Browsing category algebra

The Attack of the Mathematical Zombie: $(a+b)^2$

An occasional series highlighting common errors that refuse to die. “It just… won’t stay dead!” he said, as the Mathematical Zombie moved closer. “$(a+b)^2 = a^2 + b^2$”, it said. “Brains! $(a+b)^2 = a^2 + b^2$.” “But… it doesn’t!” he said. “You have to multiply out the brackets!” “$(a+b)^2 =

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A multiplication pattern

Long ago on Wrong, But Useful, my co-host @reflectivemaths pointed out the ‘coincidence’ that $7\times8 = 56$ and $12 \times 13 = 156$ - a hundred more. In fact, it works for any pair of numbers that add up to 15: $x(15-x) = 15x - x^2$, and $(x+5)(20-x) = 100

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Proving a nice pattern

Over on reddit, noncognitivism posted a nice sequence s/he had come across: $4 + 1 = 5 = \sqrt{ (1)(2)(3)(4) + 1 }$ $(4+6) + 1 = 11 = \sqrt{ (2)(3)(4)(5) + 1}$ $(4+6+8) + 1 = 19 = \sqrt{ (3)(4)(5)(6) + 1}$ $…$ $(4+6+ … + (2k+2)) + 1

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The Mathematical Pirate and the Formula Triangle

The Mathematical Pirate took one look at the piece of paper attached to the dock. “They’re BANNING formula triangles?! By order of @srcav?!” He swished his sword around. “Let me figure out where he lives, I’ll show him.” “He lives… inshore, cap’n” said the $n$th mate. “It’s too dangerous.” “Can

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The Mathematical Pirate’s Guide to Factorising Cubics

“Yarr,” said the Mathematical Pirate. “Ye’ll have plundered a decent calculator, of course?” “Er… well, I bought it from Argos, but… aye, cap’n! A Casio fx-83 GT PLUS!” “A fine calculator,” said the Mathematical Pirate. “One that offers you at least three ways to factorise cubics.” “Really!? I thought you

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A student asks: How do you simplify surds?

A student asks: How could I simplify a sum like $(\sqrt 3+\sqrt 2)(\sqrt 3-\sqrt 2)$? Great question! The trick is to treat it like it's an algebraic bracket, like this: $(x + y)(x - y) = x^2 + yx - xy - y^2$ But then you've got $+yx -xy$ in

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Why the Maclaurin series gives you Pascal’s Triangle

The Mathematical Ninja, some time ago, pointed out a curiosity about Pascal's Triangle and the Maclaurin1 (or Taylor2 ) series of a product: $\diffn{n}{(uv)}{x} = uv^{(n)} + n u'v^{(n-1)} + \frac{n(n-1)}{2} u'' v^{(n-2)} + ...$, where $v^{(n)}$ means the $n$th derivative of $v$ - which looks a lot like Pascal's

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The Dreadful Truth About The Table of Joy

"But I don't liiiiike fractions," said the student. He also didn't like the look of the poker in the Mathematical Ninja's hand, which was beginning to glow red. "Sure you do," said the Mathematical Ninja. "Do I?" "How do you do percentages?" He swished the poker around a bit, as

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There’s More Than One Way To Do It: Arithmetic Series

You've got the formulas in the book, of course. $u_n = a + (n-1)d$ $S_n = \frac n2 \left(a + L\right) = \frac n2 \left(2a + (n-1)d\right)$ This is somewhere the book and I have a serious disagreement: as a mathematical document, it ought to define its terms. $a$ is

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Secrets of the Mathematical Pirate: Switcheroos

"Don't tell the Mathematical Ninja," said the Mathematical Pirate. The student shook his head enthusiastically. "Narr!" "You've got $ \frac {7}{x} = 14$. Ask yourself: what would the Mathematical Ninja do?" "The Mathematical Ninja would do something that looked extremely dangerous and terrifying, but was completely under control." "Correct!" said

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