Abacas

Introduction to Numbers

real : rational : irrational : transcendental : imaginary : integers : squares : sums


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Types of Number

All the numbers that people use in normal day-to-day activities are called Real Numbers. The first Real Numbers that you learn about are the positive integers (1, 2, 3, 4, etc). Next come fractions (1/2, 2/3, 1/4, etc). The integers are really forms of fractions (1/1, 2/1, 3/1, etc). There are also negative numbers. These are written with a minus sign in front of them (-1, -3/4, etc...).

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:

Number
in a/b Format
In Decimal
Format
77.0
1/40.25
-2/11-0.181818...

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 (√) 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:

Surd Format Decimal Format
√2 1.4142135...
1 / √5 0.4472136...
4√3 / 3√7 0.6879881...

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 π (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

In the essay, Pascal's Triangle, we looked at Newton's Binomial Series. This series was often infinite, sometimes convergent, sometimes divergent. The Binomial Series is by no means the only infinite series in mathematics. There are many others.

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.

Exp Series

This is a nice regular series. Each term is of the form xr / 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.

Value of x Value of Exp(x)
-10 0.00004540...
-5 0.00673793...
-1 0.36787944...
0 1
1 2.71828183...
5 148.413...
10 22026.46...
15 3269017.372...

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 π. It crops up in Probability Theory, Statistics, Trigonometry, Logarithms, the building of suspension bridges, as well as growth and decay. Along with π, e crops up in many physics formulas.

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


Imaginary Numbers

Up to now we have been discussing Real Numbers. There are also 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:

What is the square root of -1?

No real number when multiplied by itself gives -1! However, √-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.

i = √-1 by definition
i2 = i × i = -1
i3 = i × i × i = i × (-1) = -i
i4 = i2 × i2 = -1× -1 = 1
i5 = i × i4 = i × 1 = i

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

2 + i
-3 + 2i
1 - 5i

Numbers like these are called Complex Numbers.

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:

1
-1/2 + (√3/2)i
-1/2 - (√3/2)i

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.


A Strange Relationship

I have just introduced three strange numbers, π, e and i. Quite unexpectedly there is a simple yet elegant relationship between them.

e= -1

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.


The Integers

Integers are whole numbers. These may seem a little dull after the exotic numbers discussed above but there is a lot more to them than first appears. The integers beginning with 1 are called the Natural Numbers.

Sum of Natural Numbers

The sum of the natural numbers is the value obtained when a selection of the numbers (beginning from 1) are added together. For example the sum of the first 5 natural numbers is 15 because:

1 + 2 + 3 + 4 + 5 = 15

If the sum of a longer number sequence is required (say the sum of the first 100 numbers) this process can become tedious. There is, luckilly, a formula for summing the first n natural numbers. The symbol Σn (the Greek capital Sigma followed by n) is used to denote the sum of the first n natural numbers. The formula can then be written:

Σn = n × (n + 1) / 2

Using it for the first five natural numbers we obtain: 5 × (5 + 1) / 2 = 5 × 6 / 2 = 30 / 2 = 15.

We can now use it to find the sum of the first 100 natural numbers by setting n as 100 to give:

100 × (100 + 1) / 2 = 100 × 101 / 2 = 10100 / 2 = 5050. Easy!

Odd and Even Numbers

Numbers are even if they are exactly divisible by 2. Examples of even numbers:

2 18 24 40 76 98 154

Two even numbers added, subtracted or multiplied will always give another even number. For example: 2 + 6 = 8; 40 - 22 = 18; 12 × 8 = 72.

Numbers are odd if they are not exactly divisible by 2. Examples of odd numbers:

1 19 21 49 77 91 153

Two odd numbers added or subtracted will also always give an even number. For example: 3 + 7 = 10; 21 - 3 = 18.

Two odd numbers multiplied will always give an odd number. For example: 11 × 7 = 77; 3 × 21 = 63.

Square Numbers

Square Numbers are integers that are the square of smaller integers. For example 4 is 22 and 9 is 32 so the first few square numbers are:

1 4 9 16 25 36 49 64 81 100 121 144 169

Note that the sequence of square numbers alternates between odd and even.

Another interesting fact is that this series of square numbers can be produced by adding successive odd numbers. For exqample, the sum of the first two odd numbers (1, 3) is 1 + 3 = 4 (a square number). The sum of the first three odd numbers (1, 3, 5) is 9 (another sqauare). This is shown in the table below:

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

Sum of Square Numbers

The sum of the first four squares is 30 because:

12 + 22 + 32 + 42= 1 + 4 + 9 + 16 = 30

There is a formula for allowing us to sum the squares of the first n natural numbers. The symbol Σn2 is used to denote the sum of the first n square numbers. The formula can be written:

Σn2 = n × (n + 1) × (2n + 1) / 6

Using it for the first four natural numbers we obtain: 4 × (4 + 1) × (2 × 4 + 1) / 6 = 4 × 5 × 9 / 6 = 30.

We can now use it to find the sum of the first 100 square numbers by setting n as 100 to give:

100 × (100 + 1) × (2 × 100 + 1) / 6 = 100 × 101 × 201 / 6 = 338350.

Perfect Numbers

These are numbers whos factors (apart from itself) add up to the number itself. The smallest Perfect Number is 6. Its factors (numbers that divide into 6 exactly) are 1, 2, and 3 (excluding 6 itself). The number 6 is perfect because:

1 + 2 + 3 = 6

The next Perfect Number is 28. Its factors are 1, 2, 4, 7, 14 (and 28). And

1 + 2 + 4 + 7 + 14 = 28

The first two Perfect Numbers have been known since ancient times. The next two are 496 and 8,128 and they were discovered by Nichomachus in Alexandria (Egypt). The fifth was not discovered until modern times and has the value, 33,550,336. In the year 2016, 49 of these numbers are known with the largest containing 44,677,235 digits. None is odd but there is no proof that perfect numbers must be even.

Apart from 6, every Perfect Number can be written as a series of consecutive odd cubes:

13 + 33 = 28

13 + 33 + 53 + 73 = 496

Amicable Numbers

Amicable Numbers are pairs of numbers. The factors of one number (apart from itself) add up to the other. For thousands of years the only pair known were 220 and 284.

The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 and these numbers when added give 284.

The factors of 284 are 1, 2, 4, 71, 142 and when these are added give 220.

The next pair of Amicable Numbers were found in 1636 (17,296 and 18,416). Two years later the pair 9,363,584 and 9,437,056 were found. 60 of these numbers were known by 1760. The overlooked pair, 1184 and 1210, was discovered in 1866 by a 16 year old mathematician in Italy.

40,871,144 Amicable Numbers were known by the end of 2015.

Prime Numbers

Prime Numbers are numbers with no factors, apart from 1 and themselves. The first few are:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ...

All other number are called Composite Numbers. These can all be made up of products of Prime Numbers. For example,

60 = 2 × 2 × 3 × 5

110 = 2 × 5 × 11

360 = 2 × 2 × 2 × 3 × 3 × 5

Euclid proved that the number of Prime Numbers was infinite in the 3rd Century BC. Formulas are known that produce some Prime Numbers but no definitive formula is known that produces all the Prime Numbers exclusively. The largest prime Number known in 2016 has 22,338,618 digits.

Prime Numbers are used in cryptography. When two large Prime Numbers are multiplied together, it is very difficult to factorise the product and regain the original Prime Numbers. This is the basis of secure codes and ciphers.

Prime Numbers have given rise to Skewes' Number, the largest number ever to have come out of a mathematical proof. There is a formula for predicting the number of Primes up to a given number. The number of Primes found is consistantly less than what the formula predicts. Mathematicians have proved that when a high enough number is reached, the number of Primes should exceed the prediction. This switch should occur somewhere below Skewes' Number. In other words, by the time Skewes' Number is reached the number of Primes will be greater than predicted. The value of this number is:

10101034

Skewes' Number is far, far larger than the number of all the particles in the observable Universe.

That must be a good place to stop!

© 2000, 2016 KryssTal


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