The integers are the whole numbers that you see as tick marks on the number line, but in between each of these integers, you have these large spaces that you can’t describe with just integers.
You have to use other numbers to represent the space in between the integers and to do that, we have decimals, and we have fractions.
Next lesson: Prime Numbers, and Rational Numbers
Any number between two integers can be represented both as a fraction and as a decimal. So to answer one of the most frequently asked questions: decimals and fractions are NOT integers.
To demonstrate this, we’re going to enlarge essentially this section of the number line right here. So, this section of the number line is now going to be represented right here. We have zero over here, and one over here, and we have this massive space in between where there are no integers.
To describe this, we have decimals, and fractions. We’ll describe fractions first. Fraction is essentially a way to divide up this section inside of here into equally-spaced segments. So, if we put one right in the middle, we’ve cut it in half, or into two pieces.
So, two pieces – so we have a two on the bottom of our fraction. If we just go one of those two pieces, we have one over two, or one-half (1/2), and that is what this point here is. If we want to go further, we can divide it up into four segments. Now we have four equal segments.
If we decide we want to use one, two, three of those segments, we can write this as a fraction with a four on the bottom, because there’s four parts, and we’ve gone three of them. So, we write this as three-fourths (3/4).
You may notice that if you go two-fourths (2/4), that’s the same as going one-half (1/2). So, one-half (1/2) and two-fourths (2/4) are the same thing. This is kind of the idea of fractions. You divide up the space between the integers into equal parts, and you put that number on the bottom, or the denominator of the fraction.
And then, the number of those parts that you’re counting is the number on the top of the fraction. And now, I’ve divided this up into two, and then four, because that’s easy to draw, but you can divide up the space between the integers into as many sections as you like. You can divide it up into three, you can divide it up into five, seven, ten, any number of parts, and that’s the number that’ll go on the bottom. This provides a good transition into decimals. A decimal representation is essentially dividing up the space between zero and one into 10 different segments. Alright, so you’ve got the space between zero and one divided up into ten parts.
With decimals, instead of writing this as three-tenths (3/10), which is a valid way to write this particular point as a fraction, you would write this as 0.3, because we use the decimal system, every number place to the right of the decimal means you divided up the space between the two integers into 10 equal parts, and this is how many of the parts we’re talking about. So, we take the integer immediately beforehand, and then count forward three of the ten parts, and that gives us 0.3, which is the same as
three-tenths (3/10). And so, you can take the space between any of these integers, for example, if you wanted to take the space between three and four, and count six-tenths (6/10) of the way from three to four, you would write this as 3.6, because you’ve gone six-tenths (6/10) of the way from three to four. And so, you can describe any number on the number line using decimals or fractions.
If you want to get even more specific than tenths, you can divide up, for instance, if you wanted to go somewhere between. .3 and .4, you can divide this space up here into ten equal parts, and then, how many of those parts you went forward from .3, you would write that as another decimal here, so .3, and however many parts.
That is a quick overview of integers, decimals, and fractions.