The Concepts Behind These Tools
Roots and logarithms are inverse questions. A square root asks "what number, multiplied by itself, gives this?" A logarithm asks the mirror-image question: "to what power must I raise this base to reach this number?" Our log tool defaults to base 10 but accepts any base, because the answer depends entirely on the base — log₁₀(1000) is 3, while log₂(1000) is about 9.97. Cube roots differ from square roots in one respect worth knowing: every real number has a real cube root, including negative numbers, whereas negative numbers have no real square root at all.
Coordinate geometry is subtraction and division. The slope-intercept calculator finds slope as rise over run — the change in y divided by the change in x between two points — then solves for the intercept to produce y = mx + b. The midpoint calculator is even simpler: average the x values, average the y values. The cross product tool steps up to three dimensions, producing a vector perpendicular to both inputs, which is the operation behind surface normals in graphics and torque in physics.
The Pythagorean theorem is a special case, not a universal one. Our Pythagorean calculator solves a² + b² = c², finding the hypotenuse or a missing leg — but only for right triangles. That constraint is the most commonly forgotten part of the theorem, and applying it to a non-right triangle produces a confidently wrong answer.
Averages disagree on purpose. Mean, median and mode are three different definitions of "typical", and the gap between them is informative rather than annoying. Mean is sensitive to outliers; median is not. That is precisely why income is reported as a median and rainfall as a mean. Our tool returns all three at once so you can see when a dataset is skewed — when mean and median diverge sharply, something is pulling the average.
Spread needs two different tools. Standard deviation measures average distance from the mean, and our calculator distinguishes population from sample — the sample version divides by n−1 rather than n, a correction that stops small samples from systematically understating true spread. The IQR takes a different approach entirely: the range of the middle 50%, found via Q1 and Q3. It is outlier-resistant by construction, which is why box plots are built on IQR rather than standard deviation.
Percent error compares against a known truth. The percent error calculator measures the gap between an experimental and a theoretical value as a proportion of the theoretical one — the standard way lab work is graded, and the reason a 1-unit error means something very different at a scale of 10 than at a scale of 10,000.