How Math Works

The Tower of Math: Numbers

Numbers pose a difficulty for humans. Sure, some of us have more of a gift for math than others, but every one of us reaches a point in our mathematical education where things become hard. Learning your multiplication tables is difficult because the human brain never evolved to handle such advanced computations as 17 x 32 = 544. After a certain point, our mathematical education is largely an exercise in rejiggering ill-adapted brain circuits [source: Dehaene].

Number sense may come naturally to us, but mathematical literacy comes only with time. Likewise, humanity's use of mathematics has steadily grown over the ages. Like science itself, math isn't the product of one mind but rather a steady accumulation of knowledge throughout human history.


Think of math as a tower. Natural human height is finite, so if we're to reach higher into the air and see out farther across the landscape, we'll need to build something external to ourselves. Our mental abilities to understand math are equally finite, so we build a great tower of number systems and climb upward to the stars.

To break down the basic structure of this tower, let's first look at the raw materials. These are the basic types of numbers:

Integers: You probably know these as whole numbers, and they come in both positive and negative forms. Integers include the basic counting numbers (1-9), negative numbers (-1) and zero.

Rational numbers include integers but also encompass simple fractions that can be expressed as a ratio of two integers. For example, 0.5 is rational because we can also write it as 1/2.

Irrational numbers: These numbers can't be written as a ratio of two integers. Pi (the ratio of the circumference of a circle to its diameter) is a classic example, as it can't be written accurately as a ratio of two integers and has been calculated to trail off decimal points into the trillions.

Rational and irrational numbers both fall under the category of real numbers or complex numbers. And yes, there are also imaginary numbers that exist outside the real number line, and transcendental numbers, such as pi. There are many other different numbers types as well, and they, too, play a part in the structure of our tower.

On the next page, we'll look at some of the core branches of mathematics.