Browsing category algebra

Ask Uncle Colin: Why does the line with equation $10y+36x=16.5$ have a gradient of -3.6?

Dear Uncle Colin, I've got a line with equation $10y+36x=16.5$. That equation has no negative numbers in it, yet its gradient is apparently negative. I don't understand why. -- Silly Line, Only Positive Equation Dear SLOPE, It looks like we're in misconception-land! In fact, you can write the equation of

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Ask Uncle Colin: Simultaneous Equations

Dear Uncle Colin, Could you please tell me how to solve simultaneous equations? I have a rough idea, but I get confused about it. -- Stuck In Mathematical Examinations/Qualifications Hello, SIMEQ! Here’s how I attack linear simultaneous equations, such as: $5x + 6y = -34$ (A) $7x + 2y =

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A STEP expansion

A STEP question (1999 STEP II, Q4) asks: By considering the expansions in powers of $x$ of both sides of the identity $(1+x)^n (1+x)^n \equiv (1+x)^{2n}$ show that: $\sum_{s=0}^{n} \left( \nCr{n}{s} \right)^2 = \left( \nCr{2n}{n} \right)$, where $\nCr{n}{s} = \frac{n!}{s!(n-s)!}$. By considering similar identities, or otherwise, show also that: (i)

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Further Decimal Curiosities

This post is based on work by Mark Ritchings; I know of no finer1 maths tutor in Bury. A few weeks ago, I pointed in the vague direction of a few decimal curiosities -- fractions that spit out lovely patterns in their decimal expansions. Having found one that generated the

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Ask Uncle Colin: Is my friend crazy?

Dear Uncle Colin, A friend of mine told me that $1 + 2 + 4 + 8 + ... = -1$. Is he crazy, or is there something going on here? -- Somehow Enumerating Ridiculous Infinitely Extended Sum Dear SERIES, There are a couple of 'proofs' of this non-fact that

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Ask Uncle Colin: A logarithmic coincidence?

Dear Uncle Colin, I noticed that $2^{\frac{1}{1,000,000}} = 1.000 000 693 147 2$ or so, pretty much exactly $\left(1 + \frac{1}{1,000,000} \ln(2)\right)$. Is that a coincidence? Nice Interesting Numbers; Jarring Acronym Dear NINJA, The easiest way to see that it's not a coincidence is to check out $3^{\frac{1}{1,000,000}} $, which

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Ask Uncle Colin: a disguised quadratic

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can. Dear Uncle Colin, How do I solve $3x^{\frac{2}{3}} + x^{\frac{1}{3}}-2

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Getting your $x$s in one basket

A reader asks: I need to solve $\frac ac \frac {NP}{N_0 + N} = mP$ for $N$, and I don't know where to start. Help! I had a maths teacher in the early 90s who loved nothing more than making the class groan with bad jokes. If she showed up

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Simplifying

A student asks: How do I simplify horrible product and quotient rule expressions? The example they gave was differentiating: $f(x) = (2x - 3)^4 (x^2 + x + 1)^5$ First up, a careful bit of product rule: $u = (2x - 3)^4$, so $\diff{u}{x} = 8(2x- 3)^3$ $v = (x^2

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The Mathematical Ninja and The $n$th Term

Note: this post is only about arithmetic and quadratic sequences for GCSE. Geometric and other series, you're on your own. Quite how the Mathematical Ninja had set up his classroom so that a boulder would roll through it at precisely that moment, the student didn't have time to ponder. He

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