Expanded form explained
Three ways of taking a number apart, all called expanded form, plus a fourth term that means opposite things in two countries.
The three expanded forms
Expanded form breaks a number into what each column contributes. Three versions of it are taught, and which one is wanted depends on the exercise.
- Expanded form: 1,000 + 500
- Expanded factor form: (1 × 1,000) + (5 × 100)
- Expanded word form: one thousand + five hundred
They say the same thing at different levels of detail. The first shows the values, the second shows where each value came from, and the third puts it in words.
| Number | Expanded | Factor form | Word form |
|---|---|---|---|
| 1500 | 1,000 + 500 | (1 × 1,000) + (5 × 100) | one thousand + five hundred |
| 4507 | 4,000 + 500 + 7 | (4 × 1,000) + (5 × 100) + (7 × 1) | four thousand + five hundred + seven |
| 90 | 90 | (9 × 10) | ninety |
| 2026 | 2,000 + 20 + 6 | (2 × 1,000) + (2 × 10) + (6 × 1) | two thousand + twenty + six |
Zeros are left out
A column with a zero in it contributes nothing, so it does not appear. 4,507 expands to 4,000 + 500 + 7, with no term for the empty tens column. This trips people up, because the zero does important work in the original number even though it is worth nothing.
The standard form trap
Worth knowing before an exam, because the same words mean opposite things in different countries.
- In British schools, standard form means scientific notation: 1,500 is 1.5 × 10³. A number between one and ten, multiplied by a power of ten.
- In American schools, standard form means the ordinary digits: 1,500. What a British pupil would call standard notation.
So “write 1,500 in standard form” has two different right answers depending on where the question was set. The expanded form tool prints both and labels which is which.
Scientific notation in one line
Move the decimal point until exactly one non-zero digit is in front of it, then count how many places you moved it. That count is the power of ten. 1,500 becomes 1.5 with the point moved three places, so it is 1.5 × 10³. For a number below one you move the other way and the power is negative: 0.0015 is 1.5 × 10⁻³.
Why it is taught
Expanded form is place value made visible. Written out as a sum, it becomes obvious why column arithmetic works, why a digit moving one place left multiplies it by ten, and why an amount that is out by a factor of a thousand is usually one misplaced comma rather than a wrong digit.