Order of Operations

The rule

Without an agreed order, 2 + 3 × 4 could be 14 or 20. The convention that settles it is taught as PEMDAS in the United States and BODMAS or BIDMAS in Britain and much of the Commonwealth. The acronyms differ; the rule does not.

So 8 ÷ 2 × 4 is 16, not 1 - you take the division first only because it comes first on the page, not because division outranks multiplication. Likewise 10 − 3 + 2 is 9, not 5.

Worked through, step by step

1. Evaluate 2 + 3 × 4²

   2 + 3 × 4^2

   1. Exponents first: 4² = 16, leaving 2 + 3 × 16.
   2. Multiplication before addition: 3 × 16 = 48, leaving 2 + 48.
   3. Finally the addition.

   = 50

2. Evaluate (6 + 2) × 3 − 4 ÷ 2

   (6 + 2) × 3 - 4 ÷ 2

   1. Brackets first: 6 + 2 = 8, leaving 8 × 3 − 4 ÷ 2.
   2. Multiplication and division rank equally, so work left to right: 8 × 3 = 24.
   3. Then 4 ÷ 2 = 2, leaving 24 − 2.

   = 22

3. Evaluate 100 ÷ 5 ÷ 2

   100 ÷ 5 ÷ 2

   1. Both operations are division, so strictly left to right.
   2. 100 ÷ 5 = 20.
   3. 20 ÷ 2 = 10. Reading right to left would give 40, which is wrong.

   = 10

4. Evaluate 2 × (3 + (4 − 1)²)

   2 × (3 + (4 - 1)^2)

   1. Work the innermost bracket first: 4 − 1 = 3.
   2. Apply its exponent: 3² = 9.
   3. Complete the outer bracket: 3 + 9 = 12.
   4. Then the multiplication outside.

   = 24

The cases the acronyms do not cover

Unary minus binds looser than a power

−5² evaluates to −25, not 25. Exponentiation is applied before the negation, so the expression reads as "the negative of 5 squared". If you want the negative number squared, write (−5)². This is standard across mathematics and across essentially every calculator, but it surprises people constantly.

Stacked exponents group right to left

2^3^2 is 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64. Exponentiation is the one common operator that associates right to left rather than left to right.

Implicit multiplication

This calculator accepts 2π and 3(4+1) without an explicit multiplication sign, treating them as multiplication at the normal multiplication precedence. Note that some handheld calculators give implicit multiplication a higher precedence, so that 6÷2(1+2) yields 1 on those machines and 9 here. There is no universal standard for this case - it is genuinely ambiguous notation, and the honest advice is to add brackets rather than rely on any particular machine's reading.

Functions apply to what follows them

sin(30) + 1 takes the sine of 30 and then adds 1. Because the argument is parenthesised there is no ambiguity - which is exactly why this calculator always inserts the opening bracket when you press a function key.

What this calculator actually does

Expressions are parsed by math.js, a well-established open-source expression parser, wrapped in a thin layer that handles the display symbols. The effective precedence, highest first:

• Every function on the keypad
• What each error message means
• Exponent calculator

Frequently Asked Questions