5.3 Complex Numbers and Functions: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Mathematics makes frequent use of sets of numbers: collections of numbers that share a common property. These sets are nested inside one another, forming a hierarchy from the most basic counting numbers to the most general. Understanding this hierarchy, and the standard notation used to express it, is an important part of mathematical literacy.
- The natural numbers are the counting numbers: [latex]\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}[/latex].
The name natural reflects the idea that these are the numbers that arise naturally when we count everyday objects; they are the first numbers that human beings encountered and used. The symbol [latex]\mathbb{N}[/latex] stands for Natural.
Some mathematicians include zero in this set; others do not. This is a matter of convention that varies between textbooks and countries.
- The integers extend the natural numbers to include zero and all negative whole numbers: [latex]\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}[/latex].
The name integer comes from the Latin word integer, meaning whole or untouched; integers are whole numbers, i.e. with no fractional or decimal part. The symbol [latex]\mathbb{Z}[/latex], however, comes from the German Zahlen, meaning numbers, reflecting the strong tradition of German mathematics in the nineteenth century. Every natural number is an integer, so [latex]\mathbb{N} \subset \mathbb{Z}[/latex].
- The rational numbers [latex]\mathbb{Q}[/latex] are the numbers that can be expressed as a fraction [latex]\dfrac{p}{q}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q \neq 0[/latex].
The name rational comes from the word ratio; a rational number is one that can be expressed as a ratio of two integers. Note that rational here does not mean sensible or logical in the general English sense; it refers specifically to the idea of a ratio. The symbol [latex]\mathbb{Q}[/latex] stands for Quotient. Every integer is rational, since any integer [latex]n[/latex] can be written as [latex]\dfrac{n}{1}[/latex], so [latex]\mathbb{Z} \subset \mathbb{Q}[/latex].
- The real numbers [latex]\mathbb{R}[/latex] include all rational numbers together with all irrational numbers (numbers that cannot be expressed as a fraction of two integers).
The name real denotes that these are all the numbers known to exist at that time. The symbol [latex]\mathbb{R}[/latex] stands for Real. Every rational number corresponds to a unique point on the real number line, so [latex]\mathbb{Q} \subset \mathbb{R}[/latex].
- This brings us to complex numbers [latex]\mathbb{C}[/latex], the topic of this chapter. Note that [latex]\mathbb{C}[/latex] stands for Complex. As the real line is part of the complex plane, we have [latex]\mathbb{R} \subset \mathbb{C}[/latex].
The sets described above are nested inside one another as follows: [latex]\mathbb{N} \;\subset\; \mathbb{Z} \;\subset\; \mathbb{Q} \;\subset\; \mathbb{R} \;\subset\; \mathbb{C}[/latex].
We define the imaginary unit as
\[i = \sqrt {−1}.\]
This is read out as i is equal to the square root of minus one.
Given two real numbers [latex]x[/latex] and [latex]y[/latex], we define complex numbers to be numbers of the form
\[z = x + iy,\]
where [latex]x[/latex] is called the real part and [latex]y[/latex] the imaginary part of [latex]z[/latex].
This is read out as z is equal to x plus i y.
Mathematics Focus
The form [latex]z = x + iy[/latex] is often called the rectangular form of complex numbers, which emphasises the use of a rectangular coordinate system to represent complex numbers in the plane. It is also called the Cartesian form, named after the French mathematician René Descartes. This representation of complex numbers extends the real number line into a two dimensional plane called, the complex plane, where the x-axis is the real part and the y-axis the imaginary part. This visual representation is also called an Argand diagram, named after the mathematician Jean-Robert Argand.
Note that sometimes in engineering the letter [latex]j[/latex] is used to denote the imaginary unit instead of [latex]i[/latex]. The primary reason for this is notational conflict. In electrical engineering, the letter [latex]i[/latex] (and uppercase [latex]I[/latex]) is already used to represent electric current, one of the fundamental quantities in circuit analysis. So using [latex]i[/latex] for both current and the imaginary unit would create ambiguity.
Language Focus 1
Avoid confusing the mathematical meaning of the word complex with the general meaning of the word. The general meaning of the word complex means something difficult to understand or find an answer to, because of having many different parts. For example, we can say This is a complex problem that requires careful analysis.
We need to be careful to distinguish which meaning, general or mathematical, is intended. In a mathematical context a complex exponential, means an exponential that has a complex number as its power; not an exponential that is difficult to understand. Similarly, complex logarithms refers to logarithmic functions defined on complex numbers, not logarithms that are complicated.
To avoid ambiguity, it is best to use precise language. Rather than saying a complex function (which may be interpreted either way), it is better to specify a function of a complex variable or a complex-valued function to make clear that you mean the mathematical sense, not difficulty level. This is especially important when communicating mathematical ideas in writing or to non-specialists who may interpret the general meaning first.
Finally, it is worth noting that the term imaginary numbers in mathematics is historical rather than descriptive. When imaginary numbers were first discovered, mathematicians found them puzzling and philosophically problematic, so they named them imaginary, reflecting the scepticism at that time. Complex numbers, being combinations of real and imaginary parts, inherited this terminology. However, modern mathematics treats complex numbers as fundamental and concrete objects, not as difficult or obscure concepts. So these names are now somewhat misleading.
Given a complex number [latex]z = x + iy[/latex], the complex conjugate of [latex]z[/latex] is the complex number denoted by
\[\bar {z} = x − iy.\]
This is read out as z bar is equal to x minus i y.
Given a complex number [latex]z = x + iy[/latex], we define its modulus, magnitude or absolute value to be
\[|z| = \sqrt{x^2 +y^2},\]
which is read out as the modulus of z is equal to, the square root of, x squared plus y squared.
Furthermore, we define its argument to be
\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]
which is read out as theta is equal to arctan of y over x.
Given a complex number [latex]z[/latex] with modulus [latex]r[/latex] and argument [latex]\theta[/latex], we can also write it in polar form (also called trigonometric form)
\[z = r(\cos \theta + i \sin \theta),\]
which is read out as z is equal to r times, cos theta plus i sine theta.
Note that the modulus of a complex number [latex]|z| = \sqrt{x^2 +y^2}[/latex] gives us the distance of the complex number from the origin. Furthermore, the argument is the angle that the line from the origin to the complex number makes with the positive real axis, measured anti-clockwise. It is important to note that the formula for the argument [latex]\theta = \tan^{-1}\left(\frac{y}{x}\right)[/latex] only works when both [latex]x>0[/latex] and [latex]y>0[/latex] (that is, in the first quadrant).
To illustrate this, consider the following example. Given [latex]x>0[/latex] and [latex]y>0[/latex], the complex number [latex]z=x+iy[/latex] is in the first quadrant, and the complex number [latex]w=-x-iy[/latex] is in the third quadrant. Clearly, [latex]z[/latex] and [latex]w[/latex] have different arguments. However, [latex]\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-y}{-x}\right)[/latex], so the formula cannot distinguish between these two complex numbers. This is why adjustments are needed for complex numbers outside the first quadrant:
\[ \theta = \begin{cases}
\arctan\left(\frac{y}{x}\right) & \text{if } x > 0 \\
\arctan\left(\frac{y}{x}\right) + \pi & \text{if } x < 0 \text{ and } y \geq 0 \\
\arctan\left(\frac{y}{x}\right) - \pi & \text{if } x < 0 \text{ and } y < 0 \\
\frac{\pi}{2} & \text{if } x = 0 \text{ and } y > 0 \\
-\frac{\pi}{2} & \text{if } x = 0 \text{ and } y < 0 \\
\text{undefined} & \text{if } x = 0 \text{ and } y = 0.
\end{cases} \]
Language Focus 2
The polar form of complex numbers is called this because it describes position relative to a central pole: the origin of the complex plane. Rather than specifying a complex number by its Cartesian form ([latex]x[/latex] and [latex]y[/latex]), the polar form uses distance and angle from the origin ([latex]r[/latex] and [latex]\theta[/latex], respectively).
In more advanced complex analysis, the term pole takes on a different meaning. It refers to a specific type of singularity of a complex-valued function. A singularity is a point where a function is not defined or behaves unusually. A pole is specifically, a singularity where the function approaches infinity.
A famous identity in complex numbers is Euler's identity/formula (pronounced 'oiler'), which gives us the key relationship between trigonometric functions and the complex exponential
\[e^{i \theta} = \cos \theta + i \sin \theta.\]
This is read out as e to the power of i theta is equal to, cos theta plus, i sine theta
We also have De Moivre’s theorem (pronounced 'de mwarv') which allows us to find powers and roots of complex numbers easily. It states:
\[(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta),\]
which is read as cos theta plus i sine theta all to the power n is equal to, cos of n theta plus, i sine of n theta.
An important set of functions in complex analysis is the hyperbolic functions [latex]\sinh x[/latex], [latex]\cosh x[/latex] and [latex]\tanh x[/latex]. These are defined as follows:
\[\sinh x = \frac {e^x - e^{-x}} {2}, \quad \cosh x = \frac {e^x + e^{-x}} {2}, \quad \tanh x = \frac {\sinh x} {\cosh x}.\]
- [latex]\sinh[/latex] is called the hyperbolic sine, which is read out as shine or sinch.
- [latex]\cosh[/latex] is the hyperbolic cosine, read out as cosh,
- and [latex]\tanh[/latex] is the hyperbolic tangent, and it is read out as than or tanch.
Using Euler's formula we can obtain
\[ \cos(iy) = \cosh y \quad\text{ and } \quad \sin(iy) = i\sinh y.\]
The English language allows several ways of pre-modifying nouns. By pre-modifying we mean 'putting some language before a noun to modify or qualify it in some way'. There are four standard ways of pre-modifying nouns, illustrated in the examples below:
Example set 1
- Mathematical knowledge (Adjective + Noun) adjective mathematical modifies the noun knowledge.
- Mathematics exam (Noun + Noun) noun mathematics modifies the noun exam.
- Mathematics' history (Possessive/'apostrophe-s' Noun + Noun) possessive noun form mathematics' modifies the noun history.
- The mathematics of complex numbers (Noun + of + Noun/Noun phrase) noun mathematics + of + noun phrase complex numbers (which is itself an adjective + noun string).
Example set 2
- difficult mathematics/the easiest mathematics/computational mathematics: (Adjective + Noun) adjectives: difficult/the easiest/computational, modify the noun mathematics.
- undergraduate mathematics/pen-and-paper mathematics (Noun + Noun) nouns undergraduate/pen-and-paper modify the noun mathematics..
- Euclid's mathematics/this morning's mathematics (Possessive/'apostrophe-s' Noun + Noun) possessive noun forms Euclid's/this morning's modify the noun mathematics.
- the high point of mathematics/the history of mathematics (Noun + of + Noun/Noun phrase) noun phrases the high point of /the history of modify the noun mathematics.
The grammatical rules behind these four pre-modifying structures are complex, and we will not cover them all here. A complementary approach to learning grammatical rules is simply to notice, record, and learn individual examples of these patterns. With this learning approach in mind, consider identifying examples of each of the four patterns as you read mathematical texts, recording them in a notebook, and learning some of them.
Some examples of each from the text in this section is given below:
- Adjective + Noun: imaginary unit, rectangular form, trigonometric forms.
- Noun + Noun: Argand diagram, mathematics focus.
- Possessive Noun + Noun: Euclid's identity.
- Noun + of + Noun: modulus of z, arctan of y over x.
For more practice on pre-modifying nouns, see Language Focus 3 in Section 8.3.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
