Complex numbers

The imaginary unit is i, an ordinary constant. Juxtaposition is already multiplication, so 4i is 4*i and needs no special number syntax — the same reading that makes 2e mean 2*e.

z

3 + 4 i

=3 + 4i

w

1 - 2 i

=1 - 2i

Arithmetic. Any operation with a complex operand on either side answers complex, promoting the real one.

sum

z + w

3 + 4i + 1 - 2i

=4 + 2i

difference

z - w

3 + 4i - (1 - 2i)

=2 + 6i

product

z \cdot w

(3 + 4i) \cdot (1 - 2i)

=11 - 2i

quotient

\frac{z}{w}

\frac{3 + 4i}{1 - 2i}

=-1 + 2i

scaled

2 \cdot z

2 \cdot (3 + 4i)

=6 + 8i

negated

0 - z

0 - (3 + 4i)

=-3 - 4i

i squared is where the whole thing comes from.

i^{2}

=-1 + 0i

Taking a value apart. Re and Im give real numbers; Im is the coefficient of i, so Re(z) + Im(z)*i rebuilds z.

Re (z)

Re (3 + 4i)

3

Im (z)

Im (3 + 4i)

4

conj (z)

conj (3 + 4i)

=3 - 4i

| z |

| 3 + 4i |

5

\arg (z)

\arg (3 + 4i)

0.927295

Whole powers only, by repeated multiplication. A fractional power of a complex number needs a branch cut, which this language does not choose.

z^{2}

{(3 + 4i)}^{2}

=-7 + 24i

z^{3}

{(3 + 4i)}^{3}

=-117 + 44i

z^{0}

{(3 + 4i)}^{0}

=1 + 0i

z^{(0 - 1)}

{(3 + 4i)}^{(0 - 1)}

=0.12 - 0.16i

A complex value stays complex. Nothing demotes when the imaginary part happens to be zero, so a result's type never depends on its value.

{real}_{again}

z - 4 i

3 + 4i - 4 i

=3 + 0i

Units. Both parts share one dimension, because they are components of one measurement rather than two measurements.

Z

(3 + 4 i) Ω

=(3 + 4i) Ω

Z

=(0.003 + 0.004i) kΩ

| Z |

| (3 + 4i) Ω |

5 Ω

Re (Z)

Re ((3 + 4i) Ω)

3 Ω

The argument is an angle, so it is dimensionless whatever Z is measured in.

\arg (Z)

\arg ((3 + 4i) Ω)

0.927295

The impedance of a series RLC branch at one frequency, which is what the corpus uses complex arithmetic for.

R

32.3383 Ω

L

4.58533 \times 10^{−5} H

C

4.56546 \times 10^{−8} F

f

110000 Hz

Zrlc

R + 2 \cdot π \cdot f \cdot L \cdot i + \frac{1}{2 \cdot π \cdot f \cdot C \cdot i}

32.3383 Ω + 2 \cdot π \cdot 110000 Hz \cdot 4.58533 \times 10^{−5} H \cdot i + \frac{1}{2 \cdot π \cdot 110000 Hz \cdot 4.56546 \times 10^{−8} F \cdot i}

=(32.3383 + 1.79395e-5i) Ω

| Zrlc |

| (32.3383 + 1.79395e-5i) Ω |

32.3383 Ω

\frac{\arg (Zrlc) \cdot 180}{π}

\frac{\arg ((32.3383 + 1.79395e-5i) Ω) \cdot 180}{π}

3.17844 \times 10^{−5}

A branch of impedances, as one value

A network is a collection of impedances, and until now each one had to be its own name. A vector holds them, sums to the series impedance, and comes apart elementwise into magnitudes and angles — which is what a table of the branch would show.

The same branch, split into the three impedances it is made of, using the resistance, inductance and capacitance already defined above.

Z_{R}

R + 0 i \cdot Ω

32.3383 Ω + 0 i \cdot Ω

=(32.3383 + 0i) Ω

Z_{L}

0 Ω + 2 \cdot π \cdot f \cdot L \cdot i

0 Ω + 2 \cdot π \cdot 110000 Hz \cdot 4.58533 \times 10^{−5} H \cdot i

=(0 + 31.6915i) Ω

Z_{C}

0 Ω + \frac{1}{2 \cdot π \cdot f \cdot C \cdot i}

0 Ω + \frac{1}{2 \cdot π \cdot 110000 Hz \cdot 4.56546 \times 10^{−8} F \cdot i}

=(0 - 31.6915i) Ω

branch

[\begin{matrix} Z_{R} \\ Z_{L} \\ Z_{C} \end{matrix}]

=[(32.3383 + 0i) Ω, (0 + 31.6915i) Ω, (0 - 31.6915i) Ω]

Z_{series}

sum (branch)

sum ([(32.3383 + 0i) Ω, (0 + 31.6915i) Ω, (0 - 31.6915i) Ω])

=(32.3383 + 1.79395e-5i) Ω

magnitudes

| branch |

| [(32.3383 + 0i) Ω, (0 + 31.6915i) Ω, (0 - 31.6915i) Ω] |

=[32.3383 ohm, 31.6915 ohm, 31.6915 ohm]

angles

\arg (branch)

\arg ([(32.3383 + 0i) Ω, (0 + 31.6915i) Ω, (0 - 31.6915i) Ω])

=[0°, 90°, -90°]

Scaling the whole branch is one multiplication — two of everything in parallel halves each impedance.

halved

\frac{branch}{2}

\frac{[(32.3383 + 0i) Ω, (0 + 31.6915i) Ω, (0 - 31.6915i) Ω]}{2}

=[(16.1691 + 0i) Ω, (0 + 15.8458i) Ω, (0 - 15.8458i) Ω]

What a complex vector does not do, it says. Ordering complex numbers has no meaning, so sort and min refuse one rather than answering emptily, and a matrix of complex numbers is not built at all.