When we hear of complex numbers, one thing comes to our mind and it is something complex. One thing I would like to tell that complex numbers are not as complex as it seems to be. Even Euler wanted it to be taught in schools. It is a combination of symbols, idea and presentation. the evolution of complex numbers is attributed to the equation x2 + 1 = 0. As we can see from the above equation that the square of a number is negative. And we know that the square of any integer cannot be negative. To solve such type of equations we use imaginary numbers. The basic of imaginary numbers is √-1. We represent it by i. One of the most interesting thing in the complex numbers is the product of two complex number i. i×i = √-1 × √-1 is not 1 as can be thought. The reasoning of the above is √-1 × √-1 = √(-1×-1) = √1 = 1. But we must keep in mind that now the number can not be thought as per the rules of real algebra. Any negative number n under the root sign can not be thought as a combination of a number, sign and a root. But it is to be interpreted as a sign i√n. Where the number n denotes the magnitude of the negative number. So the new reasoning is i×i = i2 = -1.
A complete imaginary number consists of real as well as imaginary parts.But one thing which comes to our mind is what is the representation of these numbers. To show such kind of numbers we use Argand plane or complex plane. A complex number has two parts the real part and the imaginary part. The real part is denoted on the real axis and imaginary part is denoted on the imaginary axis. The real part is equivalent to the x-axis and the imaginary part is equivalent to the y-axis of the Cartesian Coordinates.
Euler established a relationship between e, i, cos θ and sin θ. He found that they form a definition and it is
eiθ = cos θ + i sin θ Much of the mathematics of complex numbers is result of identity. We find that eiθ = cos θ + i sin θhas no simple proof but as we expand eiθ we find that its representation is like this
eiθ = 1 + iθ + i2θ2 + i3θ3 + i4θ4 + ...
= 1 + iθ − θ2 − iθ3 + θ4 + ...
= (1 − θ2 + θ4 ...) + i(θ − θ3 + ...)
We know that (1 − θ2 + θ4 ...) is expansion of cos θ and (θ − θ3 + ...) is the expansion of sin θ
= cos θ + i sin θ
Similarly by induction we can prove that (cos nθ + i sin nθ) = (cos θ + i sin θ)n.
There are many such results which form the basis of complex numbers. But the origin of complex numbers if the result of deep knowledge of mathematics as it takes ideas from different fields.
A complete imaginary number consists of real as well as imaginary parts.But one thing which comes to our mind is what is the representation of these numbers. To show such kind of numbers we use Argand plane or complex plane. A complex number has two parts the real part and the imaginary part. The real part is denoted on the real axis and imaginary part is denoted on the imaginary axis. The real part is equivalent to the x-axis and the imaginary part is equivalent to the y-axis of the Cartesian Coordinates.
Euler established a relationship between e, i, cos θ and sin θ. He found that they form a definition and it is
eiθ = 1 + iθ + i2θ2 + i3θ3 + i4θ4 + ...
= 1 + iθ − θ2 − iθ3 + θ4 + ...
= (1 − θ2 + θ4 ...) + i(θ − θ3 + ...)
We know that (1 − θ2 + θ4 ...) is expansion of cos θ and (θ − θ3 + ...) is the expansion of sin θ
= cos θ + i sin θ
Similarly by induction we can prove that (cos nθ + i sin nθ) = (cos θ + i sin θ)n.
There are many such results which form the basis of complex numbers. But the origin of complex numbers if the result of deep knowledge of mathematics as it takes ideas from different fields.
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