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Occasional Random thoughts from us.
Why 0.999... Is Exactly Equal to 1
Not "very close to" 1. Not "infinitely close to" 1. Exactly 1. Here are four proofs, why the result feels wrong, and what it reveals about what numbers really are.
Why Is the Derivative of x² Equal to 2x?
Grow a square by a sliver and watch where the new area goes: two thin strips and a corner too small to matter. Here is the picture, the limit, and the 1734 critique that made calculus rigorous.
Why Does the Quadratic Formula Work?
The quadratic formula isn't a magic incantation to memorize. It's what you get when you complete a literal square, an idea from 9th-century Baghdad. Here is the derivation, step by step.
The meaning of Pi
As a child, you may recall learning about Pi in math class. Remembering the term may have triggered a smell of your grandmother’s baked pies. However…
Why Does the Area of a Circle Equal πr²?
Slice a pizza into thin wedges, line them up, and you get a rectangle that's πr wide and r tall. Here are three ways to see why a circle's area is πr², including Archimedes' original.
Why Is the Sum of the Angles in a Triangle 180°?
On a flat page, a triangle's angles always add to 180°. On a globe, you can draw a triangle with three right angles. Here is the proof, and why it depends on one famous assumption.
Teaching math in today’s century
If you’re a parent raising a child in the 21st century in the public-school system, you’re most definitely no stranger to the idea that the way math is…
Why Does the Pythagorean Theorem Work?
a² + b² = c² has hundreds of known proofs, including one by a U.S. president and one by two high school students. Here are three of the clearest, and why the theorem is really about area.
Why Does Cross Multiplication Work?
Cross multiplication is just multiplying both sides of an equation by the same thing, done in one move. Here is why it works, where it doesn't, and the 1,500-year-old "rule of three" behind it.
Why Does Dividing by a Fraction Mean Multiplying by Its Reciprocal?
"Keep, change, flip" gets the right answer, but why? Here are three explanations, from counting pizza slices to algebra, and a study showing how rarely even teachers could explain it.
Why Does Anything to the Power of Zero Equal 1?
Multiplying 5 by itself zero times gives 1, not 0. Here are three explanations, from patterns to exponent rules to the "empty product," plus the long-running argument about 0 to the power 0.
Why Is a Negative Times a Negative Positive?
The novelist Stendhal gave up on math partly because nobody could explain this rule to him. Here are four explanations, from patterns and debts to a short proof and a 180° rotation.