Operators

Package: ARITHMETIC_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

An operator is the sign that joins two numbers into a sum: + to add, * to multiply, and so on. Solve reads the usual mathematical symbols, and it reads a word form for each, so a line can be written whichever way reads more naturally.

2 + 2 * 10 // 22
(2 + 3) * 4 // 20
10 - 4 // 6
20 / 4 // 5
2^10 // 1,024
17 mod 5 // 2

% is the percent operator, not modulo. Writing 17 % 5 is a parse error because 17 % is already a complete expression. Use mod or modulo.

mod gives the remainder left over after dividing, the 2 in “17 is three fives and 2 over”. A remainder by zero has none, since the division never ends, and neither does a remainder of an infinite number, so both are refused by name.

5 mod 0 // ERROR: 5 mod 0 has no value: nothing is left over from a division by zero, because it never ends.

Dividing by zero follows the same idea. A number divided by something ever closer to zero grows without limit, so 5/0 answers infinity, with the sign the division would have. Zero divided by zero has no such limit: every number times zero is zero, so every number is an equally good answer, and none is given. It is refused by name rather than answered with NaN (not a number), and so is an infinity divided by an infinity, for the same reason.

5/0 // ∞
-5/0 // -∞
0/0 // ERROR: 0 divided by 0 has no single answer: every number times 0 is 0, so no one quotient is right.

The boundary: only a division is refused. An infinity less an infinity (1/0 - 1/0) is still NaN, and a list divided cell by cell ([0, 1] / 0) keeps a NaN cell, since one cell of a list has no room for a refusal.

^ is the only operator that groups from the right. A tower of powers is worked out from the top down, as in mathematics: 2^3^2 means 2^(3^2), which is 2^9. Everything else groups from the left, so 10-3-2 is (10-3)-2.

2^3^2 // 512
(2^3)^2 // 64

A minus sign at the front of a number belongs to the number, so it is applied before a power: -2^2 is (-2)^2, which is 4. That is how a spreadsheet reads it, and not how a maths textbook does, where -2^2 is -(2^2), which is -4. Neither reading is wrong, so write the brackets whenever it matters. A minus between two numbers is subtraction and waits for the power as usual.

-2^2 // 4
-(2^2) // -4
0 - 2^2 // -4

A fractional power is a root: 8^(1/3) is the cube root of eight, 2. A negative number has a real root when the root is odd, since -2 cubed is -8, and none when it is even, since no real number squared is negative. The engine reads the power as a fraction when it is written as one (1/3) or typed as a decimal (0.2 is a fifth), so an odd root answers, and an even one is refused by name rather than answered with NaN (not a number).

(-8)^(1/3) // -2
(-32)^0.2 // -2
(-8)^(2/3) // 4
(-1)^0.5 // ERROR: (-1)^0.5 has no real value: a negative number to a fractional power has one only when the fraction's denominator is odd, as in (-8)^(1/3).

The boundary: ^ stays in the real numbers. The square root of a negative number has an exact complex answer, and sqrt(-1) gives it, i. A power that is not a fraction at all, such as log(2), has no real value on a negative number either, and is refused the same way.

Operators in words

Most operators have a word form, which is often how a line reads more naturally.

8 times 9 // 72
2 plus 3 // 5
5 minus 3 // 2
10 divide by 2 // 5
3 multiplied by 4 // 12

Multiplication and powers have the longer spellings people say aloud as well: times by and multiply by are *, and to the power of is ^.

6 times by 7 // 42
6 multiply by 7 // 42
2 to the power of 10 // 1,024

with adds and without subtracts, which reads well for a running total.

40 with 2 // 42
40 without 2 // 38

add A to B is the sum written as an instruction. Without its to, the word add is the plus sign it always was. A to followed by a unit is still a conversion, and A to B on its own, with no add in front, is the percentage change from A to B (see percentages).

add 3 to 10 // 13
add $5 to $10 // $15.00
add 3 and 4 // 7
3 to 10 // 233.33%

subtract A from B is the difference written the same way, and takes A away from B, in the order the sentence says. take and remove read the same. Without a from, each of these words is the minus sign it always was.

subtract 3 from 10 // 7
take $5 from $20 // $15.00
subtract 3 km from 10 km // 7.00 km
subtract 3 // -3

The boundary: the from must stand at the top of the line, outside any brackets, since a from inside them belongs to what is bracketed.

The words mul, exponent and prime are not operators. They were once spellings of * and ^, which made them unusable as names (even as :exponent), so they are ordinary names now, and 7 is prime asks whether 7 is prime (see number theory).

exponent = 3 // 3
2 ^ exponent // 8

Symbols pasted from elsewhere

A word processor, a chat client or a web page often replaces a typed hyphen with a minus sign (−) or an en dash (–), and the multiplication and division signs turn up in copied text and on some keyboards. Solve reads all four as the operator they look like, so a line pasted from a document still calculates.

10 − 3 // 7
10 – 3 // 7
3 × 4 // 12
12 ÷ 4 // 3

The mathematical symbols a phone keyboard or a pasted formula carries are read too: ≤ and ≥ compare as <= and >= do, √ is a square root, ∞ is infinity and π is pi. The root binds as tightly as a minus sign, so √16 + 9 is the root of sixteen plus nine, and 2√3 is two times the root of three.

3 ≥ 2 // true
3 ≤ 2 // false
√16 + 9 // 13
√(9 + 16) // 5
2√3 // 3.46
2π // 6.28
1/∞ // 0

An infinity is written ∞ wherever it appears, with a unit or a currency as much as on its own: dividing a distance by zero, or multiplying one past about 1.8e308, the largest number that can be held, gives ∞ km, and it converts like any other distance. It is never written as JavaScript’s own word for it, Infinity, which a note could not read back.

∞ km // ∞ km
-∞ m // -∞ m
5 km / 0 // ∞ km
1e308 * 10 km // ∞ km
∞ km in m // ∞ m
$1e308 * 10 // $∞
1 / 0 as sci // ∞

π is pi only while nothing in the note is named π, so a note that already uses it as a variable keeps its own value. The word infinity is not read, since it is ordinary English in a line of prose.

The em dash (—) is not an operator. It is a sentence mark, and a line that carries one is read as prose.