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Although every number on the number line is unique, they all have certain characteristics in common. But two useful numbers, the irrational π (pi) and the imaginary i, have unique characteristics that help expand the reach of mathematics.
Going Greek: Pi (π)
The symbol π (pi — pronounced "pie") is a Greek letter that stands for the ratio of the circumference of a circle to its diameter. The approximate value of π is 3.1415926535. . . . Although π is just a number — or, in algebraic terms, a constant — it's important for a couple reasons:
- Geometry just wouldn't be the same without it. Circles are one of the most basic shapes in geometry, and you need π to measure the area and the circumference of a circle. So whether you want to know the area of the crop circles that showed up in your cornfields or just the area of your round kitchen table, π comes in handy.
- Pi is an irrational number, which means that no fraction that exactly equals it exists. Beyond this, π is a transcendental number, which means that it's never the solution to a polynomial equation (the most basic type of algebraic equation). Thus, even though π emerges from a very simple operation (measuring a circle), it contains deep complexity that numbers like 0, 1, –1, 42, and 99.9 don't share.
The imaginary number i
An imaginary number is any real number multiplied by the square root of –1.
To understand what's so strange about imaginary numbers, it helps to know a bit about square roots. The square root of any number is a second number that, when multiplied by itself, gives you the first number. For example, the square root of 9 is 3 because 3 x 3 = 9; the square root of 9 is also –3 because –3 x –3 = 9.
The problem with finding the square root of –1 is that it isn't on the real number line. If it were on the real number line, it would be a positive number, a negative number, or 0. But when you multiply any positive number by itself, you get a positive number. And when you multiply any negative number by itself, you also get a positive number. Finally, when you multiply 0 by itself, you get 0.
If the square root of –1 isn't on the real number line, where is it? For thousands of years, mathematicians believed that the square root of a negative number was simply meaningless. They banished it to the mathematical non-place called undefined (which is also where they kept fractions with a denominator of 0). In the 19th century, however, mathematicians began to find these numbers useful and found ways to incorporate them into the rest of math.
Mathematicians designated the square root of –1 with the symbol i. Even though numbers multiplied by i are called imaginary, mathematicians today consider them no less real than the real numbers. And the scientific application of imaginary numbers to electronics and physics has verified that these numbers are more than just figments of someone's imagination.
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