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A detailed tutorial on the absolute value of a complex number. Step by step tutorial including several examples on the absolute value of a complex number for reference.

The absolute value of a complex number is a little different than the absolute value of a real number, because complex numbers deal with imaginary numbers. However, the answer is still a non-negative real number, just like the numbers you deal with in other math classes every day. Say that a complex number z is equal to a + bi, where i is an imaginary number. The |z| is equal to the square root of a^2 plus b^2. In other words, square both a and b, add them together, and find the square root in order to have to absolute value of a complex number z.

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A detailed tutorial on invariants and the property of invariance. Step by step tutorial including several examples of invariants for reference.

Invariants are any function or number that displays the property of invariance. Invariance is when a function or number can go through several transformations without changing, or without going outside of its set parameters. The set parameters differ depending on the function or number. Some examples of invariant functions and numbers are the absolute value of a complex number, the degree of a polynomial, and certain parts of a square matrix

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A detailed tutorial on what a scalar is. Step by step tutorial including several examples of scalars and how they relate to vectors for reference.

A scalar is a number that relates vectors on a vector space through the process of scalar multiplication. A scalar can be taken from any set of numbers, including rational, algebraic, real, and complex sets of numbers. The scalar is always a real number. A scalar is a single component, and things such as vectors, matrices, and tensors can be reduced to a scalar.

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A detailed tutorial on the definition of an operand. Step by step tutorial including several examples of an operand for reference.

An operand can be any number. However, a number is only called an operand when there is some kind of operation being performed on it. There are simple operands and complex operands. A simple operand is what people call an operand – just one number. A complex operand is an operand that consists of an operation inside it, and therefore has at least 2 operands inside the first operand.

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A detailed tutorial on quaternions. Step by step tutorial including several examples and a visual example of what a quaternion is for reference.

Quaternions form a four-dimensional normed division algebra over real numbers. The original quaternions that were described in mathematics had a slightly definition – they formed a noncommutative number system that extends the complex numbers. However, the most recent definition, the one that is used in mathematics today, is the first one that was given. Quaternions are often denoted as the letter H, often written in the same script style as the number sets R and N. H in this case stands for Hamilton, named after the mathematician who first introduced quaternions to math.

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A detailed tutorial on the conjugate zeros theorem. Step by step tutorial including several examples of the conjugate zeros theorem for reference.

The conjugate zeros theorem states that if a + b * i is a zero of a polynomial with real coefficients, then so is a – b * i. The conjugate zeros theorem can be proved by taking any function in this form and setting it equal to zero. The conjugate zeros theorem makes many equations easier to solve, especially complex equations when you get to higher levels of math.

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A detailed tutorial of a lissajous curve. Step by step tutorial including several visual examples of lissajous curves for reference.

A Lissajous curve represents the graph of a system of parametric equations, which can be mathematically expressed as . This also decribes complex harmonic motion. The way that the figure appears is very sensitive to the ratio a / b, so the figure can appear in many different forms.

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A detailed tutorial on Mandelbrot sets and identifying Mandelbrot sets. Step by step tutorial including a several visual examples of a Mandelbrot set for reference.

A Mandelbrot set is defined as a set of points in the complex frame, the boundary of which forms a fractal. This can be mathematically defined as the set of complex values c for which the orbit of zero under iteration of a complex quadratic polynomial remains bounded.

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A detailed tutorial on Fourier transforms. Step by step tutorial including several examples of Fourier transforms for reference.

A Fourier transform is an operation that transforms one complex-valued function of a real variable into another. The domain of the original function is typically referred to as the time domain, because it is a representation of time. The domain of the new function represetns frequency. The Fourier transform itself is often called the frequency domain representation of the original function because of this.

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A detailed tutorial on the solving of problems using the binomial theorem. Stepby step tutorial including several examples of how to solve problems using the binomial theorem for reference.

The binomial theorem is something you should all be familiar with – it is the alternative to the F.O.I.L. technique. It is used when you are given a binomial that is raised to a power. The simplest version of it is expressed like this:

This can also be expressed as a factorial notation, in the form: