Complex number

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This article/section deals with mathematical concepts appropriate for a student in mid to late high school.

A complex number is a number composed of two parts - a real component and an imaginary component, of the form $a+bi$, where a and b are real numbers and $i^{2}=-1$.

Whereas the real numbers can be represented as all the possible points on an infinitely extended number line, to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the abscissa) and the imaginary components on the vertical axis (the ordinate). This representation is known as the Argand diagram.

The complex numbers form an algebraically closed field but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the real numbers. One notable consequence, and a very natural way of seeing the necessity of complex numbers is the fact that all matrices of full rank over a vector space over real numbers repesent transformations, which, after a base transformation, are equivalent to a diagonal matrix of the same size with complex entries on the diagonal. Thus, any linear linear equation of motion of arbitrary order and dimension of real numbers can be represented in this way and be decomposed into eigenvectors (or modes). The evolution of the system is fully described by the complex amplitudes.

Many functions used in real analysis can be extended in to complex numbers using Taylor series. This is the subject of complex analysis.

The complex numbers are defined as a 2-dimensional vector space over the real numbers. That is, a complex number is an ordered pair of numbers: (a, b). The familiar real numbers constitute the complex numbers with second component zero. That is, x corresponds to (x, 0).

The second component is called the imaginary part. Its unit basis vector is called $i$. The first component is called the real part. Its unit basis vector is just 1. Thus, the complex number $(a,b)\,$ can also be written $a+bi\,$.

The complex numbers form a field, with the mathematical operations defined as shown below.

Arithmetic operations

The field operations are defined as follows:

Addition

Addition is just the standard addition on the 2-dimensional vector space. That is, add the real parts (first components), and add the imaginary parts (second components).

Letting the complex numbers w and z be defined by their respective components:

$w=(a,b)=a+bi\,$

$z=(c,d)=c+di\,$

we have:

$w+z=(a+c,b+d)=a+c+(b+d)i\,$

Multiplication

Multiplication has a special definition. It is this definition that gives the complex numbers their important properties.

$w*z=(a*c-b*d)+(a*d+b*c)i\,$

When we multiply $(0+1i)*(0+1i)\,$, this definition gives $i^{2}=-1\,$

Writing out the product $w*z\,$ in the obvious way, we get the same answer:

$(a+bi)*(c+di)=a*c+b*d*i^{2}+(a*d+b*c)i=(a*c-b*d)+(a*d+b*c)i\,$

This means that, using $i^{2}=-1\,$, one can perform arithmetic operations in a completely natural way.

Division

Division requires a special trick. We have:

${\frac {w}{z}}={\frac {a+bi}{c+di}}\,$

To get the individual components, the denominator needs to be real. This can be accomplished by multiplying both numerator and denominator by $(c-di)\,$. We get:

${\frac {w}{z}}={\frac {(a+bi)*(c-di)}{(c+di)*(c-di)}}={\frac {(a*c+b*d)+(b*c-a*d)i}{c^{2}+d^{2}}}\,$

This division will fail if and only if $c^{2}+d^{2}=0\,$, that is, $c\,$ and $d\,$ are both zero, that is, the complex denominator $z\,$ is exactly zero (both components zero). This is exactly analogous to the rule that real division fails if the denominator is exactly zero.

Fundamental theorem of algebra

The complex numbers form an algebraically closed field. This means that any $n^{th}\,$ degree polynomial can be factored into n first degree (linear) polynomials. Equivalently, such a polynomial has n roots (though one has to count all multiple occurrences of repeated roots. This statement is the Fundamental Theorem of Algebra, first proved by Carl Friedrich Gauss around 1800. The theorem is not true if the roots are required to be real. But when the roots are allowed to be complex, the theorem applies even to polynomials with complex coefficents.

The simplest polynomial with no real roots is $x^{2}+1\,$, since -1 has no real square root. But if we look for roots of the form $a+bi\,$, we have:

$(a+bi)^{2}+1=a^{2}-b^{2}+1+2abi=0\,$

For the imaginary part to be zero, one of a or b must be zero. For the real part to be zero, a must be zero and $b^{2}\,$ must be 1. This means that $b=\pm 1\,$, so the roots are $(0,1)\,$ and $(0,-1)\,$, or $\pm {}i\,$.

These two numbers, $i\,$ and $-i\,$, are the square roots of -1.

Similar analysis shows that, for example, 1 has three cube roots:

• $1\,$
• $-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\,$
• $-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i\,$

One can verify that $(x-1)(x+{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i)(x+{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i)=x^{3}-1\,$

For a given field, the field containing it (strictly speaking, the smallest such field) that is algebraically closed is called its algebraic closure. The field of real numbers is not algebraically closed; its closure is the field of complex numbers.

Polar coordinates, modulus, and phase

Complex numbers are often depicted in 2-dimensional Cartesian analytic geometry; this is called the complex plane. The real part is the x-coordinate, and the imaginary part is the y-coordinate. When these points are analyzed in polar coordinates, some very interesting properties become apparent. The representation of complex numbers in this way is called an Argand diagram.

If a complex number is represented as $x+yi\,$, its polar coordinates are given by:

$r={\sqrt {x^{2}+y^{2}}}\,$

$\theta =\tan ^{-1}{\frac {y}{x}}\,$

Transforming the other way, we have:

$x=r\cos \theta \,$

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The radial distance from the origin, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\,} , is called the modulus. It is the complex equivalent of the absolute value for real numbers. It is zero if and only if the complex number is zero.

The angle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta\,} , is called the phase. (Some older books refer to it as the argument.) It is zero for positive real numbers, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi\,} radians (180 degrees) for negative ones.

The multiplication of complex numbers takes a particularly interesting and useful form when represented this way. If two complex numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1\,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2\,} are represented in modulus/phase form, we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 = r_1 (\cos\theta_1 + i\sin\theta_1)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2 = r_2 (\cos\theta_2 + i\sin\theta_2)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 * z_2 = r_1 * r_2 * ((\cos\theta_1 \cos\theta_2 - \sin\theta_1 \sin\theta_2) + i (\sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2))\,}

But that is just the addition rule for sines and cosines!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 * z_2 = r_1 * r_2 * (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))\,}

So the rule for multiplying complex numbers on the Argand diagram is just:

Multiply the moduli.

Add the phases.

It is a common belief that complex numbers have a weaker connection to physical reality than real numbers. Observables in Physics for example weight, energy, pressure etc. are usually represented as real numbers, and the SI system of units relies on real numbers. However, the transformation between a SI base unit, e.g. an inductance/capacitance/resistance value and a complex impedance is arbritrary and set by convention, and the "natural" representation depends on the measurement method. As a matter of fact, a number of measurement devices (network analysers, lock in amplifiers) directly output real and imaginary component (where the imaginary component is obviously a real voltage/current value). Also, the index of refraction is often expressed as a complex number whose imaginary component indicates absorption loss as light propagates through the medium.

Polar notation

The complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+bi} can also be written in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho e^{i\theta}} , where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^2=a^2+b^2} is the square of the number's magnitude

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If a line is drawn on the complex plane (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} and the angle it makes to the real (horizontal) axis will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} . This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\theta}} is equivalent to an anticlockwise rotation through an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} in the complex plane.

Complex Numbers as Matrices

The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form

[a -b]

[b a],

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+bi} mapping as a function f to the above matrix. We can see that F and F' are isomorphic because: The function f is clearly 1-to-1 and onto, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x+y)=f(x)+f(y)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x*y)=f(x)*f(y)} .

See Also

• Arithmetic with complex number