Decimals — place value, reading and writing: Question bank

Figure — Decimals — place value, reading and writing

Definition Two kinds of zero — read this first

Several traps below hinge on where a zero sits, so let's fix the vocabulary once:

A trailing zero sits at the far right end, after the last non-zero digit: the last $0$ in $0.30$ . It fills an already-empty smaller place, so it changes nothing.
An internal zero sits between the point and a non-zero digit: the $0$ in $0.06$ . It is a placeholder that pushes the digits after it into smaller places, so it changes the value.

A note on notation: this page cloze-hides a few key answer words with ==double equals== — in Obsidian these render as a highlight you can quiz yourself on. When you read them, treat the highlighted word as the "hidden answer." I also write "divided by $10$ " or " $/10$ " in words rather than the symbol "÷", to keep every step unambiguous.

True or false — justify

True or false:

0.7

is larger than

0.65

True. Line up places by rewriting

0.7 = 0.70

. Now the tenths compare first:

0.7

has

7

tenths,

0.65

has only

6

tenths, and one whole tenth (

0.1

) already outweighs the

5

hundredths (

0.05

) that

0.65

has left over. So we decide it at the tenths and never need the hundredths.

True or false:

0.3 = 0.30 = 0.300

True. Watch the fractions grow their denominators together:

\frac{3}{10} = \frac{30}{100} = \frac{300}{1000}

. Each trailing zero multiplies top and bottom by

10

, which is multiplying by

1

— so the amount is untouched, only the way of writing it changed. See Comparing and Ordering Decimals.

True or false:

0.09

is larger than

0.1

because

9 > 1

False. Compare the biggest place first:

0.1

has

1

tenth,

0.09

has

0

tenths. Since one tenth (

0.10

) is already bigger than nine hundredths (

0.09

), the tenths settle it:

0.1 > 0.09

. Judging by "

9 > 1

" wrongly treats the digits as whole numbers.

True or false: adding a zero in the middle of a decimal never changes its value.

False. That is an internal zero. In

0.6

the

6

is in tenths (

\frac{6}{10}

); inserting a zero to get

0.06

shoves the

6

into hundredths (

\frac{6}{100}

), which is ten times smaller. Only trailing zeros are free.

True or false: the decimal point is itself a place with a value.

False. It is only a marker that says "the ones place ends here." It holds no digit and no value — like the fence in the figure, it separates but occupies no ground of its own.

True or false: every decimal place is a power-of-ten fraction.

True. Tenths

= \frac{1}{10}

, hundredths

= \frac{1}{100}

, thousandths

= \frac{1}{1000}

— the denominators are

10, 100, 1000

, i.e.

1 0^{1}, 1 0^{2}, 1 0^{3}

. See Fractions.

True or false: "

3.5

" and "three and five hundredths" name the same number.

False.

3.5

is three and five tenths (

\frac{5}{10}

); "three and five hundredths" puts the

5

one place further right (

\frac{5}{100}

), giving

3.05

. The fraction-name fixes which place the last digit lands in.

True or false: whole numbers grow ×10 leftward and decimals shrink by dividing by

10

rightward using the same rule.

True. It is one continuous Base-10 Number System rule extended past the ones place; the point just marks where we cross below

1

. Left:

1 \to 10 \to 100

; right:

1 \to \frac{1}{10} \to \frac{1}{100}

Spot the error

Find the mistake: "

0.45 > 0.5

because

45 > 5

The error is comparing them as whole numbers. Fix it by lining up places: rewrite

0.5 = 0.50

, then compare tenths (

5

4

). Five tenths beats four tenths, so

0.5 > 0.45

Find the mistake: "Seven and five hundredths

= 7.5

"Hundredths" fixes the last digit in the 2nd place. Since only "five" was named for that place, the tenths place is empty and must be filled with

0

7.05

. Writing

7.5

wrongly puts the

5

in tenths.

Find the mistake: "

0.256

is read 'two hundred fifty-six point'."

Digits after the point are read individually — "zero point two five six" — because each lives in a different shrinking place (

\frac{2}{10}, \frac{5}{100}, \frac{6}{1000}

), not one shared place worth

256

Find the mistake: "Three hundred four thousandths

= 3.04

"Thousandths" puts the last digit in the 3rd place, so it is

\frac{304}{1000} = 0.304

. The internal zero keeps

3

in tenths and

4

in thousandths;

3.04

would instead be three-and-four-hundredths, a completely different number.

Find the mistake: "In

34.256

the digit

2

is worth

2

The

2

sits in the tenths place, so its value is

2 \times 0.1 = 0.2

, not

2

. Place value always multiplies the digit by the place's worth, and here that worth is a tenth.

Find the mistake: "

0.5

means five-hundred-somethings, it's a big number."

The point makes it less than one:

0.5 = \frac{5}{10} = \frac{1}{2}

, exactly half of one whole. It sits between

0

and

1

, so it is smaller than

1

, not larger.

Find the mistake: "To make

0.3

bigger, just add a zero:

0.30

Adding a trailing zero changes nothing —

0.30 = 0.3

because

\frac{3}{10} = \frac{30}{100}

. To actually grow the value you must change a digit, e.g.

0.4

Why questions

Why do we read digits after the point one at a time instead of as a group?

Because each digit lives in a different, shrinking place (tenths, then hundredths, then thousandths). Reading "

256

" would falsely claim they share one place worth

256

, when really they are

\frac{2}{10} + \frac{5}{100} + \frac{6}{1000}

— three separate, ever-smaller amounts.

Why does the last digit's place name the whole fraction?

The last place is the smallest, so it sets the common denominator that all the earlier digits can be rewritten over. In

0.256

the last digit is thousandths, so everything becomes

\frac{200 + 50 + 6}{1000} = \frac{256}{1000}

Why must each place right of the point be exactly one-tenth of the place to its left?

To keep the single unbroken "divide by

10

each step" pattern of the Base-10 Number System. If one step divided by something else, the places would no longer be powers of ten and our reading and comparing rules would break.

Why write a leading zero in

0.6

instead of just

.6

The lone point is tiny and easy to miss, so a reader might see "

6

" and think six. The leading zero makes the boundary obvious; the value is identical, since a zero in the ones place adds nothing.

Why does

0.60

equal

0.6

but

0.06

does not?

0.60

the extra zero is trailing — it fills an empty hundredths place, adding

0

hundredths, so nothing changes. In

0.06

the zero is internal: it occupies the tenths place and shoves the

6

down into hundredths, making the number ten times smaller.

Why do we compare decimals from the left, not the right?

Because the leftmost place after the point (tenths) is worth the most, and a single unit there (

0.1

) outweighs any combination of the smaller places to its right. So a difference in tenths settles the comparison before we ever look further right.

Edge cases

What number is

0.0

It is simply zero. Every place after the point holds

0

, so no pieces are counted — its value equals the whole number

0

5.0

a decimal or a whole number?

Both descriptions fit: the

.0

means zero tenths, so its value equals the whole number

5

; the trailing zero is decoration and can be dropped.

What does

0.000

mean, and how big is it?

All three fractional places are empty, so it is exactly

0

— three trailing zeros add no tenths, hundredths, or thousandths.

Between

0

and

1

, is there a "first" decimal just after

0

No. You can always go smaller —

0.1, 0.01, 0.001, \dots

— because each new place divides by

10

forever, so there is no smallest positive decimal.

Which is larger,

0.9

0.10

0.9

. Compare tenths first:

9

tenths vs

1

tenth. The extra digit in

0.10

is a trailing zero and adds nothing, so

0.9 > 0.10

Where does the ones digit go in

7.05

, and why is it not "after the point"?

The

7

is the whole part, on the left of the fence; only pieces smaller than one live right of the point. See Place Value in Whole Numbers.

- 0.4

smaller or larger than

- 0.5

- 0.4

is larger. For negatives the sign flips ordering:

- 0.4

is closer to

0

on the number line, and being nearer zero means greater. So

- 0.4 > - 0.5

, even though

0.4 < 0.5

without signs.

- 0.3

the same as

0.3

No. The minus sign puts it on the opposite side of

0

- 0.3

is three tenths below zero,

0.3

is three tenths above. They are equally far from

0

but are different numbers.

What does the repeating decimal

0.333 \dots

equal, and why can't we write it with finitely many digits?

It equals the fraction

\frac{1}{3}

. Dividing

1

3

never leaves a remainder of

0

— you keep getting a remainder of

1

— so the digit

3

repeats forever; no finite string of tenths, hundredths, … lands exactly on a third.

0.9999 \dots

(nines forever) less than

1

==No — it equals exactly

1

.== The nines never stop, so there is no gap left to fill; the difference from

1

would have to be smaller than every decimal place, which means it is

0

Connections

Decimals — place value, reading and writing — the parent these traps drill
Comparing and Ordering Decimals — left-to-right place comparison behind the true/false items
Fractions — why every place is a power-of-ten fraction, and where repeating decimals come from
Rounding Decimals — relies on the same place names
Base-10 Number System — the master divide-by-ten rule the edge cases stress-test