Noesis
1.1.6 · D2Arithmetic & Number Systems

Visual walkthrough — Order of operations — BODMAS - PEMDAS with nested brackets

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We will derive the whole ladder using ONE running example and watch it collapse to a single number:

Every step has three parts: WHAT we do, WHY we do it, and a PICTURE of it.

Step 1 — Counting is the ground floor

WHAT: we start by naming the simplest operation — combining amounts.

WHY: you cannot understand why one operation "beats" another until you know which one is the foundation. Everything else will turn out to be a shortcut built on top of this. See Arithmetic operations — addition, subtraction, multiplication, division.

PICTURE: below, two heaps of dots slide into a single heap. Nothing is hidden — the total is just how many dots there are.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 2 — Multiplication is repeated addition (one floor up)

WHAT: we unfold into three identical additions.

WHY this matters for order: the symbol is a package. Inside the box is . Until you open the box (multiply it out to ), you are holding a sealed parcel — you don't yet know the number. That single fact is the seed of the whole hierarchy.

PICTURE: the red box wraps three copies of a pile. The label on the box (“”) is just a promise; the dots inside are the truth.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 3 — Why multiply before you add

WHAT: we justify the rule " before " — not as a decree, but as a consequence.

WHY: if you added the loose to the visible first, you'd be reaching inside the sealed package and stealing one of its copies — which changes what the box even means. That's cheating.

PICTURE: the loose dot “2” waits outside; the red box must be unpacked before the two can merge.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 4 — Exponents are repeated multiplication (top floor)

WHAT: we climb one more floor. Multiplication was repeated addition; a power is repeated multiplication.

WHY it sits above : by the exact same sealed-box logic, is a box-inside-a-box. It packs up multiplications, which themselves pack up additions. To be honest to the shorthand you must unwrap the outermost shorthand first — the power — before you can even use it in a multiplication.

PICTURE: a two-layer box. The outer label “” hides two copies of being multiplied; open it to reveal .

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 5 — The ladder, and our example falling down it

WHAT: we now evaluate end to end, obeying the ladder we derived.

Term by term:

  • — highest shorthand, opened first.
  • — next box down, opened next.
  • — the loose finally merges.

WHY this order and no other: each arrow opens the most tightly packed box available. You can never merge with a box that is still shut.

PICTURE: the expression as a stack of nested boxes; we peel them from the inside out, and the accent-red pointer marks which box we open at each moment.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 6 — Brackets: the override switch

WHAT: we add the one tool that can override the natural ladder.

WHY we need it: the ladder gives one fixed reading. Brackets let a human choose a different reading on purpose. They don't break the rule — they sit on a tier above powers.

Compare, same digits, one bracket difference:

PICTURE: the red bracket forcibly seals “” into a single value before anything else runs. See also Fractions — numerator and denominator as grouping.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 7 — Edge case: the unary minus is not a power

WHAT: we handle the sign, the most common degenerate case.

WHY: the exponent only "sees" the digit sitting immediately beneath it — the minus is a separate, lower-tier operation applied after. Only a bracket can hand the minus to the base.

PICTURE: two rows. Top: the exponent grabs only the red , minus applied last → . Bottom: the bracket makes the whole red the base → .

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

Step 8 — Edge case: same-tier ties go left → right

WHAT: we resolve what happens when two operations sit on the same rung.

WHY: the ladder only tells you which tier wins. Inside a tier there is a genuine tie, and the tie-breaker is the reading direction of the page: left to right. The same applies to vs (e.g. ).

PICTURE: an arrow sweeps left to right; the red operation is whichever one it reaches next.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets

The one-picture summary

Everything above, compressed: the ladder on the left, our example collapsing box-by-box on the right, and the two override/tie rules pinned below.

Figure — Order of operations — BODMAS - PEMDAS with nested brackets
Recall Feynman retelling — say it to a 12-year-old

Think of an expression as a set of boxes inside boxes. Plain adding is just loose dots on the table. A times sign is a box that hides a stack of equal piles. A power is a box hiding a stack of multiplications — a box inside a box. You're only allowed to combine numbers you can actually see, so you must open the most tightly-sealed box first: powers, then times/divide, then add/subtract. Brackets are you, the boss, drawing a box wherever you want and shouting "open THIS one first!" — they beat everything. Two warnings: the little minus in is not part of the power, so the power opens first and you get ; and when two jobs are the same size (like and , or and ), there's no winner, so you just read them left to right like words on a page. Follow the boxes and every person on Earth gets the same number.

Recall Quick self-test

Evaluate ::: (power , then , then ). Evaluate ::: (bracket seals first). Evaluate ::: (power binds before the minus). Evaluate ::: (bracket puts the sign in the base). Evaluate ::: (same tier, left to right). Evaluate ::: (same tier, left to right).

Connections