The positional system with base 10

Twenty-one is not the same number as 12. That's obvious right? But why not? Both numbers consist of the same digits. The digits are in different orders, and that matters.

But to be more accurate, it's not exactly the order matters. It's the position that dictates the value assigned to the digits. The position to the far right denotes the number of units you have. The position to the left denotes the number of tens. Since there are no digits before the tens, we know that there are zero hundreds, zero thousands, and so on.

If we add 10,000 to 21, we get 10,021 but we can't leave empty spaces between the numbers, because whoever reads it won't know if we mean the numbers 1 and 21 or 10,000 and 21, so we fill the positions in between with zeroes. If you only have 2 hundreds but no tens or ones, you fill up the other positions with zeroes. Otherwise we won't know if you mean 2 or 20 or 200. Fill up with zeroes until the decimal comma. Decimal comma?

There is no decimal comma here. If there is no decimal comma written out, you can write one. It is always directly behind the position of the units. To the right of the decimal comma, each position also has a special value, but here they become smaller. First there are tenths, next, hundredths, thousandths, and so on.

Can you find a pattern here? Each place is worth ten times more than the one to its right, and one tenth than the one to its left. Take the number 42, four tens and two ones. If we move both numbers one place to the right, the number is worth one tenth as much. Four point two is one tenth of 42.

If we instead move them in place to the left, they become worth ten times more. Four hundred twenty is 10 times 42. They become worth exactly 10 times more for every place because we have 10 digits. No, not like that. 10 is not a digit.

It is a number consisting of two digits. But 0 is a digit, so we now have 10 digits. It might seem strange to go from 0 to 9, but take a look. In every position, you can count ten different digits before you increase the value in the next position. Here, the number is 10 where there are zero ones.

We can then count up one step at a time until we have nine ones. Now we've used all ten digits, and it's time to change the tens to two and the ones are zero again. The number system we use is called base ten or the decimal number system. It comes from the Latin decimus which means tenth. In the decimal number system, a number is worth ten times more when you move it one place to the left and a tenth times less when you move it one place to the right.

If we had another base, 8 or 12 for example, a number's value would increase by 8 or 12 times for every place you moved the digits to the left and to be worth and eight or a twelfth every time you moved them a place to the right. Take a look at the lesson on the binary number system as well. The binary number system is base two. There a number's value is doubled every time you move it a place to the left and divided by two for every place to the right.