A full circle has 360 degrees. Not 100, not 400, not 256. The Babylonians picked 360 roughly 4,000 years ago, and we never changed it. The reason is hiding inside the number itself.
360 = 2^3 x 3^2 x 5
Three distinct primes, exponents 3, 2, and 1. The divisor count formula gives (3+1)(2+1)(1+1) = 24 divisors. That is a lot. You can split a circle into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths, twelfths - all landing on whole-degree marks. Try that with 100 (only 9 divisors) or 256 (also 9). The Babylonians did not know the formula, but they knew 360 divided evenly by almost everything they needed.
The calculator that reveals this takes about two seconds. Type 360, click Factorize, read the answer.
Factorization is not just a school exercise
The same trick that explains 360 degrees powers modern encryption. RSA - the algorithm behind HTTPS, online banking and digital signatures - works because multiplying two large primes is fast but reversing that multiplication is computationally infeasible. A 600-digit product of two 300-digit primes would take billions of years to factorize with current hardware.
On a smaller scale, factorization shows up every time you simplify a fraction. Want to reduce 84/360? Factor both numbers: 84 = 2^2 x 3 x 7, 360 = 2^3 x 3^2 x 5. The shared factors are 2^2 x 3 = 12. Divide both by 12 and you get 7/30. No guessing, no trial and error.
Numbers worth factorizing
| Number | Factorization | Divisors | Why it matters |
|---|---|---|---|
| 12 | 2^2 x 3 | 6 | A dozen - packs, eggs, months |
| 60 | 2^2 x 3 x 5 | 12 | Seconds in a minute, minutes in an hour |
| 1024 | 2^10 | 11 | One kilobyte in computing |
| 720720 | 2^4 x 3^2 x 5 x 7 x 11 x 13 | 240 | Smallest number with 6 distinct prime factors |
The pattern is clear: numbers with many small prime factors have disproportionately many divisors. That makes them useful as bases for measurement systems. And factorization is the tool that exposes the structure.
Your taxi bill is a straight line
A completely different kind of math, but equally practical. A taxi charges $3.50 base fare plus $2.10 per mile. The total cost for x miles is:
f(x) = 2.10x + 3.50
That is a linear function. Two numbers define the entire relationship: the slope (2.10 - cost per mile) and the y-intercept (3.50 - the fare before you move). At 5 miles, the bill is $14.00. At 10 miles, $24.50. The relationship is perfectly predictable because the rate of change never varies.
The linear function calculator takes the slope and intercept and gives you the zero (where the line crosses the x-axis), the y-intercept point, and whether the function is increasing or decreasing. For the taxi example, the zero is at x = -1.67 miles - no practical meaning, but mathematically it is the point where the fare would hypothetically hit $0.
Linear functions appear everywhere once you start looking. Battery drain (constant rate per hour), straight-line depreciation (fixed annual loss), water tank emptying (steady flow rate). If the rate of change is constant, the function is linear.
When the line curves: quadratic functions
Not everything changes at a constant rate. Throw a ball upward and gravity accelerates it downward - the height follows a parabola, not a straight line. Revenue depends on price times quantity, and both vary - the relationship is quadratic.
A quadratic function f(x) = ax^2 + bx + c produces a parabola. The calculator extracts five properties from three coefficients:
- Discriminant (delta = b^2 - 4ac) - tells you how many roots exist
- Roots - where the parabola crosses the x-axis (zero, one, or two points)
- Vertex - the lowest or highest point on the curve
- Axis of symmetry - the vertical line through the vertex
- Direction - opens up (a > 0) or down (a < 0)
For f(x) = x^2 - 6x + 8: discriminant is 4 (positive, so two roots), roots at x = 2 and x = 4, vertex at (3, -1). The vertex sits below the x-axis, which makes sense - the parabola dips between its two roots.
When does the discriminant matter?
| Discriminant | Roots | What it looks like |
|---|---|---|
| delta > 0 | Two distinct roots | Parabola crosses x-axis twice |
| delta = 0 | One repeated root | Parabola touches x-axis at vertex |
| delta < 0 | No real roots | Parabola floats above or below x-axis |
A projectile launched at 20 m/s upward from 1.5 m height follows h(t) = -4.9t^2 + 20t + 1.5. The vertex gives the maximum height (21.9 m at t = 2.04 s). The positive root gives the landing time (t = 4.15 s). Two pieces of information from three coefficients.
Three tools, one thread
Prime factorization breaks a number into its building blocks. Linear functions describe constant-rate processes. Quadratic functions describe processes where the rate itself changes. Different tools, but they share a common thread: reducing complexity to a handful of numbers that explain everything.
360 has 24 divisors because its factorization packs three primes into five total factors. Your taxi costs $14 because the slope is 2.10 and the intercept is 3.50. A ball peaks at 21.9 meters because a = -4.9, b = 20, and c = 1.5.
Math is not about memorizing formulas. It is about knowing which tool to reach for.
Tools discussed in this article
- Prime Factorization Calculator - break any number into its prime factors, list all divisors, and count them
- Linear Function Calculator - analyze f(x) = ax + b to find the zero, y-intercept, and slope direction
- Quadratic Function Calculator - analyze f(x) = ax^2 + bx + c to find discriminant, roots, vertex, axis of symmetry
More Math tools
- Percentage Calculator
- Pythagorean Theorem Calculator
- Fraction Calculator
- Average Calculator
- Standard Deviation Calculator
- Proportion Calculator
- Powers & Roots Calculator
- Logarithm Calculator
- GCD & LCM Calculator
Check for a specific number
- Factorization of 100 - prime factors
- Factorization of 360 - 24 divisors
- Is 199 prime? - quick check