Decimals

We will write some special fractions, like 910, 48100 and 8751000, in the number system we use every day: the decimal number system.

For example, we will write:

First, let's quickly remember our number system. When we write 6378:

• The ones place: I have 8 ones.
• The tens place: I have 7 tens.
• The hundreds place: I have 3 hundreds.
• The thousands place: I have 6 thousands.

If you look at the place values, each place is 10 times the place on its right. Or, looking the other way, each place is the one on its left divided by 10.

We also need a way to show pieces smaller than 1 in this number system. For smaller units, I divide 1 into 10 equal pieces, and I get tenths (110).

And this is how I show it. For example, 3.8:

• The ones place is made of ones: I have 3 ones.
• The tenths place: I have 8 of 110.

This number is between 3 and 4. We read it as "three and eight tenths".

There is no whole part, so we need to write zero for the whole part: 0.

There are 6 pieces of 110, so we write 6 in the tenths place:

We read it as "zero and six tenths".

So what if we need even smaller pieces? To get a new place, we need to divide by 10 again.

Now the pieces are not the same size. To name a piece as a fraction, the whole must be cut into equal parts. So I divide the other pieces into 10 as well.

When I divide each piece into 10 again, 1 whole is made of 100 equal pieces. Each piece is 1100.

Let's show 245100:

45 pieces of 1100.

Where do I write the digits?

45100 is 45 pieces of 1100. We can think of it like this:

There are no wholes, so I write zero for the whole part: 0.

36100 means 36 pieces of 1100.

In 0.36: 3 of 110, that is 310, and 6 of 1100.

Why don't we write it all in just one place?

You might think: why don't we write 36 in the hundredths place only?

When we write whole numbers, we group them; we don't show them in a single place. For example, when we show 375, we don't show it with one symbol or one place: we group in tens.

It works the same way here: we group every 10 pieces of 1100 and count them as 110.

There are no wholes, so I write 0. There are 5 pieces of 1100.

The correct way:

• Hundredths: I have 5 of 1100.
• Tenths: I have no 110.
• Ones: I have no ones.

Let's put all the places together. Under each place, you can see its value. Every place is the place on its left divided by 10.

• We divided 1 into 10 and got 110.
• We divided 110 into 10 and got 1100.
• We divided 1100 into 10 and got 11000.

If I need even smaller pieces, with the same logic I divide the previous unit by 10.

Let's show 4761000: it is 476 pieces of 11000.

I have no wholes, so I write 0 for the whole part.