Math Solver — Algebra, Geometry, Trigonometry & Statistics

The places arithmetic quietly goes wrong

A calculator gives you an answer. Whether it is the answer you wanted depends on a few conventions that are easy to get wrong and rarely explained. These are the ones that produce confidently incorrect results.

Order of operations, and the case where there is no right answer

The standard sequence — brackets, exponents, multiplication and division left to right, then addition and subtraction left to right — settles almost everything. Two details within it are commonly missed:

• Multiplication and division have equal precedence and are evaluated left to right. In 8 ÷ 2 × 4 the division happens first, giving 16, not 1. The same applies to addition and subtraction.
• A minus sign before a squared term is not squared with it. -3² is −9, because the exponent binds tighter than the negation. (-3)² is 9. Spreadsheets are a notable exception and evaluate -3^2 as 9, which is a real source of disagreement between a spreadsheet and a calculator.

Then there is the expression that circulates online every few years, usually written 6 ÷ 2(1+2). It has no settled answer, and that is the actual point. Some conventions treat implicit multiplication — the invisible operator in 2(3) — as binding more tightly than division, giving 1. Others treat it as ordinary multiplication evaluated left to right, giving 9. Physical calculators disagree with each other for exactly this reason. The expression is ambiguous notation rather than a test of arithmetic, and the professional response is to add brackets and remove the question.

Why computers get 0.1 + 0.2 wrong

Ask almost any programming language for 0.1 + 0.2 and it returns 0.30000000000000004. This is not a bug. Binary floating point cannot represent 0.1 exactly, in the same way decimal cannot represent one third exactly — you get an extremely close approximation, and the tiny errors become visible when they accumulate.

The practical consequences are worth knowing. Never test two calculated decimals for exact equality; check that the difference is smaller than some tolerance instead. And for money, work in whole cents as integers wherever you can, because a rounding discrepancy that appears after a thousand transactions is painful to trace.

Rounding is not one rule

School teaches round-half-up: 2.5 becomes 3. Most programming languages and many financial systems use round-half-to-even instead, so 2.5 becomes 2 and 3.5 becomes 4. It looks arbitrary and is not — rounding halves consistently upward introduces a small systematic bias, and over a large number of values that bias compounds. Alternating to the nearest even number cancels it out.

Round once, at the end. Rounding at each intermediate step introduces error that grows with every operation, which is how a column of individually correct figures ends up not matching its own total.

Mean, median, and which one is lying to you

The mean is the sum divided by the count. The median is the middle value when the data is sorted. They agree on symmetric data and diverge sharply on skewed data, and the difference between them is often the most informative thing in a dataset.

Take five salaries: 40,000, 45,000, 50,000, 55,000 and 500,000. The mean is 138,000 — a figure nobody in the group earns and which describes none of them. The median is 50,000, which describes the group well. This is why median is used for house prices and household income: a small number of extreme values drags the mean somewhere unrepresentative.

When a mean and a median differ substantially, the gap is telling you the distribution is skewed. Quote both, or quote the median and say so.

The n versus n−1 question in standard deviation

There are two standard deviation formulas and picking the wrong one is a genuine error, not a rounding preference.

• Population standard deviation divides by n. Use it when your numbers are the entire group you care about — every student in the class, every transaction last month.
• Sample standard deviation divides by n−1. Use it when your numbers are a sample and you want to estimate the spread of the larger population they came from.

Dividing by n−1 (Bessel's correction) exists because a sample's own mean sits closer to its own values than the true population mean does, so the population formula systematically underestimates the real spread. The correction compensates. On large datasets the difference is negligible; on ten data points it is around 5%, which is enough to matter. Most statistical software defaults to the sample formula — check which one a tool is giving you before comparing results across tools.

Reading a quadratic before solving it

For ax² + bx + c = 0, the discriminant b² − 4ac tells you what kind of answer to expect before you compute one:

• Positive — two distinct real roots; the parabola crosses the x-axis twice.
• Zero — one repeated real root; the parabola touches the axis at its vertex.
• Negative — no real roots; two complex ones. The parabola never reaches the axis.

If a word problem produces a negative discriminant, that is usually a signal the problem was set up wrongly rather than an invitation to work in complex numbers.

Percentage error and why the base matters

Percentage error is the absolute difference divided by the true value, times 100. Dividing by the measured value instead is a common slip that produces a different, wrong number.

A related trap: an absolute error means very different things at different scales. Being off by 5 is negligible on a measurement of 10,000 and catastrophic on a measurement of 6. Quoting the percentage rather than the raw difference is what makes an error figure comparable.

Habits that catch mistakes

• Estimate before you compute. Knowing roughly what the answer should be is what catches a misplaced decimal point, and a misplaced decimal is the most common and most expensive arithmetic error there is.
• Check units. If the inputs are in minutes and the answer should be in hours, the number is wrong regardless of the arithmetic. Carrying units through a calculation catches whole classes of error.
• Substitute the answer back. For an equation, putting the solution into the original is a complete check that costs seconds.
• Read the working, not just the result. Where this page shows steps, they are there so you can see where an answer diverges from your expectation — which is the part that teaches you something.

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