What is a number line? You might reasonably think that it is a line with numbers on it, like this:
But when mathematicians draw a number line, the numbers are not actually on the line. A number's location is marked like this:
On a number line, a number represents a point on the line, and every point on the line represents a number. So you might think a number line would look like this:
However, this is not quite correct. The numbers need to be in order from smallest to largest, or, as a mathematician might say, from least to greatest. The next line shows this progression from right to left.
But the preceding example is still not the number line used in mathematics. The conventional way to draw a number line is horizontal, with numbers increasing from left to right.
But something is wrong with this line. Look at 0, 1, and 2. Notice how the distance between 0 and 1 is bigger than the distance between 1 and 2.
Distances between numbers should be consistent. The difference between 0 and 1 is 1, the same as the difference between 1 and 2. Therefore, the distance between 0 and 1 should be same as the distance between 1 and 2. Let's try again.
But this line seems to have lost some numbers. Actually, once the size of the unit was chosen (as the distance between 0 and 1) it was not possible to fit all of the numbers previously shown on this number line.
You can pick any unit you like for a number line. That is why we speak of a number line, rather than the number line. You can draw many number lines, but they all have the characteristics discussed so far. You can make the unit on a number line one millimeter or one mile. However, you should probably choose a unit that works for what you are doing.
For example, you would have had to have chosen a small unit to fit in all the numbers from −15 to 8,654,320. The unit chosen for the last number line drawn in the preceding discussion allows most of your numbers to fit and is large enough to show the difference between 0 and .1235, the two closest numbers.
It is clear that the distance from 0 to 1 is one unit on a number line, but what about the distance from 1 to 0? Going backwards, the distance is the same, but 0 − 1 = −1.
For this situation on a number line, you can use the absolute value function, so it does not matter which direction you go when subtracting numbers to measure distance.
Applying the absolute value function gives
So a number line is a line with numbers represented by points on the line located at a distance from 0 equal to the number in some arbitrarily chosen unit (typically, the distance from 0 to 1). The unit distance between two points on the number line is the absolute value difference between the numbers representing those two points.
Number lines have arrows on both ends. The arrows show that a number line goes on forever in both the positive and negative directions. You can always take a very large counting number, such as 123,456,789,102,131,141 and make a still larger counting number by adding 1. In the same way, all the numbers on a number line keep going without stopping or having a largest number. Any number that can have a point on a number line is called a real number, and all the numbers in the set of real numbers have a corresponding point on a number line. Such numbers as π and are real numbers, and you can find points that correspond to them on a number line. Imaginary numbers, such as are not real numbers, and you cannot find a point for them on a real number line such as the ones described here.
see also Absolute Value; Integers; Number System, Real; Numbers, Complex.
Stanislaus Noel Ting
Davis, Philip J. The Lore of Large Numbers. New York: Random House, 1961.
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