Chapter 16: Mathematical Operations

C provides a robust suite of mathematical functions through the <math.h> library. However, as a systems language, C also requires you to understand the underlying representation of numbers, specifically floating-point values.

I. Standard Math Functions (`<math.h>`)

To use these functions, you must #include <math.h> and link with the math library (using the -lm flag in GCC).

Function	Action	Example
`pow(x, y)`	x raised to power y	`pow(2, 3) = 8.0`
`sqrt(x)`	Square root	`sqrt(16) = 4.0`
`abs(x)` / `fabs(x)`	Absolute value	`fabs(-5.5) = 5.5`
`ceil(x)`	Round up	`ceil(4.2) = 5.0`
`floor(x)`	Round down	`floor(4.8) = 4.0`
`sin`, `cos`, `tan`	Trigonometry	Uses radians

#include <stdio.h>
#include <math.h>

int main() {
    double base = 2.0, exp = 3.0;
    printf("%.1f ^ %.1f = %.1f\n", base, exp, pow(base, exp));
    return 0;
}

II. Floating-Point Mechanics (IEEE 754)

Most modern systems represent float and double using the IEEE 754 standard. A floating-point number is stored in three parts:

Sign Bit: 0 for positive, 1 for negative.
Exponent: Scales the number.
Mantissa (Significand): Stores the actual digits.

Precision Issues

Because floating-point numbers are stored in binary, they cannot represent some decimal fractions (like 0.1) exactly. Never compare floats using ==. Use a small epsilon instead:

if (fabs(a - b) < 0.00001) { /* Equal */ }

III. Integer Overflow and Underflow

If you add 1 to the maximum value of a signed integer (INT_MAX), it "wraps around" to the minimum value (INT_MIN). This is Undefined Behavior in C and is a frequent cause of security bugs (like buffer overflows).

IV. Random Number Generation (`<stdlib.h>`)

rand(): Returns a pseudo-random integer between 0 and RAND_MAX.
srand(seed): Seeds the random number generator. Usually seeded with the current time: srand(time(NULL)).

V. Complex Numbers (C99)

The <complex.h> header adds support for complex and imaginary numbers, essential for engineering and scientific applications.

#include <complex.h>
double complex z = 1.0 + 2.0 * I;

VI. Performance Tip: Fast Inverse Square Root

C is famous for "hacks" that bypass standard math libraries for speed. The Quake III Arena fast inverse square root is a legendary example of using bit manipulation and pointer casting to calculate 1/sqrt(x) significantly faster than the standard library.

Mathematical Operations