Intuition The one core ideaA string of numbers and symbols like 2 + 3 × 4 list of jobs , and the whole topic of BODMAS/PEMDAS exists to answer one question: in what order do we do the jobs? Everything below is the toolkit of symbols you must recognise before you can even ask that question — because you cannot order jobs you cannot read.
This page assumes nothing . If the parent note (Order of operations — BODMAS - PEMDAS with nested brackets ) waved a symbol at you and you weren't sure, it is defined here, from the picture up.
An expression is a written recipe made of numbers joined by operation signs (like + × instructions , not yet a single number. Evaluating it means following the instructions until only one number is left.
Think of it like a to-do list. 2 + 3 × 4 "you have an add-job and a multiply-job — do them and report the final number." The whole chapter is about the word order in that sentence.
Figure s01 (described): the expression 2 + 3 × 4 "multiply first" giving 2 + 12 = 14 CORRECT . A white arrow goes right to "add first (wrong)" giving 5 × 4 = 20 WRONG . Same three numbers, two reading orders, two answers — that gap is the entire reason the rule exists.
Definition The two acronyms, expandedThey are two names for the same rule , taught in different countries. Each letter is the first letter of a tier of jobs, listed in the order you do them.
BODMAS = B rackets · O rders · D ivision · M ultiplication · A ddition · S ubtraction.PEMDAS = P arentheses · E xponents · M ultiplication · D ivision · A ddition · S ubtraction.
"Orders" and "Exponents" mean the same thing (powers and roots); "Brackets" and "Parentheses" mean the same thing . The D/M and M/D swap between the two names is harmless — those two live on one shared tier and are always done left to right, as Section 6 shows.
Mnemonic Read the acronym, but remember the tiersThe word is only a memory hook for "Brackets → Powers → ×÷ → +−." Never read "D before M" or "A before S" out of the letters — that is the classic trap the parent note warns about.
These are the four verbs of arithmetic. See Arithmetic operations — addition, subtraction, multiplication, division for their full story; here we only need to recognise them.
Definition The four signs
+ plus ) — combine amounts. Picture: two piles of blocks pushed together.− minus ) — take away. Picture: blocks removed from a pile.× times ) — repeated adding. Picture: 3 rows of 4 dots = 3 × 4 ÷ divide ) — sharing into equal groups. Picture: 12 dots dealt into 3 equal rows.
Definition A fifth way to write "times": juxtapositionWhen two things are written right next to each other with no sign between them , that gap silently means multiply . This is called implicit multiplication (or juxtaposition ).
2 ( 1 + 2 ) 2 × ( 1 + 2 ) 3 a 3 × a Algebraic expressions — evaluating and substitution ).
Why it matters here: juxtaposition is still just multiplication , so it lives on the same tier as × ÷ and obeys left-to-right. In 48 ÷ 2 ( 1 + 2 ) 48 ÷ 2 = 24 24 × 3 = 72 no special priority.
× + 3 × 4 shorthand for 4 + 4 + 4 three additions into one symbol. The topic needs this fact: to honour the shorthand, you must unpack (compute) the multiply before you fold its result into a nearby add. That single observation is the seed of the whole tier order.
Figure s02 (described): a three-rung ladder. Bottom rung — addition 4 + 4 + 4 multiplication as stacked adds 3 × 4 exponent as stacked multiplies 3 2 = 3 × 3
We need this because "Orders / Exponents" is a whole tier of BODMAS. See Exponents and powers — laws of indices for the laws; here we just decode the symbol.
Definition The power notationa n a n write a n
The small raised number n exponent (or power , or index ); the big number a base .
Picture: 3 2 = 3 × 3 square of side 3, holding 9 unit boxes.
Picture: 2 3 = 2 × 2 × 2 cube .
Common mistake The raised-number is not "timesn 3 2 not 3 × 2 = 6 3 × 3 = 9 how many copies to multiply , not a factor.
3 2 = 9 3 × 2 = 6
Worked example Powers of a fraction — where does the exponent reach?A power written on one number touches only that number, not its neighbours.
2 2 1 = ( 2 2 ) 1 = 4 1 2 first (Orders beat the division of the fraction bar), giving 4 1 not ( 2 1 ) 2 3 2 1 = 9 1 ( 1 3 ) 2 = 9 whole fraction , you must bracket it: ( 2 1 ) 2 = 2 2 1 2 = 4 1
Definition Brackets (grouping symbols)Brackets draw a fence around a chunk of an expression, meaning "treat everything inside as one finished number before you do anything outside."
They come in shapes that mean the same thing , used from inside out for readability:
( ) [ ] { }
The shapes carry no difference in power — only nesting depth. See the figure.
Figure s03 (described): the nested expression { 6 + 2 × [ 3 + ( 4 − 1 ) 2 ] } amber innermost box hugs ( 4 − 1 ) cyan box wraps the [ ] white box wraps the outer { } = 30
Intuition Why brackets beat everythingEvery tier rule (powers, then × ÷ + − default . A bracket is the human override switch : "ignore the defaults inside here, and finish this piece first, because the rest of the expression depends on its value." Nested brackets = a value inside a value; you must finish the innermost one first, because the outer one is still waiting on it.
Common mistake "Brackets means only round( ) Fix: [ ] { } fraction bar , and the root sign all grouping fences. Resolve the innermost , whatever its shape.
Definition The square-root sign x "which positive number, multiplied by itself, gives x So 9 = 3 3 × 3 = 9 vinculum ) that stretches over the top is a grouping fence with a lid : everything under the bar is computed first , as if bracketed.
9 + 16 = 25 = 5 9 + 16 9 + 16 = 3 + 16 = 19 9 + 16
We need this because the parent's Example 4 treats a fraction as grouped. See Fractions — numerator and denominator as grouping .
We need this for the parent's Example 5, the − 3 2 Negative numbers and the unary minus sign .
Definition Binary minus vs unary minus
Binary minus sits between two numbers and means subtract : 7 − 2 Unary minus sits in front of one number and means the negative of : − 3
Figure s04 (described): left half shows 7 − 2 − 3 − 3 2 = − ( 3 2 ) = − 9 ( − 3 ) 2 = 9
Definition Left-to-right within a tierWhen two jobs sit on the same tier (both × ÷ + − leftmost job first.
Picture: your eyes sweeping across the line, resolving each equal-rank job as you meet it.
Numbers and the four signs
Multiplication as repeated addition
Juxtaposition means multiply
Exponents as repeated multiplication
Root sign as reverse of squaring
Fraction bar as hidden brackets
Unary minus vs binary minus
Brackets as override fence
Tier ladder Powers over times over plus
Binding strength who grabs first
BODMAS PEMDAS reading order
Evaluation loop innermost first
Each box is a symbol or idea from this page; every arrow says "you must own this before the next makes sense." They all funnel into the one rule the parent teaches.
See also Algebraic expressions — evaluating and substitution and Calculator vs mental arithmetic — parsing expressions for where these symbols travel next.
Cover the right side; say your answer aloud before revealing.
What is an expression, in one phrase? A written list of jobs (numbers joined by operation signs) to be reduced to one number.
What do the letters BODMAS stand for? Brackets, Orders, Division, Multiplication, Addition, Subtraction.
What do the letters PEMDAS stand for? Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Are BODMAS and PEMDAS different rules? No — same rule, different country's name; Orders = Exponents, Brackets = Parentheses.
What does × Repeated addition — 3 × 4 = 4 + 4 + 4
What does it mean when two things are written with no sign between them? Juxtaposition — implicit multiplication, e.g. 2 ( 1 + 2 ) = 2 × ( 1 + 2 )
What does a n Multiply n a n a
Is 3 2 6 No — it is 3 × 3 = 9 2
What is 2 2 1 4 1 2 4 1
What does the "Which positive number times itself gives this?"; the bar brackets everything underneath.
Evaluate 9 + 16 5 9 + 16 = 25
Do [ ] { } ( ) No — all are grouping fences; only nesting depth differs.
What does a fraction bar do to its top and bottom? Brackets the whole numerator and the whole denominator separately.
What are the two jobs of the minus sign? Binary (subtract between two numbers) and unary (negative of one number).
Why is − 3 2 = − 9 The exponent binds tighter than the unary minus: − ( 3 2 ) = − 9
Why is ( − 3 ) 2 = 9 The bracket forces − 3 ( − 3 ) × ( − 3 ) = 9
When two jobs share a tier, how do you order them? Left to right, like reading a sentence.
Evaluate 8 − 3 + 2 7 3
