Decimals

Decimals #

A fraction whose bottom number is ten.

A decimal is a fraction whose denominator is ten, or ten times ten, and on down

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Cut the whole bar into 10 pieces, then color in 3 of them.

3 pieces out of 10 is 3 tenths. This can be written as a decimal: 0.3.

After the dot, digits count smaller steps: 2.3 is two wholes and three tenths.

Hundredths #

Cut every tenth into ten again.

Cutting every tenth into ten again gives you hundredths

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Cut every tenth of 0.3 into ten and you get 100 little pieces. 30 are colored.

30 out of 100 is 0.30 — the same amount, counted in smaller pieces.

Thousandths #

The third place after the point.

Cutting every hundredth into ten again gives you thousandths

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Between 0.12 and 0.13 there is room. Cut each hundredth into ten: thousandths.

Thousandths sit third after the point: 0.125 is 125 of the 1000 pieces.

Cents stop money at two places. A length is cut as finely as the ruler allows.

Comparing Decimals #

More digits does not mean bigger.

A longer decimal is not automatically a bigger one

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0.35 is less than 0.40.

Multiplying and Dividing Decimals by 10, 100, 1,000 #

The digits move; the point stays.

Multiplying or dividing by ten moves every digit one place; the decimal point stays where it is

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2.5 × 10 is not 2.50 — adding a zero fails. Each digit moves one place left: 25.

× 100 moves each digit two places left: 0.7 × 100 = 70, and a 0 holds the empty ones place.

Dividing moves the digits the other way, to the right: 45 ÷ 10 = 4.5.

3.2 ÷ 100 = 0.032: the 3 and the 2 each move two places right, and zeros hold the empty places.

A thousand moves each digit three places, and a multiple splits: × 30 is × 3, then × 10.

Lining Up the Dot #

Tenths add up like any matching pieces.

Tenths add up just like any other matching pieces

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Here are three tenths and four tenths.

Count the colored tenths: 3 + 4 = 7, so 0.3 + 0.4 = 0.7

0.5 Is a Half #

The few worth knowing by heart.

A few decimals are worth knowing by heart because they name simple fractions

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Half the bar is colored in.

Cut it into ten instead and five pieces are colored. Half is 0.5

Tenths name fifths too: $\frac{1}{5}$1/5 is 2 of the ten pieces, 0.2, and $\frac{2}{5}$2/5 is 0.4.

A quarter will not fit tenths. Cut into 100 instead: a quarter is 25 of them.

So a quarter is 0.25, and three quarters is 75 of the 100 pieces: 0.75.

Adding in Columns #

Line up the points, not the ends.

Line the points up and every digit lands in the column where it belongs

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Line up the two points, not the two ends. Look at the gap under the 5.

That gap is a hundredths cell. Fill it with a zero, which changes nothing.

Now add each column as usual, and bring the point straight down.

Subtracting in Columns #

Borrowing works across the point too.

Borrowing works across the point exactly as it works anywhere else

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The bottom number reaches into hundredths, and the top one stops at tenths.

Write the missing zero, so both numbers have a hundredths digit.

0 − 8 will not go, so borrow a tenth — one tenth is ten hundredths.

10 − 8 = 2, 2 − 1 = 1, 5 − 2 = 3. So 5.3 − 2.18 = 3.12.

Counting the Decimals #

Multiply without the points, then count them.

Multiply as though the points were not there, then count how many digits followed them

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0.3 has one decimal place, and so does 0.4. That is 2 decimal places in total.

Do 3 × 4 = 12 first. 3 is 10 times more than 0.3, and 4 is 10 times more than 0.4.

So 0.3 × 0.4 is 12 ÷ 100: each digit moves two places right, giving 0.12.

Dividing a Decimal #

The point rises straight up into the answer.

Divide exactly as usual, and bring the point straight up into the answer

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9.6 ÷ 4. Set it out just as you would set out 96 ÷ 4.

4 goes into 9 twice, with 1 left over to carry on past the point.

16 tenths ÷ 4 = 4 tenths, and the point rises straight up: 9.6 ÷ 4 = 2.4.

A whole number divides on too: 7 is 7.0, and 7 ÷ 2 carries past the point to 3.5.

Dividing by a Decimal #

Scale both up until the divisor is whole.

Scaling both numbers by the same amount leaves the answer completely unchanged

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Sharing into 0.4 of a group makes no sense. Make that 0.4 a whole number.

Division also asks how many fit: how many 0.4s make up 6?

Multiply both by ten: every digit moves one place left, so 6 becomes 60 and 0.4 becomes 4.

Both grew ten times over, so the count is unchanged: 6 ÷ 0.4 is 60 ÷ 4.

Now it is ordinary division: 60 ÷ 4 = 15, so 6 ÷ 0.4 = 15 too.

Rounding Decimals to One and Two Decimal Places #

The digit just after the cut decides.

A decimal rounds to whichever neighbor is nearer, and the halfway mark decides

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3.47 sits above the halfway mark of 3.45 and rounds up to 3.5

2.62 is below the halfway mark of 2.65 and rounds down to 2.6

Two decimal places work the same: 1.283 is below the mark of 1.285, so 1.28.