** Introduce concepts in a simple context and
then generalize them in such a way that rules and facts that are true
in the simple context remain true in the more general context.**

1, 2, 3, 4, 5, ....

We can **add** or **multiply** two natural numbers and obtain
another natural number. However, the **difference** or the
**ratio** of two natural numbers is not always a natural number.
For example, **5-2** and **12/3** are natural numbers, but
**3-5** and **3/12** are not.

- a + b = b + a The commutative law of addition.
- a * b = b * a The commutative law of multiplication.
- (a + b) + c = a + (b + c) The associative law of addition.
- (a * b) * c = a * (b * c) The associative law of multiplication.
- (a + b) * c = a * c + b * c The distributive law.

These laws are true for all and any natural numbers **a**,
**b**, **c**. Actually they hold for all numbers we will
encounter, but the whole point of building the number system is that
we do this in such a way that the above rules remain true.

The **distributive law** connects multiplication and addition and is
the most crucial, and the most misused and misunderstood law in the
above list.

..., -3, -2, -1, 0, 1, 2, 3, ....

The set of integers can be thought of as having been obtained by expanding the set of natural numbers to make subtraction always possible. Of course we have to define what we mean by the sum, difference, product, and ratio of two integers. This is done in the familiar way with the guiding principle being that the laws listed above remain valid. (That principle for example leads to the requirement that the product of two negative numbers is positive.)

The sum, difference, and product of two integers is always an integer. The ratio, however, is not, which gives rise to the next level:

The result of adding, subtracting, multiplying, or dividing
rational numbers (so long as we don't divide by zero) is another
rational number. We say that the set of rational numbers is
**closed** under addition, subtraction, multiplication, and
division.

Of course, after extending the integers to the rational numbers, we again need to define what we mean by the sum, difference, product, and ratio of two rational numbers. This is discussed in detail on the page on fractions.

It can be shown
that there is no rational number whose square equals 2. Hence the
number system needs to be extended once more. For our purposes a
real number is a decimal expression
whose digits may or may not terminate or repeat. It can also be shown
that a real number is rational if and only if its digit repeat or
terminate. Real numbers that aren't rational are **irrational**

Each number set contains the number sets it surrounds. For example the set of rational numbers contains all natural numbers (and all integers). The Figure also indicates which operations are possible in each set. For example, we can add, subtract, and multiply integers, and the result will be an integer. (The result of dividing two integers is not always an integer, for example 5/2 is not.) The examples given are in the set shown, but not in a smaller set. For example, 2/3 is a rational number. It's also a real number, but it's not an integer, and it's not a natural number.