What is 4 ÷ 2 + 3 × 6?
Unless there are parentheses (or brackets), operations of higher “precedence” are performed before operations of lower precedence.
- addition and subtraction have the lowest precedence.
- multiplication and division are of slightly-higher precedence.
- exponentiation (powers) are even higher than the above.
- pairs of parentheses, brackets, etc. are used for grouping
(and must be ( (very) carefully) “nested”); this must be done before any of the above.
For most operations of the same precedence, evaluation proceeds from left to right, but exponentiation is usually assumed to associate right-to-left:
Applying these rules to your expression, we have:
Your expression: 4 ÷ 2 + 3 * 6
is really equivalent to: (4÷ 2) + (3 * 6)
but the parentheses can be removed if you follow the precedence rules.
There is another way to write such expressions unambiguously, without using parentheses or other bracketing: instead of putting the operator in between the two operands (”infix” notation), place it in front of the two operands (“prefix” notation. If we convert (4÷ 2) + (3 * 6) to this prefix notion,
it becomes: + ÷ 4 2 * 3 6
The mathematician who developed this notation was named Łukasiewicz
but since many people found it difficult to pronounce the this Polish name,
it is often called “Polish notation”. An alternative notation, which places the operator AFTER the two operands, is called “Reverse Polish Notation” (RPN), but it probably should be called “Zciweisakul notation“.
In RPN, (4÷ 2) + (3 * 6) simply becomes: 4 2 ÷ 3 6 * + and no bracketing is ever necessary, regardless of how complicated the expression is!
With calculators (such as the HP-35) that RPN, you enter the operands first, and then you say what to do with them. If there is no operation to perform, simply push the “enter” key to store the operand in the “stack”, and then entern the next operand. When operators such as +, -, *, or / are performed, they pop two operands from the “stack” (and push the result onto the “top-of-stack”). On an HP-35, the sequence for (4÷ 2) + (3 * 6) would be:
- DIVIDE (÷)
- MULTIPLY (*)
- ADD (+)