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Cartesian form
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Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
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Imaginary number i
AHL 1.12
The imaginary number i is the square root of −1:
i=√−1⇔i2=−1🚫
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
(bi)2=b2i2=−b2<0
Conclusion: the square of any imaginary number is negative.
Complex Numbers a+bi
AHL 1.12
A complex number
z=a+bi📖
is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.
Finding complex roots of quadratics
AHL 1.12
When a quadratic
ax2+bx+c=0
has
Δ=b2−4ac<0,
it has no real roots since the square root in
x=2a−b±√b2−4ac
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
Real and Imaginary parts
AHL 1.12
For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:
Re(z)Im(z)=a=b
For example,
z=2−3i
is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
The sets ℂ, ℝ and ℝ\ℂ
AHL 1.12
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.
Product of complex numbers
AHL 1.12
The product of two complex numbers in Cartesian form is
(a+bi)×(c+di) =ac+adi+bci+bdi2=ac+i(ad+bd)+bd⋅(−1)=ac−bd+(ad+bd)i
The complex plane
AHL 1.12
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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