Key Competencies Kit
for Facing Lifelong Learning
 |
This Project has been funded with support from the European Commission. This communication reflects the views only of the author, and the Commission can not be held responsible for any use which may be made of the information contained therein. |
 |
Upon completion of this unit you shall be able to:
The natural numbers had their origins in the words used to count things, beginning with the number 1. A much later advance in abstraction was the development of the idea of zero as a number with its own numeral.
The set of natural numbers is denoted by 
and it contains all following numbers:
The basic operation acting on this set of numbers is addition, denoted by the “+” symbol. Every natural number can be obtained from the preceding natural number by adding 1, except the first natural number 0.
A few examples of additions:
Adding two numbers is practically counting together two sets of objects. The rules of addition are based on the additions between the first 10 natural numbers, also known as digits:
Every addition between numbers larger than these ten numbers is based on operations between the ten digits.
| 1 | 8 |
+ |
2 |
3 |
|
1+2+1=4 |
8+3=11 |
|
4 |
1 |
|
The red digit is “transported” to the left and used in the addition of the digits to the left.
The second basic operation on the set of natural numbers is subtraction, which is the opposite of addition, taking away a number of objects and counting the remaining items. For natural numbers, the number of objects taken must always be less or equal than the initial number of objects.
Let’s look at an example:
As we can see, if we add up the result to the number subtracted, the result is the initial number:
and this is the usual method of verification in the case of subtraction. If we want to subtract numbers that are not digits, a similar rule applies to the addition case. This time, instead of “transporting” a digit to the left, we take one away if the digit subtracted is larger than the one it is subtracted from. Let’s take a look at the following examples.
The first example is pretty straightforward, and all digits can be subtracted easily. The second looks a bit different, and is a good case of the rule described above.
5 |
3 |
- |
2 |
1 |
|
5-2=3 |
3-1=2 |
|
3 |
2 |
|
| 5 | 1 |
- |
2 |
8 |
|
(5-1)-2=2 |
11-8=3 |
|
2 |
3 |
|
Since we took away one unit from 5, the subtraction to the left will be 4-2=2.
Next, we will make the transition to the next operation between natural numbers, that is, multiplication, denoted by “×” or “∙”. Multiplication is, simply put, repeated addition of the same number. For instance, instead of writing
we can write
that is, five times four is twenty.
When multiplying numbers with more than one digit, same rules apply as for the addition: only the last digit is written, the rest is “transported” to the left and added to the result of the multiplication. Each of the digits is multiplied with the digits below, starting from left to right. Here’s an example, where the second number has just on digit:
| 1 | 3 |
× |
|
7 |
|
1×7+2=9 |
3×7=21 |
|
9 |
1 |
|
If the second number has two digits, the same operations are repeated, and the result is shifted one position to the left, as in the example below:
|
1 |
3 |
× |
|
2 |
7 |
|
|
1×7+2=9 |
3×7=21 |
|
1×2=2 |
3×2=6 |
|
|
2+1=3 |
9+6=15 |
1 |
|
We should note that every number multiplied by 0 is 0, and every number multiplied by 1 is the number itself. Multiplying a number by 10 adds a final 0.
The complete multiplication table for numbers 1 to 10 is given below:
1 x 1 = 1 |
1 x 2 = 2 |
1 x 3 = 3 |
1 x 4 = 4 |
1 x 5 = 5 |
1 x 6 = 6 |
1 x 7 = 7 |
1 x 8 = 8 |
1 x 9 = 9 |
As for the “+” operation, there is an operation that opposes multiplication, and that is the division, denoted by “÷” or “:”. The connection between the two operations is visible in the following sentence:
If 3×5=15, then 15:5=3.
That is, if we multiply a number by 5, and then we divide the result by 5, we get the number we started with. The following examples show how to divide two numbers that have more than one digit:
| 3 | 5 |
1 |
÷ |
3 |
|
|
3=3×1 |
|
|
|
1 |
1 |
7 |
= |
5 |
- |
|
|
|
|
|
3=3×1 |
|
|
|
|
|
|
2 |
1 |
- |
|
|
|
|
2 |
1 |
=3×7 |
|
|
|
|
= |
= |
|
|
|
|
In case we want to evaluate an expression more complicated, like the following
we need to establish an order in which all operations are executed. The natural order is to evaluate first the multiplications and divisions, and only then the additions and subtractions. The expression above is thus computed:
In order to group the operations in different modes, we can also make use of parenthesis, in which case we evaluate first what’s inside the parenthesis:
Parenthesis’ can be nested, and in this case we compute the innermost parenthesis and so on:
When working with parenthesis’ there are some basic rules that allow us to eliminate them during calculations, and that is the distributive property:
In general, the following rules apply, where 
are natural numbers:
In the following we shall illustrate the way natural numbers can be represented on a straight line.
0 1 2 3 4 5 6 7 a a+1
First we represent the number 0, then we choose a short segment which we call measurement unit, and position the following number, that is 1. Each of the following natural numbers is represented by moving to the right of the preceding number by one measurement unit. There is no limit to the right, as there is always a number greater than any natural number, obtained by adding 1.
One can define a relationship between two natural numbers, describing which one is greater or smaller, depending on the positions of the two numbers on the line representation.
If a number is situated to the left of a second number, it means it is smaller, otherwise, if it is positioned to the right, it is larger.
For instance 5 is greater than 3, and we write
In the same manner we can assert that 1 is smaller than 7, and write
Another way of describing the relation between two numbers is the following: the natural number a is smaller than b if there is a natural number c such that
that is we must add something to a to reach b.
We noticed that when subtracting, the first number must always be larger than the second number. If adding means moving to the right on the representation axis, then subtracting means moving to the left. If we move to the left with a number of units smaller than the current position we obtain another natural number, which is the subtraction’s result.
What would happen if we moved more to the left? It is clear that we can extend the line to the left of 0, but there are no natural numbers there, so we must define some new numbers. These numbers are similar to the natural numbers, except they have a minus to the left. For instance, -1 is the number obtained by moving one unit to the left of 0, the same way as 1 is the number obtained from zero by moving one unit to the right.
-4 -3 -2 -1 0 1 2 3
In terms of basic operations we write the former as
In a similar manner we notice that
We can also define the subtraction between any two natural numbers as the subtraction between the larger and the smaller number, in that order, with the sign of the larger. For instance:
since 17 is greater than 15, and has the minus sign.
With these rules the operations defined for natural numbers can be defined for signed numbers also, taking into account the following rules for sign:
In general
for any natural number a.
When multiplying, the following table of sign operations must be used:
| × | + |
- |
+ |
+ |
- |
- |
- |
+ |
The same rules apply when dividing two signed numbers. Now that we have defined the signed numbers, we can group them with the natural numbers in one single set of numbers, since all operations can be performed between them. This set is the set of integers, denoted by 
where
The relations between integers are similar to those between natural numbers, that is a number is lower than another if it is positioned to its left. As general rules, any negative (signed) number is smaller than any natural (positive, unsigned) number, including 0.
For two signed numbers (negative), the relationship is the following: a number 
is smaller than a number 
if the first number without the sign, 
is greater than the second number without the sign, 
.
Let us note that if we add a natural number with its signed correspondent we get 0:
