Exact answers, and where they run out

A third plus a sixth is a half, and it says so

Add ⅓ and ⅙ by hand and you put them over a common denominator: 2/6 + 1/6 = 3/6 = ½. Exact, and no decimal involved at any point. Type the same thing in:

• (
• 1
• ÷
• 3
• )
• +
• (
• 1
• ÷
• 6
• )
• =

That matters when the answer carries on into the next line. A third stored as 0.3333333333 is already wrong in the eleventh digit, and three of them add up to 0.9999999999 rather than 1. Kept as ⅓, three of them are exactly 1 — which is the answer a person would give.

One decimal point changes the whole line. Ans shows a fraction unless your input contained a decimal point or a ×10^ exponent — so (1÷3)+(1÷6) gives 1/2, while (1÷3)+0.5 gives 0.8333333333. The rule is that the calculator answers in the language you asked in, and the moment you introduce a decimal you have said you want decimals.

Why sin(π) is 0 and sin(3.141592654) is not

π is not 3.141592654. That is a ten-figure approximation, and the difference shows up the moment you do something sensitive with it. The sine of π is exactly zero — that is what π is, the point where the sine curve comes back to the axis. Ask for it in radians both ways:

• sin
• π
• )
• =

The display is not where the exactness lives. Press π and = on its own and you get 3.141592654 — ten figures, like any other number, because that is what a ten-figure display can show. Nothing symbolic appears on screen. What is kept is underneath, in the value the next function receives, and the pair of answers above is the proof: the same π that printed as digits went into sin as π.

Four ten-billionths is not an error worth worrying about on its own. It is worth understanding: the calculator holds rational multiples of π as multiples of π, so anything that lands on a clean value comes out clean — sin(π) and tan(π) are 0, cos(2π) is 1, and sin(π÷6) is exactly 0.5 rather than 0.4999999999. Type the digits instead and you are asking a slightly different question, and getting an honest answer to it.

This is a small mercy with a large cousin. The same idea is why exact fractions matter in a long calculation: every rounding is a tiny lie, and lies accumulate. Keeping the exact form until the last possible moment is the single most effective habit in numerical work, and it costs nothing.

Where the exactness stops

Ans keeps two families exactly and no others: fractions with whole-number tops and bottoms, and rational multiples of π. That list is deliberate rather than unfinished — those two cover the answers school and A-level work actually produces, and each has a compact exact form a calculator can carry through a calculation without a computer algebra system behind it. Everything else becomes a decimal at the point it is computed. So there are no surds, and there is a ceiling.

√2 is a decimal here

√2 gives 1.414213562, not a symbol. There is no exact form kept for it, so a chain of surds will drift in the last digits where a symbolic system would stay perfect. If your working needs √2 to survive as √2, it has to survive on your paper.

And there is a ceiling

The engine works up to — but not including — 10¹⁰⁰. Factorials cross that line in one step, which makes them the neatest way to see it:

An instrument that stopped quietly, or wrapped round to a small number, or offered infinity, would be worse than one that refuses. Math ERROR is a complete answer to the question "can you hold this?" — and knowing exactly where the boundary sits, at 10¹⁰⁰ inclusive, means you can work out in advance whether a calculation will reach it.