2. Number Properties

Real and Imaginary Numbers

Numbers

Real Numbers

Imaginary Numbers

But what are these "un-real numbers"? Imaginary numbers involve the square root of a negative number. We meet imaginary numbers later in the Complex Numbers chapter, and see how they are used in electronics.

Rational and Irrational Numbers

A real number is either rational or irrational.

Numbers

Real Numbers

Rational Numbers

Irrational Numbers

Imaginary Numbers

A number is rational if it can be expressed in the form , where p and q are both integers.

For example,

is rational, since 3 and 5 are integers.

Other examples of rational numbers are:

(it is equal to 5/3),

7,

0,

8.25

Rational numbers either have a

terminating decimal (that is, the decimal stops, e.g. 7.625) or
repeating decimal (e.g. 2/99 = 0.02020202... is a rational number).

Irrational numbers

π = 3.14159... is irrational, since it cannot be expressed in the form .

(BTW, π does NOT equal 22/7. This is just a handy approximation.)

An irrational number has a never ending (and never repeating) decimal.

Other irrational numbers are:

e = 2.781 828 ... (For an explanation of e, see Natural Logarithms)

√3 = 1.732 05...

Φ = 1.618 033... ("phi") (see Math of Beauty for some interesting examples)

Summary of Number Types

Numbers

Real Numbers

Rational Numbers
Can be expressed as p/q
( p, q integers)

Includes:
Fractions, decimals, whole numbers

Integers
Negative (...,-3,-2,-1)
0
Positive (1,2,3,...)

Prime numbers, composite numbers

Irrational Numbers
Cannot be expressed as
p/q
( p, q integers)

Examples:
π = 3.141592...,
e = 2.7818...,
√3 = 1.7321...

Imaginary Numbers

Complex numbers have a real part and an imaginary part.

Read more in the Complex Numbers chapter.

Example: 7 + 2j

Prime Numbers

A prime number is a positive integer which has exactly two factors, 1 and itself. The first primes are:

2, 3, 5, 7, 11, 13, ...

The other positive integers are composite and they have 3 or more factors. For example,

9 has factors 1, 3 and 9 (3 factors), so is composite.

12 has factors 1, 2, 3, 4, 6, 12 (6 factors), so is composite.

The first composite numbers are:

4, 6, 8, 9, 10, 12, ...

So the positive integers consists of the following:

Positive Integers

Not prime,
not composite
1

Prime Numbers

2, 3, 5, 7, ...

Composite Numbers

4, 6, 8, 9, ...

The number 1 is neither prime nor composite, since it has exactly one factor.

Who cares about Prime Numbers?

A lot of energy has been expended by mathematicians (ever since the time of Pythagoras) on studying prime numbers.

Recently, a very important branch of mathematics to emerge is encryption, where sensitive information is hidden from others when it is transmitted electronically (e.g. when we send credit card numbers over the Internet or by mobile phone).

Encryption works by coding the message using very large prime numbers. The device receiving the message decodes the message using the same very large prime numbers. The larger the numbers used, the better the encryption.

The largest known prime currently is