Algebra, Geometry

Types of numbers, part III

Hey guys! This is the last post about numbers, I promise! In this last one we’ll talk about some numbers that have geometrical tendencies. They are pretty spectacular!

When you say “five squared equals twenty-five”, did you ever thought why we call it ‘squared’? Well, the Greeks were big on geometry and applied it to numbers as well. Twenty-five is a square number because you can arrange twenty-five dots to form a five-by-five square. In fact, twenty-five is the fifth square number, or n=5. We grow familiar to this family of square numbers from early age and we play with it very often in our problems, only because they’re fun to analyse. They have one of the most important properties in algebra, they are always positive!

Now let me introduce to you a less well-known set: the triangular numbers (1, 3, 6, 10, 15, 21,…). They earned their name by forming triangles of dots.

Knowing these two kinds of numbers, we can easily see that some numbers are both square and triangular. I’ll show some of them below!

square triangular numbers

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Square and triangular numbers are only two of many geometric (or figurate) number sets, and the table below shows the first few along with their formulas – just replace n with any number and you will find the corresponding geometric number: Geometric numbers even exist in three dimensions; for example, there are ‘tetrahedral’ numbers that are the sums of triangular numbers and form a pyramid with a triangular base.

geometric numbers

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