Complex Numbers

Because a complex number is a two dimensional entity. The concept of less than or greater than, for ordinary numbers, is one-dimensional. It can be applied to the magnitude (absolute value) of a complex number.

Complex numbers can represent real world data, in a way that may be more complicated (or near impossible), if trying to represent with just real numbers. I cannot post links in the answer, but recently the Numberphile channel on YouTube posted a video on Quaternions. Also there used to be a really cool article with an interactive animation, showing what imaginary numbers represent on (picomonster dot com) but it no longer works.

You can use the rectangular-polar conversion on any scientific calculator to do this type of conversion. However, in this case, you should be able to do the calculation in your head: any point on the positive i-axis has an angle of 90 degrees; any point on the negative i-axis (as in this case) has an angle of 270 degrees.

A complex number is any number that can be represented in the form of a+bi, the real numbers are a and b, the imaginary number is i. Complex numbers are used in scientific and engineering fields.

A set can contain anything, or in some cases, nothing. Real numbers are an important and useful mathematical concept, and they are among the things that could be placed in various sets for various purposes.

Yes, they are.

The square roots (not sqare route!) of 99999999999999999999999 are ±316,227,766,016.8 approx.

It is: 7,400,000 or as 7.4*106 in scientific notation

The cardinality of a set is its size. For instance, since the set G contains 4 elements, then its cardinality is 4. So if the set has a finite number of elements (meaning it is a finite set), you can find its cardinality, otherwise you cannot (meaning it is an infinite set).

You can't find the amplitude of √3 - i. You can find the radius and angle of that expression, applying the Euler's formula.

√3 - i = rcos(θ) + irsin(θ)

Well...

√3 = rcos(θ)

-1 = rsin(θ)

Then:

r = 2, which gives us:

cos(θ) = √(3)/2

sin(θ) = -½

So = -π/6

Therefore, the radius if 2 and the angle is -π/6 radians.

* * * * *

The amplitude = θ = -π/6

There are two different formulas, which are commonly referred to as Euler's formula. Links are posted below.

One has to do with solid geometry, which states that F + V - E = 2, which means (number of Faces) + (number of edges) - (number of vertices) equals 2. This may be useful if you want to build a structure of a certain shape (maybe a dodecahedron, which is a 12 sided shape) and help to figure how much building materials you will need.

The other Euler's formula is used with complex numbers, which states e^(i*Θ) = cos(Θ) + i*sin(Θ), where i is the imaginary unit. Note that (Θ) must be expressed in radians to work properly.

So thinking about triangles and the unit circle, any complex number can be graphically represented by the real portion on the horizontal, and the imaginary portion on the vertical. One advantage, is complex numbers are easier to multiply, divide, and raise to powers if they are represented as e^(iΘ) rather than a + bi.

Another use of this is figuring the converse trig functions:

cos(Θ) = [e^(iΘ) + e^(-iΘ)] / 2, and sin(Θ) = [e^(iΘ) - e^(-iΘ)] / (2i)

If you can remember these, you can figure any trig identity that you may need to use, from these two.

6+5i

The value of iota square is negative one because that is how it is defined. Iota is defined as the square root of negative one, or iota square must be negative one.

Yes. The number 1 + i is imaginary but not pure imaginary, while 5i is pure imaginary.

The imaginary number (i) is defined as the square root of -1.

If you multiply i by i you get -1

Store your complex numbers in a structure of some sort that has two variables - one to store the real part and one to store the complex part.

Then to add two complex numbers, add the real parts together and add the complex part together, eg:

(2 + 3i) + (5 - 2i) = ((2 + 5) + (3 + -2)i)

= (7 + i)

how you actually do this will be entirely up to the language you are using for your programming.

It is based on definition that you have to accept...which is:

i = the square root of -1

So, if you were to square i then you would get -1.

Hope that helped.

Yes. All real numbers are considered complex numbers, with the imaginary part being equal to zero.

No inductor is perfect and has a capacitive and resistive component. As frequency increases, these components have more effect on the circuit operation. A capacitive component would be out of phase and be the imaginary value.

Complex numbers form: a + bi where a and b are real numbers.

The conjugate of a + bi is a - bi

If you multiply a complex number by its conjugate, the product will be a real number, such as

(a + bi)(a - bi) = a2 - (bi)2 = a2 - b2i2 = a2 - b2(-1) = a2 + b2

i is the Imaginary Unit, equal to sqrt(-1). So i and any real number multiplied by i will all be imaginary numbers. Here are some: i, -i, 5i, -3i, i*pi, etc.

I sense you're talking about the infinite disk, the hyperbolic disk or the Poincare

disk. The limit of the circumference is infinite and a real number and is not actually part of the hyperbolic plane.

You can certainly multiply and divide with the rectangular form, but it is somewhat easier in polar form. This is especially relevant if you want to extend to more complicated operations, such as higher powers or taking roots.

As for the polar form, any method to add and subtract them directly would probably be quite complicated, and directly or indirectly involve many of the same calculations that are done in converting from polar to rectangular, and back. Try it! (That is, try to deduce the formulas for adding two complex numbers in polar form.)

A complex number is x+iy and -6i is 0-6i or 0-i6

I would then answer yes.

A complex number of the form M /_ÆŸ (Magnitude and angle ÆŸ), can be converted to the format {a + bi} as follows: M*(cosÆŸ + isinÆŸ)