S-I+G-N Errors

What’s $1 \times 1$? $1$, right?

And $-1 \times -1$? $1$, too.

$-1 / -1$? Still $1$.

Alas, these apparently simple facts can all-too-easily morph into inescapable traps that ensnare and bewilder mathematicians of all calibres. Exacerbated by the presence of brackets and thriving in the shadows of heavy algebraic manipulation, the eternal S-I+G-N error is undoubtedly the oldest menace in the book. It is the canon event of mathematics.

The “sign error”, spelt and said as “S-I-G-N” (/esaizi:en/) to distinguish it from the otherwise audibly congruent “$\sin$ error”, is the mistake of having the wrong $\pm$ sign. Indeed we learn from a young age what happens when signs interact, but accurately manipulating them with perfect consistency would appear to be a lot more nontrivial than other algebra.

Never underestimate its potency, because no matter what situation you’re in, there is always a chance you find the familiar face of the S-I+G-N error smirking at you from the shadows. The frequency of their occurrence can, expectedly, be modelled as a strictly increasing function of the number of maths questions attempted, $Q$.

Yet remarkably, it bears seemingly zero correlation with mathematical expertise or experience (as the above differential equation clearly illustrates). S-I+G-N errors are indiscriminate in their victims, and even the most respected and admired mathematical veterans residing in the upper echelons of algebraic enlightenment are susceptible to its inescapable clutches.

Any arithmetic error will immediately invalidate a solution (except in rare cases of exceptional luck, where errors perfectly cancel out). Indeed, this is true of S-I+G-N errors – but the dread, shame and fury upon encountering one far outshines that which any arithmetic error could evoke. Appropriate reactions range from an exasperated headshake, to a hefty, table-trembling groan with one’s head in their hands.

What makes S-I+G-N errors so maddeningly irksome?

Perhaps it is their inevitability, the fact that no matter how careful we might think we’ve been, the S-I+G+N error is there to ruin a test we would otherwise have scored full marks on. Perhaps it is their humongous catastrophicality-to-significance ratio, when you’ve just spent half an hour solving an integral via quintic polynomial division into quadruple partial fractions with flawless algebra, only for the antiderivative to be ruined by a single mismatch of signs. (F.)

Nevertheless, please be assurred that S-I+G-N errors are perfectly natural. Encountering them every now and then is a part of what it means to math. Do you even math if you’ve never been hit with one? They are the torment of every seasoned mathematician; a shared experience, a common enemy.

S-I+G-N errors have very little to do with one’s “mathematical ability”. Often, they arise as the result of cognitive overload, or merely focusing on matters at hand that are more important – such as fractional arithmetic when completing the square, or drawing a integral sign that is rotationally symmetric. So take it in good humour, and humbly vow to be more careful in future. Because someday, our S-I+G-N errors will have tended to zero.