Which is larger: 0.7 or 0.65?
A learner who compares digit strings may say 0.65 is larger because 65 is larger than 7.
Place value gives a different answer.
0.7 = 7 tenths = 70 hundredths.
0.65 = 6 tenths + 5 hundredths = 65 hundredths.
So 0.70 > 0.65.
Decimal comparison is place-value comparison. The length of the digit string is irrelevant unless the positions are understood.
This is the central problem decimals create for learners who are secure with whole numbers but still treat digits as if they carry the same value wherever they appear.
The current Singapore Primary Mathematics syllabus places decimal notation, representations and place values including tenths, hundredths and thousandths in Primary 4. The same strand includes comparison, ordering and conversion between suitable fractions and decimals. That means decimal place value should be built as a number system before later arithmetic is expected to become fluent.
The base-ten system continues below one
Whole-number place value moves left by multiplying by 10:
- 10 ones = 1 ten;
- 10 tens = 1 hundred;
- 10 hundreds = 1 thousand.
Decimal place value extends the same relationship to the right:
- 10 tenths = 1 whole;
- 10 hundredths = 1 tenth;
- 10 thousandths = 1 hundredth.
Each move one place right divides the unit value by 10.
This is why 3 in the ones place means 3, while 3 in the tenths place means 0.3 and 3 in the thousandths place means 0.003.
Read decimals by place value
Take 4.308.
- 4 ones = 4;
- 3 tenths = 0.3;
- 0 hundredths = 0;
- 8 thousandths = 0.008.
So:
4.308 = 4 + 3/10 + 8/1000.
The zero matters because it preserves the hundredths place.
Without it, 4.38 is a different number.
Thousandths are not “three digits after the point”
The phrase can help locate the place, but it should not become the concept.
One thousandth is one of 1,000 equal parts of a whole:
1/1000 = 0.001.
Seven thousandths:
7/1000 = 0.007.
Forty-two thousandths:
42/1000 = 0.042.
Three hundred eight thousandths:
308/1000 = 0.308.
Equivalent decimal notation
0.5 = 0.50 = 0.500.
Why?
Five tenths = fifty hundredths = five hundred thousandths.
Trailing zeros rename the same quantity in finer base-ten units.
This is useful when comparing:
0.7 = 0.700
and
0.698.
Now the comparison is explicit:
700 thousandths > 698 thousandths.
Compare from the largest place first
Compare 2.407 and 2.39.
Whole-number parts are both 2.
Tenths:
2.407 has 4 tenths.
2.39 has 3 tenths.
Therefore 2.407 is already larger. There is no need to compare further places.
Or write 2.39 as 2.390.
Then 407 thousandths > 390 thousandths.
The number line prevents string thinking
Place 0.4 and 0.5 on a number line.
Between them lie hundredths such as 0.41, 0.42 and 0.49.
Zoom again between 0.41 and 0.42.
Now thousandths such as 0.411, 0.412 and 0.419 appear.
This shows that thousandths are positions inside intervals, not simply extra characters after a point.
Worked example: place value in 5.026
5.026 contains:
- 5 ones;
- 0 tenths;
- 2 hundredths;
- 6 thousandths.
Expanded form:
5 + 0.02 + 0.006.
As a fraction beyond the whole:
5 + 26/1000.
Worked example: order 0.507, 0.57 and 0.5075
If working only to thousandths, compare the first two directly:
0.507 and 0.570.
0.570 is larger because 5 tenths match, then 7 hundredths exceeds 0 hundredths.
The value 0.5075 extends beyond thousandths into ten-thousandths, which is mathematically valid even if outside the immediate Primary 4 syllabus precision. It lies between 0.507 and 0.508.
This is a useful boundary reminder: the decimal system does not stop where a syllabus level stops.
Common misconception 1: longer decimal means larger decimal
0.125 is not larger than 0.9.
Repair: compare tenths first or rename 0.9 as 0.900.
Common misconception 2: decimal places behave like separate whole numbers
3.42 is not “3 and 42” without unit meaning.
It is 3 ones, 4 tenths and 2 hundredths.
Common misconception 3: zeros never matter
0.5 = 0.50, but 0.5 ≠ 0.05.
Trailing zeros can preserve value; positional zeros can preserve place.
Common misconception 4: 0.004 means four hundredths
0.004 is four thousandths.
Four hundredths is 0.04.
A diagnostic ladder
- Can the learner explain 10 tenths = 1 whole?
- Can the learner explain 10 hundredths = 1 tenth?
- Can the learner explain 10 thousandths = 1 hundredth?
- Can the learner read a decimal by place value?
- Can the learner explain zeros in numbers such as 4.008?
- Can the learner rename 0.7 as 0.700?
- Can the learner compare decimals from the largest place first?
- Can the learner place tenths, hundredths and thousandths on a number line?
- Can the learner connect suitable fractions with denominators 10, 100 or 1000 to decimal notation?
How this fits Singapore Primary 4 Mathematics
The updated October 2025 MOE Primary Mathematics syllabus includes decimals up to 3 decimal places in Primary 4, including notation, representations, place values of tenths, hundredths and thousandths, comparison and ordering, and connections between suitable fractions and decimals.
This means place value is the organising concept. Arithmetic comes later and depends on it.
The deeper lesson
Whole numbers and decimals are not separate systems.
They are one base-ten structure extending in both directions from the ones place.
Once each digit is attached to its unit, decimal notation stops being a string and becomes quantity.
Final thought
0.507 is not “507 with a dot”.
It is five tenths, zero hundredths and seven thousandths.
That sentence is longer than the notation.
That is exactly why notation is useful: it compresses place value without changing it.