Real Numbers

   

Real Numbers

Numbers

There are two types of Numbers.                                                        

1. Real Numbers

2. Complex Numbers (Real numbers + Imaginary Numbers)

Real Numbers: The numbers which can be represented on the number line by a unique point are called real numbers. 

Examples: -1, -2, -3, -4 ....................... , and 1, 2, 3, 4, 5.........................  

Complex Numbers: Complex numbers are the numbers which consist of both real and Imaginary numbers.

Examples: 2 + i√3, where 2 is real part and i√3 is imaginary part. 

Real Numbers

(A)  Rational Numbers: The numbers which can be express in the form of p/q, and q≠0 such a numbers are known as real numbers. 

(B)  Irrational Numbers: The numbers which are not rational numbers but these an be represented on a number line, such numbers are known as irrational numbers. 

Rational Numbers

(I)  Integers:

      (a) Whole numbers: These are the set of positive integers from 0. Which  have not decimal and fractional part. Such numbers are known as whole numbers.

           Example: 0, 1, 2, 3, 4, 5, ........................... 

           (i)  Natural Numbers: Natural numbers counting begin from 1 to ∞                             (infinity). such numbers are called natural numbers. 

                Ex: 1, 2, 3, 4, 5 ....................................... ∞. 

          (ii) Zero: 0 

      (b) Negative integers: The numbers before 0 on number line is called as                 negative integers these are -1, -2, -3, -4 ............................. -∞. 

(II) Fractional Numbers: 

      (a) Proper fraction

      (b) Improper fraction

Natural numbers

(a) Odd Numbers: The numbers which can not be divided by 2 are called odd           numbers. 

      Example: 1, 3, 5, 7, 9, 11, 13, 15 ................................................. ∞. 

(b) Even Numbers: The numbers which can be divided by 2 are called even numbers. 

      Example: 2, 4, 6, 8, 10, 12, 14, 16 ................................................ ∞. 

(c) Prime Numbers:

(d) Composite Numbers:

(e) Co-prime Numbers:

(f)  Perfect Numbers:

Properties of real numbers

     Commutative property- If we have real numbers m,n. The general form will be m + n = n + m  for adaddition and m.n = n.m for multiplication.

     Associative property- If we have real numbers m, n, r. The general form will be  m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication

     Distributive property- If we have real numbers m,n,r. The general form will be – m (n + r) = mn + mr and (m + n) r = mr + nr

     Identity property- For addition: m + (- m) = 0

Euclid’s Division Lemma

For a and b any two positive integer, we can always find unique integer q and r such that
a=bq + r , 0 ≤ r < b

It is basic concept and it is restatement of division

a is called dividend
b is called divisor
q is called quotient
r is called remainder.

If r =0, then b is divisor of a.

Proof of Euclid's Division Lemma

Here we need to argue that q and r are no unique
Let us assume q and r are not unique i.e. let there exists another pair q₀ and r₀ i.e. a = bq₀ + r₀, where 0 ≤ r₀ < b

=> bq + r = bq₀ + r₀
=> b(q - q₀) = r - r₀ ................ (I)

Since 0 ≤ r < b and 0 ≤ r₀ < b, thus 0 ≤ r - r₀ < b ......... (II)

The above eq (I) tells that b divides (r - r₀) and (r - r₀) is an integer less than b. This means (r - r₀) must be 0.
=> r - r₀ = 0
=> r = r₀

Eq (I) will be, b(q - q₀) = 0
Since b > 0, => (q - q₀) = 0
=> q = q₀

Since r = r0 and q = q₀, Therefore q and r are unique.

HCF (Highest common factor)

Before starting on this topic, we need to prove a important theorem m which will be used in finding HCF between two numbers

Theorem

If a and b are positive integers such that a = bq + r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.

Proof : Let m be a common divisor of a and b. Then,
m| a => a = mx for some integer x
m| b => b = mq2 for some integer y
Now, a = bq + r
=> r = a - bq
=> r = mx - my q
=> r = m( x - yq)
=> m| r
=> m| r and m | b
=> m is a common divisor of b and r.
Hence, a common divisor of a and b is a common divisor of b and r

How to find HCF (Highest common factor)

Now HCF of two positive integers can be find using the Euclid's Division Lemma algorithm and above stated theorem

We know that for any two integers a,b. we can write following expression
a=bq + r , 0 ≤ r < b
If r=0 ,then
HCF( a,b) =b
If r >0 , then
HCF ( a,b) = HCF ( b,r) as already proved from above theorem

Again expressing the integer b,r in Euclid's Division Lemma, we get
b=pr + r1
HCF ( b,r)=HCF ( r,r1)
Similarly successive Euclid 's division can be written until we get the remainder zero, the divisor at that point is called the HCF of the a and b.

If You Still have any doubt, feel free to comment!