Exponents and cmath

Math and programming go hand in hand, especially in game development. We’ll cover the core concepts throughout this chapter, starting from the basics and moving on to concepts that allow us to implement things like virtual cameras and physics systems.

In this first lesson, we're starting with the fundamentals - exponents and square roots. We'll explore how these work in C++, look at different ways to implement them, and start thinking about performance considerations. This sets the stage for the more complex math concepts we'll tackle as we move forward.

Exponents

Exponentiation is the process of raising a number to the power of another number. We can think of it as multiplying a number by itself, multiple times.

The number we start with is called the base, and the number of times it is multiplied by itself is called the power, or the exponent. For example:

• $2$ to the power of $2$ is equivalent to $2 \times 2$, which is equal to $4$
• $2$ to the power of $3$ is equivalent to $2 \times 2 \times 2$, which is equal to $8$
• $3$ to the power of $2$ is equivalent to $3 \times 3$, which is equal to $9$

We generally represent this operation with a superscript notation, for example:

• $2$ to the power of $2$ is written as $2^2$
• $2$ to the power of $3$ is written as $2^3$
• $3$ to the power of $2$ is written as $3^2$

The most common form of exponentiation we will see is raising a number to the power of $2$, for example, $5^2$.

An operation like this, where the exponent is $2$, is commonly called squaring. The result of such an operation is called the square of the base number.

For example, the square of $5$ is $25$, because $5^2 = 25$.

Order Of Operations

In the order of operations, exponentiation happens before multiplication, but after brackets/parentheses. For example:

• $2 \times 2^2 = 2 \times 4 = 8$
• $(2 \times 2)^2 = 4^2 = 16$

<cmath> and std::pow()

To access the standard libraries math utilities, we need to include <cmath>:

1#include <cmath>

After including cmath or math.h the pow function is now available within the std namespace:

1#include <iostream>
2#include <cmath>
3
4int main() {
5  std::cout << "2 to the power of 3 is "
6    << std::pow(2, 3);
7}

12 to the power of 3 is 8

SDL_pow() and SDL_powf()

When we’re using SDL, alternative to std::pow() are available in the form of SDL_pow() which accepts and returns double values, and SDL_powf() which accepts and returns float values:

1#include <iostream>
2#include <SDL.h>
3
4int main() {
5  std::cout << "2 to the power of 3 is "
6    << SDL_pow(2, 3);
7  std::cout << "\n3 to the power of 4 is "
8    << SDL_powf(3, 4);
9}

12 to the power of 3 is 8
23 to the power of 4 is 81

Square Root

Finding the square root of a number is, in effect, reversing the process of squaring. For example, the square root of 25 is the number that, if squared, would result in 25.

In other words, the square root of $25$ is the value of $x$ such that $x^2 = 25$. We saw, from the previous section, that an answer is $5$, because $5^2 = 25$. Therefore, the square root of $25$ is $5$

Square roots are often denoted using the radical sign, which looks like this: $\sqrt{25}$

Expressions of any length can appear under a radical sign. That expression should be resolved before calculating the square root. For example: $\sqrt{50 + 25 \times 2} = \sqrt{100} = 10$

1#include <cmath>
2#include <iostream>
3
4int main() {
5  std::cout << "The square root of 25 is "
6    << std::sqrt(25);
7}

1The square root of 25 is 5

1#include <SDL.h>
2#include <iostream>
3
4int main() {
5  std::cout << "The square root of 25 is "
6    << SDL_sqrt(25.0);
7  std::cout << "\nThe square root of 36 is "
8    << SDL_sqrtf(36.0f);
9}

1The square root of 25 is 5
2The square root of 36 is 6

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