Natural Number with Funs and Facts

This post contains the fun and facts of Real Numbers, Rational Numbers, Irrational Numbers and Complex Numbers and important Question-Answers for all competitive viewers and students who try to know the mathematics in easy way. Natural Number with Funs and Facts

What is a Real Number?

Real Number is a collection of all the rational and irrational number. In the number system, it performs all the arithmetic operation such as Addition, Subtraction, Multiplication and Division. It is represented by capital letter ‘R’. Complex number is not a real number. It is in the form of (x + iy). Where x and y are real number. As shown in figure below:

What is a Rational Number?

Rational Number is a real number, which is in the form of p/q where p and q are integers. q is not equal to zero. It can be represented by ‘Q’. In other word, you can say that all the real number is a rational number which is in the form of p/q where q is not equal to zero, except irrational number.

How to identify the rational number?

Some conditions are there:

Rational number is in the form of p/q. where p and q are integers. q≠0.
It can be in fraction.
It can be either negative or positive or zero.
It can be in decimal form.
It can be an integer.
It can be a whole number.
It can be a natural number.
It can be represented on a number line.
It has a multiplicative inverse.

Properties of rational number

If we add, subtract, multiply and divide of two rational number. It gives as a result of rational number.
It can perform all the arithmetical operation such Addition, Subtraction, Multiplication and Division.
If we add zero to a rational number, it gives the same number itself.
√2=1.41421356… it is a non-terminating number. Hence, it can’t be written in the form of p/q. So, it is an irrational number.
Π is an irrational number, but, 22/7 is a rational number.
e=2.171828… is also an irrational number.

How to find the rational number between two rational numbers?

Method 1:

Take an example 2/3 and 4/5
To make it in decimal form, 0.6 and 0.8
So, 0.7 is a rational number in between 0.6 and 0.8

Method 2:

Take an example 3 and 4
To add both the numbers, 3+4=7
To divide the sum 7 by 2, to get 3.5
3.5 is a rational number between 3 and 4

Similarly, for fraction

Take an example 2/3 and 4/5
To add both the numbers 2/3+4/5=22/15
To divide the sum 22/15 by 2, to get 11/15
11/15 is a rational number between 2/3 and 4/5

What is an irrational number?

Irrational numbers are a real number but, it can’t be expressed in a fraction or ratio. Its arithmetical operation are very complicated. Irrational number is neither terminating nor recurring. It is always in the form of under root. √2 is an irrational number. i.e. √5, √7, √11,…. In other word, we can say that the irrational number can’t be expressed as the ratio of integers. It is generally represented as ‘P’.

Properties of irrational number

The sum of two irrational number gives an irrational number
The sum of a rational number and an irrational number give an irrational number.
The same two irrational number multiply with each other gives a rational number. √5x√5=5
The multiplication of a Non zero rational number and an irrational number give an irrational number.
The least common multiple of any two irrational number may or may not exist.
The difference of any two irrational number gives an irrational number.
The difference of an irrational number and a rational number give an irrational number.
√2=1.41421356… it is a non-terminating number. Hence, it can’t be written in the form of p/q. So, it is an irrational number.
Π is an irrational number, but, 22/7 is a rational number.
e=2.171828… is also an irrational number.

What is complex number?

Complex numbers are expressed in the form of a+ib. where a and b are the real number and i is an imaginary number called ‘iota’. The value of ‘i’=√-1. The square of an imaginary number is -1. i.e. (i)² = (-1). The main application of complex number in periodic motion such as waves, alternating current, sine, cosine etc.

There are following examples as below:

0+i; ‘0’ is a real part and ‘i’ is an imaginary part.
2+3i; 2 is a real part and 3i is an imaginary part.
3-4i; 3 is a real part and -4i is an imaginary part.
√3+2i; √3 is a real part and 2i is an imaginary part.
2+√3i; 2 is a real part and √3i is an imaginary part.

Notation and Classification

It is denoted by ‘z’. where ‘a’ is real number and ‘ib’ is an imaginary number.

What is real number?

Real number such as positive, negative, zero, integer, irrational, rational, fraction etc. For example: 1, -1, 0, ¾, √3, are real number. it is expressed as real().

What is imaginary number?

The number, which is not a real number is an imaginary number. When we square an imaginary number gives a negative value. i.e. √-3, √-5…..It is expressed as img().

For example: √-7 =√(-1)x√7 = i√7;

Conjugate of complex number

Find the conjugate value of complex number

Conjugate of (3 + 4i) = (3 – 4i)
Conjugate of (4 – 5i) = (4 + 5i)
Conjugate of (-7i) = (+7i)
Conjugate of a complex number always opposite sign of imaginary part.

Cosine formulae of complex number

Both are conjugate of each other.

Graphical Representation

Power of (i): Natural Number with Funs and Facts

Changing the value of ‘i’ when power increased.

(i)⁰ = (i)⁴= 1
(i) = √-1
(i)² = √-1x√-1 = -1
(i)³ = √-1x√-1x√-1 = -i
(i)⁴ = √-1x√-1x√-1x√-1 = -1x-1 = 1
(i)⁴⁺¹ = i
(i)⁴⁺² = -1
(i)⁴⁺³ = -i
(i)⁴⁺⁴ = (i)^4×2 = 1

Similarly, k =1, 2, 3, 4……

(i)^4k+0 = 1
(i)^4k+1 = i
(i)^4k+2 = -1
(i)^4k+3 = -i

Repeat the process after i, -1, -i , 1

Find the value of ‘i’. Natural Number with Funs and Facts

(i)⁵⁶ = (i)^4X14+0 = 1
(i)¹³⁵ = (i)^4×33+3 = (i)³ = -i
(i)²⁶⁹ = (i)^4×67+1 = (i) = i

Absolute value of complex number

Let z = (a + ib) be a complex number and its conjugate (a – ib).
Modulus of z = |z|
|z| = √(a² + b²)

Absolute value of real number

|5| = 5
|-5| = 5
Absolute value of any real number gives always positive.

We can apply Pythagoras theorem,

|z|² = |a|² + |b|² = a² + b²

Find the modulus of z = 3 + 4i

|z| = √{(3)² + (4)²} =√25 = 5

Properties of complex number

If sum of two complex number and its product are real, then these two complex numbers are always conjugate of each other.
The sum of two conjugate complex number is always real.
The product of two conjugate complex numbers is always real.
If imaginary part of complex number is zero then, the complex number is purely real number.
If real part of complex number is zero then, the complex number is purely imaginary.
If a+ib = 0 then, a=0 and b=0
The complex number obey the commutative law of addition and multiplication. z₁+ z₂ = z₂ + z₁ and z₁.z₂ = z₂.z₁
Similarly, complex number obey the associative law of addition and multiplication. (z₁+z₂)+z₃ = z₁+ (z₂+z₃) and . (z₁.z₂).z₃ = z₁.(z₂.z₃)
It obeys the distributive law also, . (z₁+z₂).z₃ = z₁.z₃+ z₃.z₂

Q: Choose the real number?

-3, √5, 7, 2/3, 3+i5, 1.6, 0, i7, 9

Q: Choose the rational number?

5, √4, 0.5, -2/3, 4/5, i3, √2, √7, (3+i7)

Q: Choose the irrational number?

5, √19, 0.5, -2/3, 4/5, i3, √2, √7, (3+i7)

Q: Find the value of ‘i’.

(i)¹⁵⁸, (i)¹⁰⁰⁵, (i)²⁰¹⁰, (i)⁴⁷⁸⁹, (i)⁴⁵⁶⁷

Q: Find the absolute value of the following.

(3 + 5i), (6i), (6 + 0i), (3 + 7i), (√2 + i)

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