Why is zero at the zero degree equal to one?


In this article, we will not talk about physics and space, but more about mathematics. But the topic is still interesting and not trivial. We all know from school that the zero raised to the zero power is equal to one. But why is it so? Let’s figure it out together!

Generally speaking, the fact that zero in the zero degree is equal to one at first glance contradicts the definition of the degree. What does the number X to the power of Y mean? This means that the number X is multiplied by itself Y times.

Those. X³ = X × X × X. As you know, multiplying by zero always results in zero. Those. zero to any power greater than zero will also be zero. For example 0⁵ = 0 × 0 × 0 × 0 × 0 = 0. Where then does the unit come from?

Let’s take a sequence of numbers: 4, 3, 2, 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4 … 0.1, 0.01 and plot the function y = xˣ. As in school, we calculate for each x the value of y and enter it on the plate.

It is easy to see that the values decrease at first. The larger the x, the larger the y, and vice versa. But as it approaches zero, around 0.3, the values suddenly begin to grow, and as it approaches 0, the value of the function asymptomatically approaches unity.

And although, strictly speaking, when x is equal to zero, the function y = xˣ undergoes a break in some branches of mathematics, it is agreed that zero in the zero degree equals one.

Such an agreement is convenient in several branches of mathematics at once, for example, in arithmetic, combinatorics, set theory, etc., and allows you to greatly simplify many rather complex calculations.

In general, in higher mathematics, in particular in mathematical analysis, the value of zero to the zero degree is considered undefined and in each specific case the uncertainty is resolved analytically, for example, by finding the limit of the natural logarithm of an expression containing zero to the zero degree.