# C++ Complex Number — 01

**2023-03-10**Last modified:

**2023-03-16**

**Daniele Pugliesi**on Wikimedia

**Table of contents**

## Complex Number

## Functions

`real`

: Real part of complex`imag`

: Imaginary part of complex`abs`

: Absolute value of complex`arg`

: Phase angle of complex`norm`

: Norm of complex`conj`

: Complex conjugate`polar`

: Complex from polar components`proj`

: Complex projection.

## Norm vs Abs

In essence, the absolute value of a real number captures the idea
of its magnitude, i.e., its distance to the origin

When you try to generalise this to multi-dimensional vectors,
the most natural extension that captures the idea of distance to
the origin is the Euclidian norm, also known as the

Just like the absolute value chooses to ignore the algebraic sign of a real number in order to capture its magnitude, the Euclidian norm chooses to ignore the direction of the vector and captures its magnitude.

However, one can choose other norms in multi-dimensional space,
as long as some rules are satisfied (these are spelled out on the
Wolfram page above). Two notable ones are the

The

This is now changing thanks to the available computational capabilities, which let us find minima and maxima of objective functions through searches within the desired space, even when there is no nice closed form solution or quite as nice analytical properties to the norm, when compared to the Euclidian norm.

```
auto printZ = [](const auto& label, const auto& Z) {
fmt::print("{:30}=> {}{:+}i\n", label, Z.real(), Z.imag());
};
auto printR = [](const auto& label, auto R) {
fmt::print("{:30}=> {}\n", label, R);
};
std::complex<double> z3_4(3.0, 4.0); // 3 + 4i
std::complex<double> z1_2{1.0, 2.0}; // 1 + 2i
std::complex<double> z4_3{4.0, 3.0}; // 4 + 3i
printZ("z3_4", z3_4);
printZ("z1_2", z1_2);
// Addition
auto sum = z3_4 + z1_2; // 4 + 6i
printZ("z3_4 + z1_2, sum", sum);
// Subtraction
auto diff = z3_4 - z1_2; // 2 + 2i
printZ("z3_4 - z1_2, difference", diff);
// Multiplication
auto prod = z3_4 * z1_2; // -5 + 10i
printZ("z3_4 * z1_2, product", prod);
// Division
auto quot = z3_4 / z1_2; // 2.2 - 0.4i
printZ("z3_4 / z1_2, quotient", quot);
// Conjugate
auto conj = std::conj(z3_4); // 3 - 4i
printZ("conj(z3_4), conjugate", conj);
// Absolute value (modulus)
double abs_z3_4 = std::abs(z3_4); // 5
printR("abs(z3_4), modulus", abs_z3_4);
// The norm value of a complex number is its squared magnitude, defined as
// the addition of the square of both its real and its imaginary part
// (without the imaginary unit). This is the square of abs(x).
double norm_z3_4 = std::norm(z3_4); // 25
printR("norm(z3_4), norm", norm_z3_4);
// The phase angle of a complex number is the angle the theoretical vector to
// (real,imag) forms with the real axis (i.e., its arc tangent). It returns
// the same as: atan2(x.imag(),x.real());
double arg_z4_3 = std::arg(z4_3);
printR("arg(z4_3), phase angle in rad", arg_z4_3);
// The projection of z onto the Riemann sphere.
printZ("proj(z1_2)", std::proj(z1_2));
printZ("proj(z3_4)", std::proj(z3_4));
std::complex<double> inf1{INFINITY, 50};
std::complex<double> inf2{-INFINITY, 50};
std::complex<double> inf3{0, -INFINITY};
printZ("proj(inf1)", std::proj(inf1));
printZ("proj(inf2)", std::proj(inf2));
printZ("proj(inf3)", std::proj(inf3));
```

## Possible output

```
z3_4 => 3+4i
z1_2 => 1+2i
z3_4 + z1_2, sum => 4+6i
z3_4 - z1_2, difference => 2+2i
z3_4 * z1_2, product => -5+10i
z3_4 / z1_2, quotient => 2.2-0.4i
conj(z3_4), conjugate => 3-4i
abs(z3_4), modulus => 5
norm(z3_4), norm => 25
arg(z4_3), phase angle in rad => 0.6435011087932844
proj(z1_2) => 1+2i
proj(z3_4) => 3+4i
proj(inf1) => inf+0i
proj(inf2) => inf+0i
proj(inf3) => inf-0i
```