Place value & the base-ten system
The idea
Our whole number system runs on one clever trick: where a digit sits decides what it is worth. With only ten digits, 0 through 9, you can write any number ever needed, because each spot in a number is worth ten times the spot to its right. In 473, the 4 does not mean four — it means four hundreds. The 7 means seven tens, which is seventy. Only the 3 means plain old three.
Picture numbers as bundles. Ten loose straws make one bundle of ten. Ten bundles of ten make one big pack of one hundred. Writing 473 is just a tidy record: 4 packs, 7 bundles, 3 loose straws. A common mix-up is reading the 7 in 473 as just seven. It is really 70, because it lives in the tens place. Whenever a number puzzles you, unpack it into its hundreds, tens, and ones, and it usually stops being puzzling.
Worked example
A school collects pencils for art class. Helpers fill 4 big boxes with 100 pencils each, 7 small bags with 10 pencils each, and have 3 loose pencils left over. How many pencils did they collect in all?
- Count the big boxes first: each one holds 100 pencils, so 4 boxes hold 4 × 100 = 400 pencils.
- Now the bags: each bag holds 10 pencils, so 7 bags hold 7 × 10 = 70 pencils.
- Add the groups from biggest to smallest: 400 + 70 = 470, and then 470 + 3 = 473 pencils.
- Look closely at the answer 473: the digit 4 records the boxes of 100, the 7 records the bags of 10, and the 3 records the loose pencils. The written number is simply a tidy list of the groups.
Answer. The school collected 473 pencils — 4 hundreds, 7 tens, and 3 ones.
Check your understanding
- Why is the 5 in 52 worth more than the 5 in 25, even though it is the very same digit?
- What happens to the value of a digit when it moves one place to the left in a number?
- How would you use bundles of straws to show why ten ones can be traded for one ten?
- What would be hard about writing big numbers if every spot in a number were worth the same amount?