complex numbers
They must be imaginary, I don't see them
just as real numbers can be located on the number line, complex numbers can be located in the plane, because they have a real part and an imaginary part. So,
a+bi
is located in the plane by a pair of coordinates: a units along the horizontal axis, and b units vertically from there.
Not sure how a long discourse here will be any more useful than the discussion in your text. Or, do a web search, and you will find many articles explaining them.
The reason they come up in trig is that any point in the plane can be located at some distance from the origin, and in some direction, given by an angle θ. All your normal trig functions can be applied to θ.
What are complex numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, which is defined as the square root of -1. In this form, 'a' is known as the real part of the complex number, and 'b' is the imaginary part.
To understand complex numbers, it is important to understand the concept of the imaginary unit 'i'. Since the square root of a negative number is not a real number, mathematicians introduced the imaginary unit 'i' to represent the square root of -1. This gives us the ability to work with numbers that contain both real and imaginary parts.
To perform calculations with complex numbers, you can use various mathematical operations such as addition, subtraction, multiplication, and division. When adding or subtracting complex numbers, you simply add or subtract the real parts and the imaginary parts separately.
For example, to add two complex numbers (a + bi) and (c + di), you add the real parts (a + c) and the imaginary parts (b + d) separately. Similarly, you can subtract the real and imaginary parts.
Multiplying complex numbers involves using the distributive property and simplifying by multiplying each term with every term in the other complex number. Division of complex numbers can be done by multiplying both the numerator and denominator by the conjugate of the denominator.
There are also other operations that can be performed on complex numbers, such as finding the magnitude or absolute value, finding the complex conjugate, and finding the argument (or angle) of a complex number.
To summarize, complex numbers are numbers that include both real and imaginary parts. They are expressed in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit. The operations on complex numbers, such as addition, subtraction, multiplication, and division, can be carried out by performing separate operations on their real and imaginary parts.