arithmetic operations on integers we can look at generalizations given below to make our work simple.

−a + −b = −(a + b)

a + 0 = a

a − b = a + −b

−a · b = a · −b = −(a · b)

−a · −b = a · b

√4 can mean either 2 or −2.

The rational number system is a set of numbers used to represent the fractions. It allows the division to be done by all numbers except zero. It is written in the form a / b. where b cannot be equal to zero.

Polynomials are usually not called as numbers but their properties are very much similar to numbers.

Algebraic numbers is the system which includes all the rational numbers and is included in the set of real numbers.

The real numbers are those numbers which really exists. The set of real numbers is usually represented by R. Real numbers can also be defined as those ones which can be written in the decimal form. All the numbers we have studied till date are called as the real numbers, but now what are imaginary numbers. When the root of a  negative number is taken then the result is known as an imaginary number.

For Example – the number 24 is real because it can be written as 24.0. ½ is also a real number because it can be written as 0.5. even the number 1/3 is also real because it can be written as 0.3333.... whereas complex numbers or imaginary numbers are those numbers which are written in the form a + i b. where i is the imaginary unit i.e. a number whose square is minus one.