Noesis
1.1.17 · D2Arithmetic & Number Systems

Visual walkthrough — Operations on decimals — all four operations

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Before any symbol appears, let us agree on one picture.

Step 1 — A decimal is columns of shrinking squares

WHAT. Take the number . We draw it as boxes. One big box = one whole unit. A strip cut into equal slices = one tenth each. So = three whole boxes + four tenth-strips.

WHY. Before we can say why points line up or places add, we must first see that a digit's meaning depends entirely on which column it sits in. The same digit "4" is worth four wholes, or four tenths, or four hundredths — purely by position. That positional worth is the hero of the whole story (see Place Value and Powers of Ten).

PICTURE.

So the very first truth is: a digit's value (its face) (its column's power of ten). Hold that.

Step 2 — Turn the whole thing into ONE fraction

WHAT. We rewrite with no dot at all, as a single fraction. Multiply the face-value picture out:

WHY. A dot is awkward — you cannot "add" or "multiply" a dot. But a fraction over a power of ten is a plain whole number () sitting over a plain whole number (), and we already know every rule for those. Converting the dot away is what makes all four operations collapse into familiar whole-number arithmetic (this is the bridge in Fractions to Decimals conversion).

PICTURE. The four tenth-strips get glued next to thirty tenth-strips (each whole box re-sliced into ten): identical little strips, each worth .

Step 3 — Addition: same denominator or bust

WHAT. Add . Translate each using Step 2, but first make the denominators match:

WHY. You may only add fractions that are cut into the same size pieces. is cut into tenths; into hundredths. Tenths and hundredths are different-size pieces — apples and oranges. Multiplying top and bottom of by (that's the padding zero: ) re-cuts it into hundredths without changing its value, because .

PICTURE. Watch the tenth-strips of each split into ten hundredth-tiles so both numbers are measured in the same tiny tile — then the piles slide together.

Step 4 — Subtraction is the same picture run backwards

WHAT. . Pad to a common denominator, then subtract numerators:

WHY. Subtraction is "take away," but you can only take away like-sized pieces. Same padding trick, same common-denominator reason. Nothing new — which is the point: and are one rule.

PICTURE. From a pile of hundredth-tiles we physically remove tiles; remain.

Step 5 — Multiplication: why the places ADD

WHAT. . Translate both: Multiply the fractions straight across — tops together, bottoms together:

WHY the bottoms combine as . Multiplying two powers of the same base means the exponents add: . That is the reason the decimal places of the two factors add to give the answer's places. Here place places places — so we shift the point hops into , giving . (Same exponent-adding law powers Scientific Notation.)

PICTURE. An area rectangle: width , height . The grid tiles are each in area, and there are of them.

Step 6 — Division: make the divisor whole, then share

WHAT. . Write it as a fraction and scale top and bottom by the same power of ten:

WHY , and why both. The divisor has decimal places, so turns it into the whole number — now ordinary Long Division algorithm applies. We must multiply the top by the same because : scaling numerator and denominator equally leaves the value untouched. Scale only one and you've changed the answer.

PICTURE. Left: sharing among -sized groups (awkward, fractional groups). Right: the identical question rescaled to — the shape of the answer, , is the same, only the units grew on both sides.

Degenerate check — dividing by a whole number. If the divisor is already whole, and you scale by (do nothing): . Just divide and keep the point directly above: . Verify: ✓.

Step 7 — The degenerate & edge cases you must never trip on

WHAT / WHY / what it looks like, gathered:

  • A trailing zero is free. because — you re-cut the pieces smaller but the total shaded amount is identical.
  • Multiplying by a number shrinks. is smaller than . The picture: the height is a thin sliver, so the area is tiny. Not an error — expected.
  • A product can lose a digit. ; the final is dropped to write . The place count () was still correct; trailing zeros just aren't written.
  • Dividing by something grows. . Sharing into tiny groups yields many groups. Again expected, not a slip.

The one-picture summary

One diagram, four operations, one law () doing all the work:

Recall Feynman retelling — say it to a friend with no symbols

A decimal is just a whole number wearing a disguise: erase the dot and remember how many steps it fell to the right — that's the "over ten, over a hundred" hidden underneath. To add or subtract, first make both numbers wear the same disguise (same number of steps, i.e. line up the dots and pad with zeros), then treat them as plain whole numbers and re-hang the dot straight down. To multiply, strip both dots, multiply the plain numbers, and count how many steps the two disguises had together — that many steps is where the dot goes back, because shrink-by-ten times shrink-by-ten shrinks by a hundred (the little exponents add up). To divide, slide both numbers the same number of steps until the one you're dividing by is a clean whole number, then just share it out the ordinary way. Every rule you memorised is that one fraction-over-ten idea, seen four times.

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