In Math, integers are whole numbers. You can see the integers as tick marks placed on the number line. In between all these integers, however, you can see those large spaces. You can’t describe those with just integers.
So we need to use other numbers for representing those spaces between the integers. To do that, we use decimals and we use fractions.
Next lesson: Fractions, Decimals, and Percentages
Well, any number in between two integers may be represented both as a decimal and as a fraction. So this is the answer to a very frequently asked question: fractions and decimals are NOT integers.
To make this clear, we will enlarge this section of this number line, right here. This section on the number line will now be represented here. We can see the zero over here and the one over here. And then we can see this massive space between 0 and 1 where you see no integers.
To describe this space, we use fractions and decimals. I’ll go over fractions first. A fraction is basically one of the ways to divide this section up into equally-spaced portions or segments. So when we put one right here, right in the middle, we cut the space in half, or you can say into two equal pieces.
So we have two pieces. We have a two at the bottom of this fraction. If we take one of these two pieces, it gives us one over two, 1 over 2, or one-half (this is 1/2), and this is exactly what this point right here is. Now, if we will go further, we may divide the space into four equal segments. So now, we are having four segments, all equal.
If we want to use 1, 2, 3 of these segments, we may write this in the form of a fraction. The fraction has a 4 on the bottom because there are four equal parts, and we will use three of them. So here, we will write this: three-fourths, 3 over 4, or 3/4.
You may have noticed that if we’ll go two-fourths (or 2/4), it will be the same as if we would go one-half (or 1/2). So 1/2 (one-half) and 2/4 (two-fourths) are one and the same thing. Now, this is what you’ll see with fractions. You can divide up spaces between integers into several equal parts and that you’ll put that number at the bottom of the fraction (also referred to as the denominator).
Then, the number of parts that you count is the number that goes on the fraction’s top. So here, I divided this into two, and here into four, because it’s easier to draw. You can, however, divide up that space between those integers into just as many segments or sections as you wish. You may divide the space up into three segments, you may divide the space up into 5, 7, 10, or any number of segments and that is the number that goes on the bottom.
This is providing a nice transition to the next thing: decimals. With decimal representation, we basically divide up this space between 0 and 1 into ten (10) segments. So we have that space between 0 and 1 divided into ten segments.
When we use decimals, rather than writing this as 3/10 (three-tenths, which is actually a valid way for writing this as a fraction), we’re going to write this as 0.3. Because we’re using the decimal system, all number places to the right side of the decimal are meaning that we’ve divided the space between the integers in ten (10) equal parts and that this represents the number of parts that we’re talking about. We’re taking, so to speak, the integer immediately beforehand. Then we’re counting forward 3 of the 10 parts, which gives us 0.3. This, again, is the same as 3/10 (three-tenths).
So, we can do this with the space between each one of these integers. For example, suppose you want to take the space between 3 and 4 and want to count 6/10 (six-tenths) of the part from 3 to 4, you would have to write that as 3.6 because you went 6/10 (six-tenths) of the entire way from 3 to 4. You see. You may describe this way any number on the number line when you’re using fractions or decimals.
When you would like to get even more specific than you did with tenths, you may divide it up even further. If you, for instance, would like to go, say, somewhere between the points .3 and .4, you may divide that space up here, again, into 10 equal parts as well. And then, depending on the number of those parts that you went from .3, that’s the point that you would write as one more decimal here. Now, that is .3, plus the number of your parts.
That was a quick overview of what you need to know about integers, decimals, and fractions.