Browsing category ninja maths

Announcing: Ninja|Alpha

After several months of high-intensity development, we're very happy to announce the launch of Ninja|Alpha. Go ahead! Ask it anything you like.

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How the Mathematical Ninja approximates $\ln(5)$

"Isn't it somewhere around $\phi$?" asked the student, brightly. "That number sure crops up in a lot of places!" The Mathematical Ninja's eyes narrowed. "Like shells! And body proportions! And arrawk!" Hands dusted. The Mathematical Ninja stood back. "The Vitruvian student!" The student arrawked again as the circular machine he

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How the Mathematical Ninja approximates $\sin(55º)$

"$0.819$," said the Mathematical Ninja, in as weary a voice as the student used to say "I suppose you're going to tell me how." The nunchaku looked a little rusty, and the axe was in need of a good sharpening. The throwing knives could have done with a clear, and

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How do you estimate the normal distribution for large $z$?

In working out a recent blog post, I had cause to find the probability, in a standard normal distribution, of $z < -23$. Beyond "that's a REALLY SMALL NUMBER"1, I was stumped. Could I get anywhere close mentally? A good question. Starting from the definition of the normal distribution, $p(z)

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The Mathematical Ninja And The Cubes

The Mathematical Ninja glanced at the Rubik Cube and paused. "And $45^3$ is..." A reach for the calculator. A flurry of colour. "Ow!" "91,125," said the Mathematical Ninja, catching the cube on the rebound and swizzling it solved. "Only needed 12 that time." The student sighed. "Go on, then. I

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The Mathematical Ninja and the Tangents Near 1

“Forty-two degrees,” said the Mathematical Ninja, as smugly as possible while still using degrees. The student’s hand had barely twitched towards the calculator. “Go ahead, punk,” said the Mathematical Ninja. “Make my day.” “Righto,” said the student, and tapped in $\tan^{-1} \left( 0.9 \right)$, carefully closing the bracket. “41.987. That’s

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The Mathematical Ninja and the Nineteenths

"Look," said the student, "we all know how this goes down. A nasty-looking fraction comes out of the sum, I reach for the calculator, you commit some act of exaggerated violence and tell me how you, o wondrous one, can do it in your head." "You're not as dumb as

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Why $\phi^n$ is nearly an integer

This article is one of those 'half-finished thoughts' put together late at night. Details are missing, and -- in a spirit of collaboration -- I'd be glad if you wanted to fill them in for me. The estimable @onthisdayinmath (Pat in real life) recently posted about nearly-integers, and remarked that

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Brutal simultaneous equations

I recently became aware of the IYGB papers, available from Madas Maths. Like the Solomon papers, they're intended to stretch you a bit -- they're ranked by difficulty from standard to extremely hard. My student, being my student, demanded we go through one of the extremely hard ones. There were

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On the square root of a third

While I'm no Mathematical Ninja, it does amuse me to come up with mental approximations to numbers, largely to convince my students I know what I'm doing. One number I've not looked at much1 is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it's $\tan\left(\frac{\pi}{6}\right)$2 . Ninja-chops taught me all about

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