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Section B.18 Obtaining the Complex Numbers from the Reals

By now, the following discussion should be transparent. The complex number system \(\complexes\) is just the cartesian product \(\reals\times\reals\) with

  1. \((a,b) = (c,d)\) in \(\complexes\) if and only if \(a=c\) and \(b=d\) in \(\reals\text{.}\)

  2. \((a,b)+(c,d)=(a+c,b+d)\text{.}\)

  3. \((a,b)(c,d)=(ac-bd, ad+bc)\text{.}\)

Now the complex numbers of the form \((a,0)\) behave just like real numbers, so is natural to say that the complex number system contains the real number system. Also, note that \((0,1)^2=(0,1)(0,1)=(-1,0)\text{,}\) i.e., the complex number \((0,1)\) has the property that its square is the complex number behaving like the real number \(-1\text{.}\) So it is convenient to use a special symbol like \(i\) for this very special complex number and note that \(i^2=-1\text{.}\)

With this beginning, it is straightforward to develop all the familiar properties of the complex number system.

Subsection B.18.1 Decimal Representation of Real Numbers

Every real number has a decimal expansion—although the number of digits after the decimal point may be infinite. A rational number \(q=m/m\) from \(\rats\) has an expansion in which a certain block of digits repeats indefinitely. For example,

\begin{equation*} \frac{2859}{35} = 81.6857142857142857142857142857142857142857142\dots \end{equation*}

In this case, the block \(857142\) of size \(6\) is repeated forever.

Certain rational numbers have terminating decimal expansions. For example, we know that \(385/8= 48.125\text{.}\) If we chose to do so, we could write this instead as an infinite decimal by appending trailing \(0\)'s, as a repeating block of size \(1\text{:}\)

\begin{equation*} \frac{385}{8} = 48.1250000000000000000000000000000000\dots \end{equation*}

On the other hand, we can also write the decimal expansion of \(385/8\) as

\begin{equation*} \frac{385}{8} = 48.12499999999999999999999999999999999\dots \end{equation*}

Here, we intend that the digit \(9\text{,}\) a block of size \(1\text{,}\) be repeated forever. Apart from this anomaly, the decimal expansion of real numbers is unique.

On the other hand, irrational numbers have non-repeating decimal expansions in which there is no block of repeating digits that repeats forever.

You know that \(\sqrt{2}\) is irrational. Here is the first part of its decimal expansion:

\begin{equation*} \sqrt{2} =1.41421356237309504880168872420969807856967187537694807317667973\dots \end{equation*}

An irrational number is said to be algebraic if it is the root of polynomial with integer coefficients; else it is said to be transcendental. For example, \(\sqrt{2}\) is algebraic since it is the root of the polynomial \(x^2-2\text{.}\)

Two other famous examples of irrational numbers are \(\pi\) and \(e\text{.}\) Here are their decimal expansions:

\begin{align*} \pi \amp =3.14159265358979323846264338327950288419716939937510582097494459\dots\\ \end{align*}

and

\begin{align*} e\amp=2.7182818284590452353602874713526624977572470936999595749669676277\dots \end{align*}

Both \(\pi\) and \(e\) are transcendental.

Example B.50.

Amanda and Bilal, both students at a nearby university, have been studying rational numbers that have large blocks of repeating digits in their decimal expansions. Amanda reports that she has found two positive integers \(m\) and \(n\) with \(n\lt 500\) for which the decimal expansion of the rational number \(m/n\) has a block of 1961 digits which repeats indefinitely. Not to be outdone, Bilal brags that he has found such a pair \(s\) and \(t\) of positive integers with \(t\lt 300\) for which the decimal expansion of \(s/t\) has a block of \(7643\) digits which repeats indefinitely. Bilal should be (politely) told to do his arithmetic more carefully, as there is no such pair of positive integers (Why?). On the other hand, Amanda may in fact be correct—although, if she has done her work with more attention to detail, she would have reported that the decimal expansion of \(m/n\) has a smaller block of repeating digits (Why?).

Let \(f\) be a function from \(\posints\) to \(X\text{.}\) For each \(n\in \posints\text{,}\) consider the decimal expansion(s) of the real number \(f(n)\text{.}\) Then choose a positive integer \(a_n\) so that (1) \(a_n\le 8\text{,}\) and (2) \(a_n\) is not the \(n^{th}\) digit after the decimal point in any decimal expansion of \(f(n)\text{.}\) Then the real number \(x\) whose decimal expansion is \(x=.a_1a_2a_3a_4a_5\dots\) is an element of \(X\) which is distinct from \(f(n)\text{,}\) for every \(n\in\posints\text{.}\) This shows that \(f\) is not a surjection.