1.1.17 · D5Arithmetic & Number Systems
Question bank — Operations on decimals — all four operations
Before we start, one reminder that unlocks almost everything here: a decimal is just a fraction whose denominator is a power of ten. The denominator is simply the number underneath the fraction bar — the size of the equal pieces we are counting. Concretely, Count the digits after the point: that count is the power of ten in the denominator ( digit → , digits → ). When a question feels slippery, rewrite the decimals as those hidden fractions and the fog clears.
Figure 1 — stripping the points, then the multiply-only move.
Figure 2 — the place-value ruler. This is the picture behind every "align the point" answer in the Spot the error section below: when you stack over , Figure 2 shows why the padded zero in matters — it gives the hundredths column a partner so like meets like.
Figure 3 — sign rules for negative decimals. Use this picture on every item that involves a minus sign: the size of the answer comes from the ordinary decimal rules, and Figure 3 tells you which sign to attach. The number-line walk at the bottom is exactly the reasoning behind and in the questions below.
True or false — justify
True or false: .
True — padding with trailing zeros multiplies both top and bottom by ten (), so the value is unchanged; only the number of decimal places shown changes.
True or false: adding a zero in front, , leaves the value the same.
False — a zero between the point and the first digit shifts every digit one column right, so is one-tenth of . Trailing zeros are free; leading zeros after the point are not.
True or false: to add you count the decimal places and give the answer places.
False — counting-and-adding places is a multiply-only move (allowed for alone). For and you align the points (Figure 2) and the answer keeps that shared place value; the sum is , not a -place number.
True or false: multiplying two decimals always gives an answer with more decimal places than either factor.
Usually but not always — the places add to (the two place-counts summed), yet trailing zeros can be dropped. , which shows only one place, fewer than the "two" you counted.
True or false: has one decimal place because has one.
True in the counting sense (), but that place is a trailing zero, so we simply write . The rule still holds; the zero is just cosmetic.
True or false: dividing by a decimal less than makes the answer bigger than the dividend.
True — dividing by something smaller than one asks "how many small pieces fit?", and many small pieces fit, so . Division does not always shrink.
True or false: and give the same answer.
True — multiplying dividend and divisor by the same leaves the ratio unchanged, so both equal . Scaling top and bottom equally is always legal.
True or false: you may change into to clear the point.
False — that moves the point in only one number. You must scale both by , giving ; is a hundred times too small.
True or false: .
True — the two have opposite signs, so (walk it on Figure 3's number line) you subtract sizes () and keep the sign of the bigger size, which is the negative one. The answer is .
True or false: .
False — a negative times a negative is positive (Figure 3), and the places still add (), so the answer is . Only the sign rule changes for negatives; the point-bookkeeping is unchanged.
Spot the error
Someone stacks over by aligning the last digits ( under ) and gets . Where is the mistake?
They aligned digits by right edge instead of by decimal point, so tenths landed under hundredths — apples added to oranges. Align the points along Figure 2's dashed line (pad to ) to get .
A student writes . What went wrong?
They stripped the points () but forgot to add the places: decimal places, so the answer is , not . Stripping the points is only half the rule.
A student computes . Diagnose it.
Same trap — is right, but the product needs places, giving . A product of two numbers each below must be smaller than both, and is bigger than , which is an instant warning.
For a student subtracts (borrowing wrongly) instead of padding. What is the fix?
Pad the top first: , so subtract over a denominator of , giving . Never subtract numbers of different lengths without equalising place value.
A student computes but writes and gets confused borrowing. What is the fix?
Pad the interior zeros first: write so every column exists to borrow from, then over a denominator of , giving . Missing padding zeros leave you with no column to borrow across.
A student computes and, seeing the second number is bigger, writes . What is the correct answer?
Since you are taking away more than you have, the result dips below zero: subtract sizes () and keep the sign of the bigger number, giving (walk it on Figure 3's number line). Dropping the minus sign is the trap.
A student says because . Find the slip.
They cleared the point in the divisor but not the dividend. Scale both by : . The true answer is , ten times their result.
A student writes . Correct the sign.
The size is right (), but a positive divided by a negative is negative (Figure 3), so the answer is . Handle the sign separately from the digit work.
To divide a student places the point one spot too late and writes . What is the correct answer and why?
Keep the point straight above its position in the dividend: but the point sits so the answer is . Since , the quotient must be less than , confirming .
Someone converts to a fraction as . Correct them.
has two decimal places, so the denominator is : it is . The number of places tells you the power of ten in the denominator.
Why questions
Why must we line up the decimal points before adding, but ignore the points when multiplying?
Addition only lets like place values combine (tenths with tenths), which is the same as giving both numbers the same denominator — the same-size pieces underneath the fraction bar (Figure 2). Multiplication instead creates a new denominator by multiplying , so alignment is irrelevant and we strip the points, then add the place counts afterward.
Why do the decimal places add on multiplication instead of, say, multiplying?
Because each factor is a whole number shrunk by and (with the two place-counts); combining shrinks by , and exponents add when powers multiply — the same law behind Scientific Notation.
Why is it legal to multiply both the dividend and divisor by in a division?
Because — scaling top and bottom by the same amount is multiplying by , which never changes a value. It only clears the point from the divisor so long division can run.
Why does padding a decimal with a trailing zero not change its value?
Because a trailing zero just multiplies top and bottom of the hidden fraction by ten () — same value, finer pieces. A zero just after the point, though, shifts real digits into smaller columns and does change the value.
Why does the "count-and-add places" rule never apply to subtraction?
Subtraction combines like columns and keeps the shared denominator; no new power of ten is manufactured. Adding place counts would invent extra shrinking that the operation never performs.
Why can a product of two decimals be smaller than both factors?
Multiplying by a number below scales down. If both factors are below , you scale down twice, so lands below either. This is why "multiply makes bigger" is a whole-number-only intuition.
Why do the sign rules for negative decimals look identical to the sign rules for negative whole numbers?
Because the digit-work and the sign are independent jobs (Figure 3): you compute the size using the ordinary decimal rules, then attach the sign by the same "same signs → positive, different signs → negative" law that governs all numbers. The decimal point never touches the sign.
Edge cases
What is , and why is it not ?
It is — sharing nothing into groups gives nothing per group. Scaling to confirms it; zero divided by any non-zero number is zero.
What happens if you try ?
Undefined — there is no number that, times , gives , so the inverse check fails. Division by zero has no answer regardless of decimals.
What is ?
Exactly — stripping points gives , and no count of decimal places can rescue a product of zero. Zero absorbs everything.
When you multiply , why may you drop the trailing zero in ?
The rule demands places, giving , but trailing zeros after the point carry no value (), so writing is exact, not rounded.
How do you subtract across several zeros, as in ?
Pad the interior with the zeros made explicit (), then treat it as over , giving . Without the padded zeros there is no column to borrow from.
Is (endlessly repeating) a terminating decimal you can pad, and does it equal ?
No — a repeating decimal has infinitely many places, so padding never "finishes"; separately, does equal (a fact from Fractions to Decimals conversion), which is exactly why non-terminating decimals need care that the four basic rules assume away.
If a division like never terminates, how does the "clear the point" method cope?
It cannot — you cannot shift a point that never ends. Keep the exact fraction , or stop after enough digits and round (see Rounding and Significant Figures). The tidy shifting rules only fit terminating decimals.
What is , and why is it cleaner to use the fraction?
Since , dividing by it is multiplying by , so exactly. Trying to "clear the point" on an infinite decimal fails; rewriting the repeating decimal as its fraction is the safe route.
What is ?
Rewrite the repeater as a fraction: . When a repeating decimal is a factor, converting to fractions avoids an endless place-counting mess.
Connections
- Operations on decimals — all four operations
- Place Value and Powers of Ten
- Fractions to Decimals conversion
- Rounding and Significant Figures
- Long Division algorithm
- Scientific Notation