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Cartesian form
Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
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The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi.
We call a the real part of z and b the imaginary part of z:
For example, z=2−3i is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
If two complex numbers are equal, then both their real and imaginary parts are equal:
The expression x+iy is often referred to as the Cartesian form of z.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
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The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
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The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
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The following properties hold for complex conjugates:
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
We can solve complex equations involving z and z∗ by using the fact that a complex number z takes the form a+bi. Recall that for complex numbers z1 and z2,
We use this fact to equate the real and imaginary parts of both sides, which creates a solvable a system of two equations in two unknowns from one given equation.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.