Representation of Complex Numbers in the form of a+ib
`text(Introduction)`
The concept of imaginary numbers has its historical origin in the fact that the solution of the quadratic equation `ax^2+ bx +c = 0` leads to an expression `(-bpmsqrt(b^2-4ac))/(2a)` which is not found meaningful when `b^2-4acle 0` . This is because of the fact that the square of a real number is never negative. So it created the need of the extension of the system of real numbers. Euler was the first mathematician who introduced the symbol `i` for `- 1` with the properties `i^2 = -1 ` and accordingly a root of the equation , `x^2 +1 =0` also symbol of the form `a + ib` where `a` and ` b` are real numbers is called a complex number.
`text(Definition of Complex Numbers:)`
A number of the form `a + ib,` where `a, b ` are real numbers and ` i^2 = -1 ` is called a complex number. If `z = a + ib` , then `-a-` is called the real part of ` z` and `-b` - is called the imaginary part of ` z` and are denoted by `Re(z)` and `Im(z)` respectively
`text(The form of the complex number)`
A complex number, `z`, has the form `z = x + iy`
where `x` and `y` are real numbers and `i` is the imaginary unit whose existence is postulated such that
`i^2 = −1 .`
Quantity `x` is the `text(real part of z)` and `y` is the `text(imaginary part)`
`x = Re(z) \ \ \ \ \ \ \ \ \ \ \ \ \ \y = Im(z) .`
Be careful to note that `Im(z)` is a real quantity.
A real number is thus a complex number with zero imaginary part. A complex number with zero real part is said to be pure imaginary. There is one complex number that is real and pure imaginary it is of course, zero. There is no particular need therefore to write zero as `(0 + i0).`
`text(Points to consider :)`
1. The complex numbers do not possess the property of order i.e., `(a+ ib) >`or`< (c + id)` is not defined. For example, `9 + 6i > 3 + 2i` makes no sense.
2. A real number a can be written as a + i · 0. Therefore, every real number can considered as a complex number, whose imaginary part is zero. Thus, the set real numbers (R) is a proper subset of the complex numbers (C) i.e., `R subset c`
Hence, the complex number system is `N subset W subset I subset Q subset R subset C`
3.A complex number z is said to be purely real, if Im (z) = 0; and is said to be purely imaginary, if Re (z) = 0. The complex number `0 = 0 + i · 0` is both purely real and purely imaginary.
4. In real number system, `a^2 + b^2 = 0` `=> a= 0 =b.`
But if `z_1` and `z_2` are complex numbers, then `z_1^2 + z_2^2 = 0` does not imply
`z_1 = z_2 =0.`
The concept of imaginary numbers has its historical origin in the fact that the solution of the quadratic equation `ax^2+ bx +c = 0` leads to an expression `(-bpmsqrt(b^2-4ac))/(2a)` which is not found meaningful when `b^2-4acle 0` . This is because of the fact that the square of a real number is never negative. So it created the need of the extension of the system of real numbers. Euler was the first mathematician who introduced the symbol `i` for `- 1` with the properties `i^2 = -1 ` and accordingly a root of the equation , `x^2 +1 =0` also symbol of the form `a + ib` where `a` and ` b` are real numbers is called a complex number.
`text(Definition of Complex Numbers:)`
A number of the form `a + ib,` where `a, b ` are real numbers and ` i^2 = -1 ` is called a complex number. If `z = a + ib` , then `-a-` is called the real part of ` z` and `-b` - is called the imaginary part of ` z` and are denoted by `Re(z)` and `Im(z)` respectively
`text(The form of the complex number)`
A complex number, `z`, has the form `z = x + iy`
where `x` and `y` are real numbers and `i` is the imaginary unit whose existence is postulated such that
`i^2 = −1 .`
Quantity `x` is the `text(real part of z)` and `y` is the `text(imaginary part)`
`x = Re(z) \ \ \ \ \ \ \ \ \ \ \ \ \ \y = Im(z) .`
Be careful to note that `Im(z)` is a real quantity.
A real number is thus a complex number with zero imaginary part. A complex number with zero real part is said to be pure imaginary. There is one complex number that is real and pure imaginary it is of course, zero. There is no particular need therefore to write zero as `(0 + i0).`
`text(Points to consider :)`
1. The complex numbers do not possess the property of order i.e., `(a+ ib) >`or`< (c + id)` is not defined. For example, `9 + 6i > 3 + 2i` makes no sense.
2. A real number a can be written as a + i · 0. Therefore, every real number can considered as a complex number, whose imaginary part is zero. Thus, the set real numbers (R) is a proper subset of the complex numbers (C) i.e., `R subset c`
Hence, the complex number system is `N subset W subset I subset Q subset R subset C`
3.A complex number z is said to be purely real, if Im (z) = 0; and is said to be purely imaginary, if Re (z) = 0. The complex number `0 = 0 + i · 0` is both purely real and purely imaginary.
4. In real number system, `a^2 + b^2 = 0` `=> a= 0 =b.`
But if `z_1` and `z_2` are complex numbers, then `z_1^2 + z_2^2 = 0` does not imply
`z_1 = z_2 =0.`