Examples of complex numbers in the following topics:
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- A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
- The complex number $a+bi$ can be identified with the point $(a,b)$.
- A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.
- It is beneficial to think of the set of complex numbers as an extension of the set of real numbers.
- Complex numbers allow for solutions to certain equations that have no real number solutions.
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- Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately.
- Complex numbers can be added and subtracted to produce other complex numbers.
- In a similar fashion, complex numbers can be subtracted.
- Note that it is possible for two non-real complex numbers to add to a real number.
- However, two real numbers can never add to be a non-real complex number.
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- Complex numbers are added by adding the real and imaginary parts of the summands.
- Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A, and B (as shown in ).
- The multiplication of two complex numbers is defined by the following formula:
- Addition of two complex numbers can be done geometrically by constructing a parallelogram.
- Discover the similarities between arithmetic operations on complex numbers and binomials
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- Any time an $i^2$ appears in a calculation, it can be replaced by the real number $-1.$
- Two complex numbers can be multiplied to become another complex number.
- Note that this last multiplication yields a real number, since:
- We then combine these to write our complex number in standard form.
- Note that it is possible for two nonreal complex numbers to multiply together to be a real number.
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- The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate.
- The complex conjugate of the complex number z = x + yi is defined as x - yi.
- Specifically, conjugating twice gives the original complex number: z** = z .
- Moreover, a complex number is real if and only if it equals its conjugate.
- The reciprocal of a nonzero complex number $z = x + yi$ is given by
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- Division of complex numbers is accomplished by multiplying by the multiplicative inverse of the denominator.
- We have seen how to add, subtract, and multiply complex numbers, but it remains to learn how to divide them.
- For complex numbers, the multiplicative inverse can be deduced using the complex conjugate.
- We have already seen that multiplying a complex number $z=a+bi$ with its complex conjugate $\overline{z}=a-bi$ gives the real number $a^2+b^2$.
- Suppose you wanted to divide the complex number $z=2+3i$ by the number $1+2i$.
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- Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$.
- This leads to a way to visualize multiplying and dividing complex numbers geometrically.
- Sometimes it is helpful to think of complex numbers in a different geometric way.
- When written this way, it now becomes easier to multiply and divide complex numbers.
- Explain how to represent complex numbers in polar coordinates and why it is useful to do so
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- The complex conjugate of the number $a+bi$ is $a-bi$.
- The complex conjugate (sometimes just called the conjugate) of a complex number $a+bi$ is the complex number $a-bi$.
- Since the conjugate of a conjugate is the original complex number, we say that the two numbers are conjugates of each other.
- The number $\sqrt{a^2+b^2}$ is called the length or the modulus of the complex number $z=a+bi$.
- Explain how to find a complex number's conjugate and what it is used for