# 3.2 complex numbers, Plotting numbers on the complex plane (video)

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Adding and subtracting complex numbers. Video transcript. Move the orange dot to negative 2 plus 2i. So we have a complex number here. It has a real part, negative 2. It has an imaginary part, you have 2 times i. And what

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## 3.2 complex numbers, Complex Numbers

Multiplying Complex Numbers Together. Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together.

But in the complex numbers you can store one more information “angle”, and this angle in most cases used as a phase difference angle. The complex numbers can be used also to simplify the calculations by convert trigonometric functions (sin(x) and cos(x)) in equations to exponential functions (e x ) by adding an imaginary part to these equations and use Euler’s formula:

Complex numbers can be referred to as the extension of the one-dimensional number line. In the complex plane, a complex number denoted by a + bi is represented in the form of the point (a, b). It is to be noted that a complex number with zero real part, such as – i, -5i, etc, is called purely imaginary.

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2. Real, Imaginary and Complex Numbers 3. Adding and Subtracting Complex Numbers 4. Multiplying Complex Numbers 5. Complex Conjugation 6. Dividing Complex Numbers 7. Quiz on Complex Numbers Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web

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3 2 = 1 p 3 p 1 2 = 1 p 3i 2: Think: Do you know how to solve quadratic equations by completing the square? This is how the quadratic formula is derived and is well worth knowing! 1.2 Fundamental theorem of algebra One of the reasons for using complex numbers is because allowing complex roots means every polynomial has exactly the expected

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(1768-1822)). The point (a,b) represents the complex number a+ biso that the x-axiscontainsallthe real numbers, and so is termed the real axis, and the y-axis contains all those complex numbers which are purely imaginary (i.e. have no real part) and so is referred to as the imaginary axis.-4 -2 2 4-3-2-1 1 2 3 +2i 2−3i −3+i An Argand diagram 4

3.2 Numbers. A Racket number is either exact or inexact:. An exact number is either. an arbitrarily large or small integer, such as 5, 99999999999999999, or -1 7;. a rational that is exactly the ratio of two arbitrarily small or large integers, such as 1/2, 99999999999999999/2, or -3 /4; or. a complex number with exact real and imaginary parts (where the imaginary part is not zero), such as 1

Polar Form of a Complex Number The polar form of a complex number is another way to represent a complex number. The form z = a + b i is called the rectangular coordinate form of a complex number. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

## 3.2 complex numbers, Complex numbers and hyperbolic functions

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3.2.3 Multiplication Complex numbers may be multiplied together and in general give a complex number as the result. The product of two complex numbers z 1 and z 2 is found by multiplying them out in full and remembering that i2 = −1, i.e. z 1z 2 =(x 1 +iy 1)(x 2 +iy 2) = x 1x 2 +ix 1y

How to Divide Complex Numbers. A complex number is a number that can be written in the form z = a + bi, where a is the real component, b is the imaginary component, and i is a number satisfying i^{2} = -1. Complex numbers satisfy many of

Complex Numbers A complex number is a number of the form a + bi, where i = and a and b are real numbers. For example, 5 + 3i, – + 4i, 4.2 – 12i, and – – i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 – 4i, for example, the real part is 6 and the imaginary part is -4i.

For example 1 would be (1,0) and 3i would be (0,3), but you can also have numbers like 3+2i or -4-5i, which would be at (3,2) and (-4,-5) respectively This type of diagram displaying complex numbers is called an Argand Diagram.

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Complex Numbers and Exponentials Deﬁnition and Basic Operations A complex number is nothing more than a point in the xy–plane. The sum and product of two complex numbers (x 1,y 1) and (x 2,y 2) is deﬁned by (x 1,y 1) +(x 2,y 2) = (x 1 +x 2,y 1 +y 2) (x 1,y 1)(x 2,y 2)

1) Factor as you normally would to get 15-10i-18i+12i^2, which simplifies to 15-28i+12(-1) since i^2=-1. Final answer is 3-28i. 2) Multiply the whole thing by the conjugate of the denominator, 4+5i.

Enjoy these free printable sheets focusing on the complex and imaginary numbers, typically covered unit in Algebra 2.Each worksheet has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

In the above notation, notice how much a complex number looks like a surd (e.g. compare and ).The only difference is that the number under the square root sign is negative.In fact, when it comes to arithmetic, complex numbers can be treated like surds.