Any numbers that can be written in the form a/b where a and b are whole numbers are called Rational Numbers. Rational Numbers can also be written as decimals. These decimals are either finite, or infinite and repetitive. Let me give you some examples of Rational Numbers:

Many early mathematicians in the ancient world believed that Rational Numbers were the only numbers that existed. This is not so.

Any number that cannot be expressed as a fraction a/b is called an Irrational Number. Some Irrational Numbers can be expressed with root signs (Ö) in them. Numbers with root signs in them are called surds. Surds can be also written as decimals but the decimal is always infinite and never repeats. Examples of Irrational Numbers:

These are still not all the numbers that exist.

There are Irrational Numbers which have decimals that are infinite and non repeating but cannot be written as surds. The best example of this type of number is p (pronounced pi and used extensively in trigonometry) which has a value beginning with 3.14159... but cannot be written either as a fraction or a surd. These numbers are called Transcendental Numbers.

Numbers that go on for ever? Infinite decimals? What next?

Infinite Series

I would like to introduce an infinite series to you now. I am going to give this series a label, EXP(x) for now and list out its expansion below.

This is a nice regular series. Each term is of the form x^r / r! (where x⁰ = 1 and 0! = 1). It is an infinite series, but amazingly, it converges for all values of x. Furthermore, the values it gives are all Transcendental (except EXP(0) which is equal to 1).

For all values of x, EXP(x) is positive. When x is less than 0, EXP(x) is less than 1. As x increases, EXP(x) gets bigger very rapidly. Below are a few values of EXP(x) to show this behaviour.

These numbers can be plotted on a graph. The graph is called the Exponential Curve. It turns up when you study uncontrolled growth. For example, if an amoeba is put into an environment full of food and allowed to reproduce, the numbers increase exponentially, in other words, they grow as EXP(x) grows. This is Exponential Growth.

If radioactive elements are studied, the number of radioactive atoms decreases with time as the nuclei change and break down. The curve of this decay is the mirror image of the Exponential Growth curve. It is called Exponential Decay and is represented by EXP(-x). If you put -x for every x in the expansion of EXP(x) to get a series for EXP(-x). This series is also infinite and convergent for all values of x.

The Exponential Curve occurs in music also. The lengths of piano strings increase in length as the piano note changes. This increase is along a portion of the Exponential Curve.

The value of EXP(1), 2.71828183... is called e. It is one of the most important Transcendental Numbers along with p. It crops up in Probability Theory, Statistics, Trigonometry, Logarithms, the building of suspension bridges, as well as growth and decay. Along with p, e crops up in many physics formulas.

The above expansion that I've called EXP(x) is actually an expansion of e^x. That is why e⁰ = 1 and e^-x = 1 / e^x.

Imaginary Numbers

A quick question, what is the square root of 4? In other words, what number when multiplied by itself gives 4. The answer is, of course, 2 because 2 × 2 = 4. But if you remember your multiplication, -2 × -2 is also 4 (because two negatives make a positive). So we can say that 4 has two square roots, +2 and -2. Every positive number has two square roots.

Every time you square a number you end up with a positive number. So, with that fact in mind:

No real number when multiplied by itself gives -1! However, Ö-1 occurs in many engineering and electrical problems. Mathematicians have invented the answer. Without the square root of -1, a lot of mathematics wouldn't make sense.

The number that has been invented to be the square root of -1 is called i (for imaginary). In fact it is no more imaginary than any other number but the name has stuck. So let us have a look at some of the properties of this strange number.

It is possible to have a combination of real and imaginary numbers. Here are some examples.

Earlier, I said that all numbers have TWO square roots. Another general rule is that all numbers have THREE cube roots. The three cube roots of 1 are:

The first root is obvious because 1 × 1 × 1 = 1. The second pair of roots are complex numbers.

Try multiplying these out using algebra, remembering that i² = -1.

Finally

Isn't that wonderful? A transcendental number raised to a power of an imaginary number multiplied by another transcendental number gives something as simple as -1. The proof is trigonometrical.

Mathematics is always full of surprises.

© 2000 Kryss Katsiavriades

Related Pages

Trigonometry
Trigonometric Functions (Sines, Cosines, Tangents) and their uses, relations and series.

The Binomial Theorem
A series devised by Isaac Newton that is used for calculations. More on indices: roots and powers. Factorials. Combinations.

Look At Logarithms
Index and base. Logarithms defined. Base 10 and base e. Uses of logarithms in calculations. Series for logarithms.

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