Number Line
A number line provides a spatial representation of numerical ratios, facilitating the intuitive understanding of arithmetic operations and some of their fundamental properties
The numbers form ordered sets, within which precise ratios and relationships exist. For example, within the natural numbers, 4 is the next of 3 and the previous of 5; 9 is less than 27; 11 is greater than 10. These relations are fixed and immutable and can be expressed in a purely symbolic way (e.g., 9 < 27; 11 > 10).
However, our understanding of numerical proportions and properties can be improved and facilitated if the mathematical symbols that express them can be associated with a spatial representation of those connections. The number line precisely plays this role of support and facilitation to the advantage of our intuition of numerical quantities and properties.
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In its simplest representation, and based on the use that has consolidated over time, a number line is a horizontal straight line ending with two arrows representing the infinite growth of numbers in both directions. Zero is marked with a tick in the center of the line; the positive integers, corresponding to the set of natural numbers, by convention follow each other to the right of zero at regular intervals. Negative integers, on the other hand, occur at equally regular intervals to the left of zero. Therefore, moving to the right, the numbers become larger and larger; moving to the left, smaller and smaller.
But the use of the number line is not limited to the representation of positive and negative integers. More extensive sets can also be reported on it: rational numbers, i.e., numbers that can be expressed in the form of fractions (1/5, 3/10, 8/7, etc.), and real numbers, such as 𝜋, √2, e, etc.
Each real number can be mapped onto the number line, which is infinitely dense. By appropriately modifying the intervals, i.e., the spaces between numbers, it is always possible to insert a third number between any two numbers, greater than the one to its left and less than the one to his right.
A use in which a number line is handy is for representing the arithmetic operations and some of their fundamental properties at an intuitive level. For instance, the following image shows the commutative property of multiplication. If you make five jumps of three units at a time on the number line, starting from 0 and going to the right, you end up on 15. It is the equivalent of multiplying 3×5. However, if you make only three jumps, but each of them is five units large, that is, if you multiply 5×3, you still end up on 15. In the synthetic language of mathematics, 3×5=5×3. The number line allows us to grasp the essence of the commutative property and demonstrate its validity at an intuitive level.
Another common usage of a number line is to report solutions for inequalities. Let us consider, for example, the solution −3 <𝑥 ≤5. It can be shown on the number line as in the following drawing.
The red segment indicates that all numbers from −3 to 5 are part of the solution for the inequality. The empty circle at the number −3 indicates that −3 is excluded from the solution, while the filled circle at the number 5 indicates that 5 is included in the solution.
The following example shows how a disjoint solution (i.e., with non-overlapping intervals) such as 𝑥 <−4 OR 𝑥 >−1 can be reported on a number line.
The arrows at the ends of the red segments indicate that the solutions extend to infinity from both sides of the line. Only the numbers from −4 to −1 (included) are not part of the two solutions.
