Browsing category algebra

Ask Uncle Colin: My partial fractions decompose funny

Dear Uncle Colin, I recently had to decompose $\frac{3+4p}{9p^2 - 16}$ into partial fractions, and ended up with $\frac{\frac{25}{8}}{p-\frac{4}{3}} + \frac{\frac{7}{8}}{p-\frac{4}{3}}$. Apparently, that's wrong, but I don't see why! -- Drat! Everything Came Out Messy. Perhaps Other Solution Essential. Hi, there, DECOMPOSE, and thanks for your message - and your

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An innovative algebraic approach

Usually, when faced with a word problem, I take the most obvious approach and call it done. But then, sometimes, I read of an alternative approach that makes me go "Whoa." This is one of those times. Here's the problem: One day, a person went to a horse racing area.

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Ask Uncle Colin; A variable fraction

Dear Uncle Colin, I have a fraction, $\frac{x^2-x}{x-1}$, and I want to cancel it down to $x$ - but I'm not sure those are the same. Are they? - Got A Lot Of Interesting Sums Hi, GALOIS, and thanks for your message! The short answer is, yes and no. Everywhere

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Ask Uncle Colin: Why is it not 4?

Dear Uncle Colin, I have a binomial expansion of $(1+x)^\frac{1}{2}$ and need to approximate $\sqrt{5}$. Apparently you need to substitute in $x=\frac{1}{4}$, but I'd have thought $x=4$ was a more obvious choice. What gives? -- Roots Are Dangerous If Understood Sloppily Hi RADIUS, and thanks for your message! That does

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A nice bit of nrichment

My dear friend @ajk44 pointed me at a puzzle on the excellent Nrich site1, and I enjoyed it enough to share my solution. (If you don't want it spoiled, don't read beyond the blockquote.) Four jewellers had respectively 8 rubies, 10 sapphires, 100 pearls and 5 diamonds. Each gave one

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On quadratic sequences

This is a guest post from Mark Ritchings, a maths tutor in Bury. A quadratic sequence is a sequence for which the $n$th term is $an^2+bn+c$. The constants $b$ and/or $c$ might be zero but $a$ definitely isn't. The first term is $a\times 1^2 +b \times 1 + c =

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Ask Uncle Colin: A Three-Variable Simultaneous Equation

Dear Uncle Colin, I have a system of equations I can't solve! $x + y + z = 100$; $.08x +.1y +.2z = 12$; $y - z = 25$ I keep tripping up on the decimals and negative signs! -- We're Extremely Stressed Solving Equations Linear Hello, WESSEL! My best

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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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