Mathematics from zero
Place value
Look at 3 and 30. The same little symbol — a 3 — appears in both. Yet one is three and the other is thirty. Something other than the symbol is doing the work. That something is place.
After this lesson you can read any whole number by knowing what each of its digits is worth, and explain why the same digit can mean three, thirty, or three hundred.
Numbers are written with digits. A digit is one of just ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Every number you have ever seen, however huge, is built from only these ten. The trick that makes ten symbols enough is place.
Each spot in a number is a place, and places have names. Reading a number from the right: the first spot is the ones place, the next is the tens place, the next is the hundreds place. Each place is worth ten times the place to its right — one, ten, a hundred.
A digit’s value is the digit times its place. This is called its place value. In the number 247, the 7 sits in the ones place and is worth 7. The 4 sits in the tens place and is worth 4 tens, which is 40. The 2 sits in the hundreds place and is worth 2 hundreds, which is 200. Add them: 200 + 40 + 7 = 247. The number is the sum of its place values.
Zero holds an empty place. Take 30. It has 3 tens and 0 ones. The 0 is not nothing — it is a placeholder. It keeps the ones place filled so the 3 is forced into the tens place. Remove the 0 and the 3 slides back to the ones place, and you are left with plain 3. The zero is what makes thirty look different from three.
Break the number 305 into its place values.
Read it right to left. Ones place: 5 — worth 5. Tens place: 0 — worth 0 tens, an empty place. Hundreds place: 3 — worth 3 hundreds, which is 300. Add them: 300 + 0 + 5 = 305. The 0 in the middle is doing real work: it holds the tens place empty so the 3 stays in the hundreds place. Without it you would write 35 — a completely different number.
Common mistake
The most common mistake is reading a digit by its symbol alone. In 80, the 8 is not “eight” — it is eight tens, which is eighty. Always ask two questions: which digit and which place. The answer is never the digit by itself; it is the digit times its place.
In the number 70, how many tens are there? Type the count.
In the number 70, how many ones are there? Type the count.
The digit 5 sits in the tens place. What is it worth? Type the value.
In 384, which digit is in the hundreds place? Type that digit.
Build the number that has 6 tens and 2 ones. Type the number.
Why is the 0 in the number 40 important?
Numbers are written from ten digits, and a digit’s worth depends on its place. Reading from the right, the places are ones, tens, hundreds — each ten times the last. A digit’s place value is the digit times its place, and a whole number is the sum of its place values. A zero is a placeholder: it keeps an empty place so the other digits stay where they belong.