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Δβα\Delta _{\beta \alpha}
01
Linear equations
Δβα\Delta _{\beta \alpha}
02
Quadratic equations
Δβα\Delta _{\beta \alpha}
03
Functions
Δβα\Delta _{\beta \alpha}
04
Linear algebra
Δβα\Delta _{\beta \alpha}
05
Vectors
Δβα\Delta _{\beta \alpha}
06
Logarithms
Δβα\Delta _{\beta \alpha}
06
Complex numbers
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Δβα\Delta _{\beta \alpha}
07
Complex numbers
Starter
List out any occurrences where your teacher has said that something will provide either no real solutions or complex solutions. I'll give you an hint:
b2−4ac<0b^2-4ac<0
.
Number scales
When you were taught elementary math, you were given them in the form of a 1D line that, for example, had a range from
−10-10
to
00
to
1010
. This type of number scale encompasses all
R\mathbb{R}
(or real) numbers. Now, if we were to add another axis (turning our 1D line into a 2D Cartesian space), we can introduce ourselves to complex numbers. A graph will be added later (when I find a suitable library to display graphs), however if you take a sheet of paper and draw out a 2D axis, naming the
xx
-axis "
R\mathbb{R}
" and
yy
-axis "
ii
", with intervals of
ii
being the same as
1,2,3,4,...1,2,3,4,...
, you've created yourself a complex plane!
Note N06.0a - Complex number form
z=a+biz=a+bi
Where
aa
is the
R\mathbb{R}
part of the number (think, location on the "
xx
-axis")
Where
bibi
is the
ii
part of the number (think, location on the "
yy
-axis")
Complex modulus
Just like the 1D axis of the
R\mathbb{R}
scale, where you have an absolute value that's always a positive magnitude of the number from a reference point (normally 0), you have the same for complex numbers (but it's called complex modulus). If you plot a complex number using the above note, you can see that you can create a right-angled triangle, and from there use some Pythagoras to find the absolute value. Even if the
R\mathbb{R}
and
ii
parts are negative, the absolute value is still positive (remember what squaring a negative number does to it).
Note N06.1a - Complex modulus
∣z∣=a2+b2|z|=\sqrt{a^2+b^2}
Simple Pythagoras is used here to find the absolute value of a complex number
Coming soon!
Topics coming soon:
1) Modulus and geometries of complex numbers
2) Applications with quadratics