Place Value & Decimals
Why the same digit is worth different amounts depending on where it sits.
The explanation
Our number system is a set of columns, and each column is worth ten times the one to its right. In 4,382 the 3 is not just "three" — it sits in the hundreds column, so it means 300.
Decimals keep the same pattern going the other direction. Move right past the decimal point and each column is worth one tenth of the one before: tenths, hundredths, thousandths. So 0.7 is seven tenths and 0.07 is seven hundredths, which is ten times smaller.
That is the whole idea. The digit tells you *how many*, the column tells you *of what*.
Our system is base ten positional notation. A number is a sum of digits multiplied by powers of ten:
4382.7 = 4×10³ + 3×10² + 8×10¹ + 2×10⁰ + 7×10⁻¹
The exponent on 10 is the position index, counting left from the decimal point starting at 0 and going negative to the right. This is why the decimal point is not really a separator between two kinds of number — it is a marker for where the exponent hits zero.
Two consequences worth internalising. First, multiplying by 10 shifts every digit one place left because it adds 1 to every exponent. Second, trailing zeros after the decimal point add nothing, since 0×10⁻³ = 0, which is why 2.5 and 2.500 are the same number.
Worked example
Write 6.043 in expanded form and say what the 4 is worth.
- Label the columns: 6 is ones, 0 is tenths, 4 is hundredths, 3 is thousandths.
- Expanded: 6×1 + 0×0.1 + 4×0.01 + 3×0.001
Answer: The 4 is worth 4 hundredths, or 0.04.
Common mistakes
- Reading 0.07 as bigger than 0.7 because it has more digits. More digits does not mean bigger after the decimal point.
- Lining up decimals by the right edge when adding. Line up the decimal points instead.