Naturally Irrational

Humans love to categorize and group. When we are presented with a set of objects, we often give names to both the individual elements and the set itself. You’ve probably tried to impress your friends with archaic names for certain groups of animals.  These whimsical descriptions for sets of animals include a murder of crow, a parliament of owls, a congress of baboons, and a flamboyance of flamingos.

We have a similar way of naming numbers — though I’m not certain your friends will be as impressed…

Most of us are familiar with counting numbers. These natural numbers occur in nature (1, 2, 3, 4, 5, and so forth).

Whole numbers are quite the same, but include zero: {0, 1, 2, 3, 4, 5,. . . .∞}.

While these two sets may seem like the only two types of numbers we would ever need, we often come across scenarios in math where we have to describe something not so natural. Imagine for a moment that you only have $20 in your bank account, but the automatic payment you have set up withdrew $30. Barring any sort of overdraft protection, you would now owe $10, and your account would read negative ten dollars (-$10). A strawberry plant can’t produce a negative amount of strawberries; this would imply we knew how many were originally supposed to occur. We can’t have a negative amount of flamingos (our flamboyance would soon become a flop). However, this is something we occasionally have to notate in situations when we owe an amount. We can write this negative amount using integers. Integers include all of the natural and whole numbers to infinity, but also introduce negative values for each natural number and, subsequently, negative infinity.

{-∞. . . .-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,. . . . ∞}

In math, we will come across many different types of numbers, but in your beginning courses most of the numbers you come across can be described as rational numbers.

“Wait. How can a number be irrational?” Don’t worry, we’ll get to that next.

Rational numbers are all numbers that can be expressed as a quotient of integers. Do you remember how fractions are just division problems? A number out of (or divided by) another integer. If you can write the number as a fraction, it’s a rational number. Now, this would include repeating decimals, square roots of perfect squares, and even 0. We can write each as a fraction (or as the quotient of two integers).Slide1

Now, if we can write rational numbers, it stands to reason that we could also describe irrational numbers. These are numbers that may appear like repeating decimals, but do not actually terminate or have repeating blocks of numbers. Irrational numbers are numbers that have no value that we can precisely enumerate. You’ve probably heard of one of the most commonly encountered irrational numbers: π (pi). While this number may sound delicious, it is entirely irrational: 3.14159265359 . . . . Another good example of an irrational number is a square root of a number that is not a perfect square. For example, the square root of 2 (not a perfect square) is irrational while the square root of 4 (a perfect square) is not.

Our final category includes real numbers. All numbers that have been discussed here are real numbers, as each can find a home on a number line from negative infinity to positive infinity (-∞ to ∞). This includes rational and irrational numbers, though mathematicians often accept approximations for the non-repeating, but continuous numbers.

As a final thought, as you advance through your math courses, you may come across more sets of numbers than described above such as √(-1) or 5i. These imaginary numbers describe a special case and are about as flamboyant as math can possibly be!

Pink Flamingos image by – JDG


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