Complex Numbers

The reactance is taken imaginary because it does not consume any power directly but reactance like inductive or capacitive reactance provides phase shift of 90 degree, whether it provides a leading phase shift or lagging phase shift. In case of inductor the current lags voltage by 90 degree while in case of capacitor current leads by 90 degree. Thus only resistance is taken as real while combination of resistance and capacitance is called impedence....

Ashish Sharma AP ECE

HIET Shahpur

a + i b, with a and b real numbers and i = sqrt(-1)

Assuming x is your number, x * 10^n = x moved n decimal places.

When n is positive, move the decimal point n places to the right.

When n is negative, move the decimal point n places to the left.

When n is 0, do nothing.

a more detailed solution of an answer

It is: 360,000,000 or as 3.6*108 in scientific notation

complex numbers with no real part
if any complex number z can be written a + i b
then pure imaginary numbers have a=0 and b not equal to 0

No. The square root of -24 isn't even real, let alone rational because the square root of any negative number is going to be an imaginary number.

That 5 is a factor.

Zero is the additive identity in the set of real numbers; when you add zero to any number, the number does not change its identity.

Assume we have the product of these terms. Multiply these terms altogether to get 12i². Make note that i = √-1, i² = -1, i³ = -i and i⁴ = 1. Then, 12i² = 12(-1) = -12.

That is the answer to the question.

A [real] constant.

a variable

Aamir jamal;

All real numbers are complex numbers with 0 as its imaginary part.A real number is

self-conjugate.

e.g;a+0i

self conjugate =a-0i i=iota

You go 6 in positive x-direction ("right") and one in negative y-direction ("down"), there is your complex number, drwa an arrow reaching from the center to this point.

The concept of a complex number appears to be first conceived by Gerolamo Cardano (about 1545). See Related Link. The work of Euler in the 1700's helped to make them more useful, though.

A complex number x+iy is non-real if x=0 like 0+i13

If you understand what the absolute value of a complex number is, skip to the tl;dr part at the bottom.

The absolute value can be thought of as a sorts of 'norm', because it assigns a positive value to a number, which represents that number's "distance" from zero (except for the number zero, which has an absolute value of zero).

For real numbers, the "distance" from zero is merely the number without it's sign.

For complex numbers, the "distance" from zero is the length of the line drawn from 0 to the number plotted on the complex plane.

In order to see why, take any complex number of the form a + b*i, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. In order to plot this number on a complex plane, just simply draw a normal graph. The number is located at (a,b).

In order to determine the distance from 0 (0,0) to our number (a,b) we draw a triangle using these three points:

(0,0)

(a,0)

(a,b)

Where the points (0,0) and (a,b) form the hypotenuse. The length of the hypotenuse is also the "distance" of a + b*i from zero. Because the legs run parallel to the x and y axes, the lengths of the two legs are 'a' and 'b'.

By using the Pythagorean theorem, we can find the length of the hypotenuse as (a2 + b2)(1/2).

Because the length of the hypotenuse is also the 'distance' of the complex number from zero on the complex plane, we have the definition:

|a + b*i| = (a2 + b2)(1/2)

ALRIGHT, almost there.

tl;dr:

Remember that the complex conjugate of a complex number a + b*i is a + (-b)*i. By plugging this into the Pythagorean theorem, we have:

b2 = (-b)2

So:

(a2 + (-b)2)(1/2) = (a2 + b2)(1/2)

QED.

Yes, as long as the number line is entirely real.

That's an approximation of "pi", truncated after ten decimal places.

five hundred quintillion

No. 3.6427 is real and rational, but not a counting number.

-6i-8

They were discovered when Cardano solved the third degree equation.

In the formulas that arose to solve the third degree equation, Cardano needed to take the square root of negative numbers and add them up in a certain way. The strange thing that happened was that the formulas used these complex numbers, even if the solutions to the equation where all real. This baffled the mathematicians of the time, because how could these strange numbers turn out to be "real"?

Later this was considered totally correct, when the field of complex numbers was better undestood.

A diamond has two rotation symmetry. It is possible to have a diamond that does have four of rotation symmetry.

the mean or average of the set