Complex Numbers

The set of real numbers are a subset of the set of complex numbers: imagine the complex plane with real numbers existing on the horizontal number line, and pure imaginary existing on the vertical axis. The entire plane (which includes both axes) is the set of complex numbers. So any real number (such as pi) will also be a complex number. But many people think of complex numbers as something that is "not a real number".

One thousand.

#include<iostream.h> #include<conio.h> class complex { int r; int i; public: complex() { } complex(int a,int b) { r=a;i=b; } friend complex operator+(complex,complex); friend show(complex); complex operator+(complex c1,complex c2) { complex c3; c3.r=c1.r+c2.r; c3.i=c1.i+c2.i; return(c3); } show(complex c) { cout<<c.r<<"i+"<<c.i<<endl; } void main() { complex a,b,c; clrscr(); a.complex(3,6); b.complex(4,7); c=a+b; show(a); show(b); show(c); getch() }

Imaginary numbers are only ever used when you are using the square roots of negative numbers. The square root of -1 is i.

You may find imaginary numbers when you are finding roots of equations.

Basically you can have an order on a number line, but complex numbers are points on a plane. You can invent some arbitrary order, like which number has the largest real part, or the biggest absolute value, but many of the order properties of real numbers are no longer valid with such definitions.

Yes.

(5-8)+(1-11) 5-8+1-11 6-19 -13

It is the arithmetic mean or average.

The following may seem far-fetched if you are not accustomed to imaginary or complex numbers, so before I continue, let me assure you that complex numbers have many practical applications, including electricity, quantum mechanics, art, and several other more.

The imaginary number is neither a positive nor a negative number. Imagine two perpendicular axes of numbers. The directions are arbitrary, but the way it is commonly drawn, from left to right you have the real numbers - the numbers you are probably most familiar with, which include positive and negative numbers. Positive at the right, negative at the left. The number line which you may have seen already.

From top to bottom is another line, that crosses the origin - the line of the imaginary numbers. One unit up is +i, two units up is +2i, one unit down (from the origin, or zero) is -i, two units down is -2i, etc. The "imaginary unit", then, is called "i", although in electricity the letter "j" is used instead (to avoid confusion with the unit for current).

A combination of a real number and an imaginary number is called a complex number - for example, 2 + 3i. Adding and subtracting complex numbers is fairly straightforward. Just add the corresponding terms. To multiply complex numbers, multiply them as you normally multiply binomials - then use the definition i2 = -1.

It so happens that when complex numbers are used, not only do negative numbers have a square root, but any root - square root or otherwise - has a solution. In a way, this makes the complex numbers more "complete" than the real numbers.

Of course, common sense should be used. Just as negative or fractional numbers don't make sense for some real-life problems, complex numbers don't make sense for some real-life problems, either. So if, for example, the quadratic formula gives you a complex solution (or a negative solution, for that matter), analyze the original problem to see whether the specific solutions found make sense, given the problem statement.

Yes, the complex numbers, as well as the real numbers which are a subset of the complex numbers, form groups under addition.

A real number is any continuous quantity which can be represented as a point on a one-dimensional line. Real numbers are used for measuring properties of objects and phenomena in the natural and social world.

Yes. They are called surds. They are also complex.

I have compiled complex number data in the form of a curvature flow graph or

a conformal mapping of z^z, z to the z power or (x+iy)^(x+iy). x and y are in the

range of +- 10 units. It's quite messy above x=1 so I zoomed in on x<1.

Please see the related link of the graph down below in the Source and Related Links.

-12 = 4 x 3 x -1 so sqrt = 2i root 3

1. A complex formula like (z+1/z) is used to study and design airplane wings.

2. I use complex numbers to make math related art. The LINK below shows artwork

based on the formula (z-1)/(z+i)

To plot a complex coordinate number, almost ignore "i".

Find the real part, for example, 5, -8, pi, 0.98, the number that's not attached to the "i".

Then find the imaginary part, the number attached to the i.

The real part would be the x coordinate and the y would be the imaginary part.

For example: (4-7i) would be plotted on x=4 and y=-7.

Since complex numbers can be expressed as magnitude and angle, a plot of a system's response can be approximated with a Bode plot (see related link). The gain is the magnitude, and the phase shift is the angle change of the system.

The set of complex numbers encompasses real, imaginary, and combinations of the two, so it is the largest set that you are likely to encounter. There are other number systems, such as quaternion imaginaries, which you may never encounter, so I only mention it here and you can look it up for more info if you're interested.

One way is to use them in graphics images. Inversion is by the formula 1/(x+iy).

Click on the INVERSION link below for an example.

Notice the yellow dot in the left pane has been inverted outside the circle in the

right pane and the checkerboard is now inside the circle.

113 is not a complex number and so there cannot be any correct notation.

The imaginary unit number is the square root of -1 and is denoted by i

I would certainly bet on a computer. Try to solve (log z)^-3i. This is log z raised to the -3i power. A computer comes in handy.

You can draw a triangle on the complex plane, but all of the distances (side lengths) are considered 'real' distances {just like the magnitudes of individual complex numbers}. So I believe the answer is No.

It is just 11+8i.