# Zero:

The Weirdness of Nothing

#### By Kaiser Himmelberg

## The Origins of Zero

In the history of number theory, zero took a while to catch on. For example, the Egyptians had counting numbers and fractions, but no zero. The Babylonians just used a space in place of zero: e.g., 3070 would be written 3 7 . The first recorded zero probably did not appear until the first century BC. Why would it take so long for such an essential number to come about? The answer is that advanced mathematics did not come about until much later (with a few exceptions like the Pythagoreans). People would instead learn to farm, trade, and so forth. Because of minimal mathematics, numbers were used primarily for measurements and how much of something you are willing to trade, which don’t need the number zero. The notion of having nothing as a number is pretty weird.

But zero is an essential number. It is one of the cornerstones of algebra. How can you calculate x + 3 = 0? You need to have a zero before the answer can be calculated. To use the quadratic formula, you need to start with ax^{2} + bx + c = 0 in order to calculate x, which equals (-b ± √(b^{2} – 4ac))/2a. Moreover, you cannot have an x, y co-ordinate plane without zero, which means you cannot graph functions. Zero is essential for algebra and more advanced math.

The Problems of Zero

Four interesting problems arise once you have the number zero: whether it is even or odd; what happens when you divide by zero; using it in exponents; and calculating its factorial.

Zero is neither positive nor negative, but people have sometimes been confused on whether zero is even or odd. The definition of an even number is that it can be divided by two and have the quotient be an integer (e.g., 6/2 = 3, which is an integer, therefore six is even). Zero can be divided by two, and the quotient is zero: 0/2 = 0, so zero is even.

Zero gets weirder in division. When you have zero divided by any number x, the answer you get is zero—unless x itself is also zero. When you have x divided by an infinitesimal, which is the smallest number there is (essentially 1/∞), you get infinity, ∞. However, when you divide by zero, the answer is undefined. Stranger yet is zero divided by zero. 0/0 is a number divided by itself, so it seems it should equal one, but, because it is being divided into zero parts, 0/0 is indeterminate.

If you have zero to the power of any number, 0^{x}, the answer is zero (because you are multiplying zero by itself x number of times), except when x = 0. If you have any number to the power of zero, x^{0}, the answer is one—but why? The answer is a proof by definition. Mathematicians got this definition because 2^{3} = 8, 2^{2} = 4, and 2^{1} = 2: as you subtract a power on one side of the equation, you divide by two on the other. This means 2^{1 – 1} = 2/2, or 2^{0} = 1. With zero to the zeroth power, 0^{0}, does it equal zero (because it is 0^{x}) or one (because it is x^{0})? This too is undefined.

The last major problem with zero is zero factorial, 0!. A factorial is the product of a positive number times all of the positive numbers before it (e.g., 4! = 4 * 3 * 2 * 1), primarily used in probability equations. As with x^{0}, is there a pattern by which we can define zero factorial? There is. It is found with the pattern 4! = 5!/5, 3! = 4!/4, 2! = 3!/3, and 1! = 2!/2. If you continue this pattern, 0! = 1!/1, which equals one. (This pattern cannot continue into negative numbers because -1! = 0!/0, which is 1/0 and thus undefined.)

Further resources:

Kaiser Himmelberg, “Is Zero Even or Odd (With Detailed Analysis),” from the YouTube channel Experiential Learning

John Derbyshire, *Unknown Quantity: A Real and Imaginary History of Algebra*

Robert Solomon, *The Little Book of Mathematical Principles, Theories, and Things*

*Kaiser Himmelberg is a thirteen-year-old numberphile. When not reading about math, he is sure to be found calculating while bowfishing, trapshooting, and exploring our beautiful planet.*

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