Introduction
A natural integer is a positive and "without comma" number such as 1, 2, 3 ... and 0. The smallest element is 0 and there is no greater element: you can count without ever stopping.
Natural numbers are the most commonly used numbers in everyday life, they allow you to count a distinct number of objects (or units). For example, we can have 2 pens, 1 cat and a bill of 5 dollars in our pocket etc.
The set of natural whole numbers is noted N, it is a tiny part (even if it is infinite) of existing numbers. It is for example a set smaller than the one of the integers Z.
The theory stops there! We will now get to the heart of the matter. We will first see how to write whole numbers, then what mathematical operations mean and how to perform them.
Decomposing a Number - Number Units
What does the number 281 represent?
This number means that we have 2 hundreds, 8 tens and 1 unit. It may also be written: $$ \pmb{\color{darkred}{2}\color{goldenrod}{8}\color{darkgreen}{1} = \color{darkred}{200} + \color{goldenrod}{80} + \color{darkgreen}{1}}$$
For easier viewing, we can put those numbers within a table called number table:
Thousands | Hundreds | Tens | Units |
---|---|---|---|
2 | 8 | 1 |
Given an integer, the rightmost digit is the units digit, which is written in the units column. The one immediately to the left is the tens digit, which is written in the tens column. We complete the table so on from right to left, until all the numbers are written.
Note: The more a digit is located to the left of the number, the greater its value. For example, 6 hundreds is greater than 6 tens because the hundreds are more to the left than the tens.
Perform a mathematical operation
Addition
Performing an addition is adding quantities, we call it: calculating the sum of several terms (or numbers). For instance, let's count the steps we take forward: if we take one step and then two steps, we will have made 1 + 2 = 3 steps forward in total. Below is the table of the additions of the figures from 1 to 10:
The good news is that we don't need to remember these kind of tables and that we will be able to use our blank numbering table to make all kind of additions. We put the numbers one below the other with aligning the figures (number units) in a column, then we add the numbers, column by column, from the right to the left.
+ 5 2 1 9 7 |
_{1}
_{1}
_{1} + 2 6 5 1 1 2 2 |
Please note: we must not forget to carry over the retain when the sum of the digits is greater than 10 (as shown in our example on the right).
In the calculation of a sum, the order of the terms does not matter (the operation is said to be commutative): $$ 2 + 4 = 4 + 2 = 6 $$
In the calculation of a sum, we can group terms to facilitate the calculation (the operation is called associative): $$ 8 + 2 + 8 + 2 = (8 + 2) + (8 + 2) = 20 $$
Subtraction
The subtraction is an operation that calculates
the difference between two (or more) numbers.
This difference between 2 numbers is the number that must be added
to one to get the other.
Let's say I have 1 pen in my kit, how many more do I need to have 3?
The answer is : 3 - 1 = 2.
I need 2 more pens to have 3 pens in the end.
Another way of looking at subtraction is to remove quantities. For example by going down steps: if we climb 3 steps and then go down a 1 step, we will have climbed 3 - 1 = 2, that is 2 steps in total.
To subtract two numbers, the terms are laid out in the same way as for an addition, then we subtract the numbers, column by column, from the right. If the number above is less than the number below, we add 10 to the number at the top, then we transfer the retain (-1) to the left column.
_{-1} - 5 2 0 9 3 |
_{-1} - 2 6 5 5 9 2 |
In the first example, we see in the tens column that 4 is less than 5. We will then have (4 + 10) - 5 = 14 - 5 = 9. Then we carry the retain -1 on the left column.
Verification
If we have a little doubt about our result, the calculation of the verification allows us to quickly check it. We simply check that our result + the term subtracted = the first number. With the two operations carried out above, we will therefore have:
_{1} + 5 2 1 4 5 |
_{1} + 2 6 5 8 5 7 |
Perfect, we fall back on our initial numbers: the calculations are verified!
Subtract a larger number?
In the natural numbers set, this operation is impossible!
The operation is not stable in this set.
But don't panic because the next lesson, the one about integers, explains how to overcome this problem ;)
Multiplication
The multiplication is an operation which allows to calculate the product between two numbers. Multiplication corresponds to an addition repeated several times. It makes it possible to greatly simplify the writings as well as the calculations.
Let's take "3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3" (11 in total), this calculation can be written 11 * 3. We will also say that we have 11 times the number 3, for a total of 33. Below is the multiplication table for the numbers from 1 to 10:
When we multiply by 10, we simply shift each digit in the box to the left. For example 20 x 10 = 200 (the 2 of tens became the 2 of hundreds).
We are now used to it (it's always the same technique), we start by arranging the numbers one below the other by aligning the numbers in columns (numbering units), then we multiply the numbers, column by column, from the right.
Warning!!!
When the multiplier is a number made up of several digits, the calculations are carried out digit after digit with all of the digits of the top number.
Then, for the next digit of the multiplier (bottom):
We go to the line and we shift the results of a column (see the example on the right).
_{3} _{1} x 6 3 1 8 |
_{1} x 2 5 3 1 5 + 1 2 6 . 1 5 7 5 |
Please note: We must not forget to defer the retain when the product of the figures is greater than 10 (this retain must be added).
In the calculation of a product, the order of the terms does not matter (the operation is said to be commutative): 2 * 4 = 4 * 2 = 8
In the calculation of a product, we can group terms to facilitate the calculation (the operation is called associative): 3 * 2 * 3 * 2 = (3 * 3) * (2 * 2) = 36
Congratulations, we have just obtained the most fundamental bases in mathematics !!
May the math be with us!
Division
Now that we have learned how to multiply integers, dividing should be a breeze. The reason is that they follow the same rules. The division allows us to calculate the sharing problems, it is the opposite of multiplying.
Example:
There are 12 blue candies, and 4 friends want to share them, how do we divide the candies?
The answer may seem evident: if each friend takes 3 candies, we are all good. Division is splitting into equal parts or groups. It is the result of "fair sharing".
Note:
As we said, another way to think of division is as the opposite of multiplication.
Taking this example we got:
--> 12 / 4 = 3 (we gives 3 candies to each of us)
Now if we invert the operation using the result, we replace the ‘=’ with a ‘x’ sign and the ‘/’ with an ‘=’ sign:
--> 12 = 3 x 4 (if 3 candies are given to each of us, we shared a total of 12 candies)
This is, indeed, the mathematical definition of the division: $$ \pmb{a / b = a * \frac{1}{b}} $$
Using multiplication is a great way to check our division and master our math tests. But remember this rule, it is the only one to know when dealing with division:
We never divide by 0!
But... Sometimes it does not fit perfectly!
Indeed, sometimes, we have a remainder; we will address this case and more complex ones within a dedicated course: “The Euclidean division” ;)