IB Mathematics AA HL Flashcards- Complex numbers

1// Up until now, students may just be told that you simply can not do the square root of a negative. This is partly correct—there is no real number solution. However, mathematicians like to have a way for talking about these roots: imaginary numbers.

[q]

NOTATION// We start with \(\sqrt{-1}\):

\[
\sqrt{-1} = i
\]

[a]

E.G.1 // Simplify \(\sqrt{-36} + \sqrt{-49}\):

\[
\sqrt{-36} + \sqrt{-49} = \sqrt{36 \cdot -1} + \sqrt{49 \cdot -1} = 6i + 7i = 13i
\]

[q]

COMPLEX NUMBERS// Real numbers work alongside imaginary, but can’t be combined:

\[
Z = 5 + 7i
\]

This cannot be written in any simpler terms than \(5 + 7i\), because the two parts are essentially from different number systems, so cannot be combined.

[a]

ADD/SUBTRACT// When you are adding/subtracting two complex numbers, \(Z_1\) & \(Z_2\), you just combine the real parts and imaginary parts separately:

E.G.2 // \((4 – 5i) + (7 + 8i) = (4 + 7) + (-5 + 8)i = 11 + 3i\)

E.G.3 // \((4 – 5i) – (7 + 8i) = (4 – 7) + (-5 – 8)i = -3 – 13i\)

[q]

MULTIPLICATION// Treat it as you would if you were expanding something like: \((x + 2)(x – 5)\). There is one extra step though, use \(i^2 = -1\).

E.G.4 // \((4 – 5i)(7 + 8i) = (4 \cdot 7) – (5i \cdot 7) + (4 \cdot 8i) – (5i \cdot 8i) = 28 – 35i + 32i – 40i^2\)

\[
= 28 – 35i + 32i + 40 = 68 – 3i
\]

[a]

DIVISION// This is an awkward process, where you assume the answer is another complex number, call it \(x + iy\). So rearrange, then solve for the real part (\(x\)), and the imaginary part (\(y\)):

E.G.5 // Find \(\frac{2 + 3i}{1 + 2i}\):

Assume \(\frac{2 + 3i}{1 + 2i} = x + iy\).

[q]

Multiply \(2 + 3i = (x + iy)(1 + 2i)\):

\[
2 + 3i = (x + iy)(1 + 2i) = (x – 2y) + i(2x + y)
\]

[a]

Real parts: \(2 = x – 2y\), Imaginary parts: \(3 = 2x + y\)

Substitution:

\[
2 = x – 2y \quad \text{sub} \quad 3 = 2x + y \quad \Rightarrow 3 = 2(2 + 2y) + y \quad \Rightarrow 3 = 4 + 5y \quad \Rightarrow y = \frac{-1}{5}
\]

\(\therefore x = \frac{8}{5}\), ANSWER \(\frac{8}{5} – \frac{1}{5}i\)

[q]

1// Up until now, students may just be told that you simply can not do the square root of a negative. This is partly correct—there is no real number solution. However, mathematicians like to have a way for talking about these roots: imaginary numbers.

[a]

NOTATION// We start with \(\sqrt{-1}\):

\[
\sqrt{-1} = i
\]

[q]

E.G.1 // Simplify \(\sqrt{-36} + \sqrt{-49}\):

\[
\sqrt{-36} + \sqrt{-49} = \sqrt{36 \cdot -1} + \sqrt{49 \cdot -1} = 6i + 7i = 13i
\]

[a]

COMPLEX NUMBERS// Real numbers work alongside imaginary, but can’t be combined:

\[
Z = 5 + 7i
\]

[q]

This cannot be written in any simpler terms than \(5 + 7i\), because the two parts are essentially from different number systems, so cannot be combined.

ADD/SUBTRACT// When you are adding/subtracting two complex numbers, \(Z_1\) & \(Z_2\), you just combine the real parts and imaginary parts separately:

[a]

E.G.2 // \((4 – 5i) + (7 + 8i) = (4 + 7) + (-5 + 8)i = 11 + 3i\)

E.G.3 // \((4 – 5i) – (7 + 8i) = (4 – 7) + (-5 – 8)i = -3 – 13i\)

MULTIPLICATION// Treat it as you would if you were expanding something like: \((x + 2)(x – 5)\). There is one extra step though, use \(i^2 = -1\).

[q]

E.G.4 // \((4 – 5i)(7 + 8i) = (4 \cdot 7) – (5i \cdot 7) + (4 \cdot 8i) – (5i \cdot 8i) = 28 – 35i + 32i – 40i^2\)

\[
= 28 – 35i + 32i + 40 = 68 – 3i
\]

[a]

DIVISION// This is an awkward process, where you assume the answer is another complex number, call it \(x + iy\). So rearrange, then solve for the real part (\(x\)), and the imaginary part (\(y\)):

[q]

E.G.5 // Find \(\frac{2 + 3i}{1 + 2i}\):

Assume \(\frac{2 + 3i}{1 + 2i} = x + iy\).

Multiply \(2 + 3i = (x + iy)(1 + 2i)\):

\[
2 + 3i = (x + iy)(1 + 2i) = (x – 2y) + i(2x + y)
\]

Real parts: \(2 = x – 2y\), Imaginary parts: \(3 = 2x + y\)

[a]

Substitution:

\[
2 = x – 2y \quad \text{sub} \quad 3 = 2x + y \quad \Rightarrow 3 = 2(2 + 2y) + y \quad \Rightarrow 3 = 4 + 5y \quad \Rightarrow y = \frac{-1}{5}
\]

\(\therefore x = \frac{8}{5}\), ANSWER \(\frac{8}{5} – \frac{1}{5}i\)

[x] Exit text