Power Functions

00:00 Python’s math module is well known to equip you with powerful functions. At the end of this lesson, you’ll see why.

00:09 So, let’s talk about power functions. A power function in mathematics is a function of the form f(x) equals x to the power of n. The meaning of x to the power of n depends a lot on what this number n is. Here, n is a fixed number, it’s called the power, and x is called the base.

00:30 When n is a positive integer, the notation x to the power of n means that we take x and we multiply it by itself n times. If the power n is 0, then x to the 0 is defined to be 1, but only if x is non-zero. When x is zero, so in other words, 0 to the power of 0—that’s usually left undefined in mathematics.

00:56 Now let’s talk about negative integer powers. x to the power of -m, if m is a positive integer—so we have a negative power—that means 1 divided by x to the power of m. And x to the power of m means we take x and we multiply it by itself m times. Now, if in f(x) = x to the n, if the power n is not an integer and x is negative, then things get a bit tricky. In fact, they get complex, meaning that you’ll have to deal with complex numbers.

01:32 We’ll talk about complex numbers in a future lesson, but with regards to the power function, things are just a little bit more mathematical that fall outside the scope of this video course.

01:44 The math module defines a pow() function, or power function, which computes x raised to the power of y.

01:53 So it takes two positional arguments, two numbers, and it’s computing x raised to the power of y. Here, y can be a non-integer but then x must be positive.

02:06 And so an edge case is that, okay, if both x and y are finite and x is indeed negative and y is a non-integer, then the power function is going to raise a ValueError.

02:18 And lastly, one last thing to note with the power function in the math module is that 0 raised to the power of 0 as computed by the pow() function, it’s going to return 1.0. So even though in math we usually don’t define 0 raised to the power of 0—we leave it undefined—in the math module, it returns 1.0 Let’s try some examples with the power function.

02:45 Let’s say 2 to the power of 8. You see here that even though the inputs are both integers, the power function from the math module is always going to return a float.

02:56 Let’s just verify that. Even though we see it, why don’t we ask for the type? We get a float. Now, if you use the built-in power operator, or the double star operator (**), 2 to the power of 8, here, you’re going to get an integer.

03:14 And if you use the built-in power function,

03:18 you’re also going to get an integer. Now, negative exponents are okay, so if we try the same thing but 2 to the power of -8, we saw that this is going to be computing, really, 1 over 2 to the power of 8, and we can also use non-integer powers. So, for example, 2 to the power of 3.4. And same thing, we can have a negative exponent, so -3.4.

03:49 And again, this is pretty much going to be equal to math.pow(), 2 to the power of 3.4, but divide. Now, we may not get exact values here because of rounding or truncation, but let’s just give it a try. And we get True. So, again, when we’re raising a base to a negative exponent, it’s really 1 over the base to the positive exponent.

04:16 One of the edge cases with pow() from the math module is that when the base is negative, if we’ve got integer powers, then everything’s okay. So again, this is really just -3 multiplied by itself 4 times.

04:32 And the same thing if we have a negative exponent. This is going to be 1 divided by -3 to the power of 4, which is 1 over 81. But if the exponent is a non-integer and we have a negative base—so, for example, 0.5—then power is going to raise a ValueError saying that math domain error.

04:58 Now let’s do a quick test of the computational speed of the power function in the math module compared to the built-in pow() and the double star or exponentiation operator (**).

05:10 And let’s compute 10 to the power of 308 using all three different methods—so using the exponentiation operator, using the built-in pow(), and then using the pow() function from the math module.

05:23 So I went ahead and wrote some of this up already. I’m going to call it with_star, the computation of computing 10 to the power of 300 using the exponentiation operator.

05:37 And then with the built-in pow() function, pow(), 10, to the power of 300.

05:46 Now let’s do this with the power function from the math module.

05:53 And then let’s take a look at all of these, so with_star, and with_pow, and with_math_pow.

06:01 It looks like if we compare how much faster, say, with_math_pow is relative to the built-in power function—so about 7 times faster. And with_star, it’s going to be pretty close, the same, rounding off to 7.

06:19 All right, so that is the power function. Up next, we’ll take a look at exponential functions.