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Place value to 1000

Numbers · level 2, lesson 1 of 7

A question to start

The numbers 307 and 370 use exactly the same three digits: a 3, a 0 and a 7. Yet one of them is much bigger than the other. Which one? And how can the very same digits make two different numbers?

Here is an even stranger one. Write the digit 5 three times: 555. All three digits are fives. But they are not worth the same! One of them is worth five hundred, one is worth fifty, and one is worth just five.

The secret is the most useful idea in the whole history of writing numbers. It has a name: place value. In this lesson you will learn it, all the way up to a thousand.

The idea

Ones, tens and hundreds

In level 1 you saw that ten ones make a ten. Now go one step further: ten tens make a hundred.

Classrooms often use base-ten blocks to see this. A small cube is one. A long stick of ten cubes is a ten (we call it a rod). A flat square made of ten rods side by side is a hundred (we call it a flat).

110100

Count the little squares in the flat: 10 rows of 10. That is 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. So a flat really is a hundred ones, all stuck together. You do not have to count them, because you know how it was built.

What a three-digit number means

Here are 2 flats, 4 rods and 3 cubes.

2 hundreds4 tens3 ones

We write this number as 243, and we say "two hundred forty-three". Each digit tells you how many of one kind of block:

Hundreds Tens Ones
2 4 3

The place of a digit (where it sits) tells you which kind of block it counts:

  • the digit on the right counts ones;
  • the digit next to it counts tens;
  • the digit on the left counts hundreds.

So 243 is worth 200 + 40 + 3. Writing a number as a sum of its parts like this is called expanded form. It shows the job each digit does.

The same digit, a different value

Now look at 555. The three fives sit in three different places:

Hundreds Tens Ones
5 5 5
worth 500 worth 50 worth 5

The digit is the same, but its value (how much it is worth) depends on where it sits. Each place to the left is worth ten times as much as the place to its right: a ten is ten ones, and a hundred is ten tens.

Now the puzzle. In 307, the 3 is worth 300, the 0 means "no tens", and the 7 is worth 7. In 370, the 3 is still worth 300, but the 7 has moved into the tens place, so it is worth 70. So 370 is bigger. How much bigger? 70 instead of 7. You will be able to work out the exact gap in the next lesson (it is 63).

Zero holds the place

Look again at 307. The 0 says "there are no tens". Why not just leave it out? Because then 307 would become 37, and the 3 would slide into the tens place and be worth only 30. The zero is a placeholder: it keeps the place open, so every other digit stays where it belongs.

That little zero was a huge invention. Our digits and this place-value way of writing came from India, more than a thousand years ago, and reached Europe through Arabic-speaking mathematicians. That is why they are often called Hindu-Arabic numerals. In the last lesson of this level you will see some older ways of writing numbers that had no zero, and how hard they made adding.

Reading and writing numbers in words

To say a three-digit number, say the hundreds, then the tens and ones together, just like a two-digit number:

  • 243: "two hundred" + "forty-three" = two hundred forty-three.
  • 708: "seven hundred" + "eight" = seven hundred eight (no tens, so nothing is said for them).
  • 560: "five hundred" + "sixty" = five hundred sixty.
  • 915: "nine hundred" + "fifteen" = nine hundred fifteen.

In Britain and some other places people say "two hundred and forty-three". Both ways are fine.

To write a number from words, fill in the places one at a time. "Six hundred four": 6 hundreds, no tens, 4 ones, so 604. The zero goes in the tens place, because there are no tens.

A hyphen joins the tens and ones words from twenty-one to ninety-nine: forty-three, sixty-eight.

A thousand

What comes after 999? It is 9 hundreds, 9 tens and 9 ones. Add one more:

  • the ones become 10, which make a new ten, so the ones go to 0;
  • the tens become 10, which make a new hundred, so the tens go to 0;
  • the hundreds become 10. And ten hundreds make a thousand.

So 999 + 1 = 1000, one thousand. With blocks, ten flats stacked up make a big cube. Every new place is ten of the place before it: ten ones, ten tens, ten hundreds. That rule never stops, so place value works for numbers as big as you like.

Comparing three-digit numbers

To compare two numbers, start with the biggest place: the hundreds. The number with more hundreds is bigger, no matter what comes after. 412 is bigger than 398, because 4 hundreds are more than 3 hundreds, even though 98 is more than 12. Why? Because the most that "98" can add is 99, and that is still less than one more hundred.

If the hundreds are the same, compare the tens. If the tens are the same too, compare the ones.

We use two signs for comparing: > means "is greater than" and < means "is less than". The open side faces the bigger number, like a hungry mouth: 412 > 398, and 398 < 412.

Many ways to make the same number

243 is 2 hundreds, 4 tens and 3 ones. But you could break one of the hundreds into 10 tens. Then you have 1 hundred, 14 tens and 3 ones. It is still the same number (100 + 140 + 3 = 243), just packed differently. You can also say 243 is 24 tens and 3 ones. Breaking a bigger block into ten smaller ones, or bundling ten small ones into one bigger one, is called regrouping. It is the secret behind subtraction in the next lesson.

Worked examples

1. Reading the places. In 586, what is each digit worth?

The 5 is in the hundreds place: 500. The 8 is in the tens place: 80. The 6 is in the ones place: 6. In expanded form, 586 = 500 + 80 + 6. In words: five hundred eighty-six.

2. Words to digits. Write "nine hundred twelve" in digits.

9 hundreds, then "twelve" is 1 ten and 2 ones. So the number is 912.

3. A zero in the middle. Write "four hundred seven" in digits.

4 hundreds, no tens, 7 ones: 407. Without the zero it would be 47, which is only forty-seven.

4. Comparing. Which is bigger, 629 or 692?

Both have 6 hundreds. Compare the tens: 629 has 2 tens, 692 has 9 tens. So 692 > 629.

5. Regrouping. How many tens are there in 350, all together?

3 hundreds is 30 tens (each hundred is 10 tens), and there are 5 more tens. So 350 is 35 tens. Check: 35 tens is 350 ones, because 35 rods of 10 make 350.

Try it

  1. What is the 7 worth in each number: 703, 370, 37?
  2. Write in expanded form: 482, 609.
  3. Write in digits: three hundred fifty-one; eight hundred; five hundred six.
  4. Write in words: 274, 830.
  5. Put these in order from smallest to biggest: 518, 158, 581, 185.
  6. What number is 10 more than 395? 100 more than 395?
  7. A number has 6 hundreds, 0 tens and 13 ones. Write it as a normal three-digit number. (Careful!)
  8. Using the digits 4, 0 and 9 once each, make the biggest and the smallest three-digit number you can.
Answers
  1. In 703 the 7 is worth 700. In 370 it is worth 70. In 37 it is worth 7.
  2. 482 = 400 + 80 + 2. 609 = 600 + 0 + 9, which we usually write as 600 + 9.
  3. 351; 800 (8 hundreds, no tens, no ones); 506 (no tens, so a zero holds the tens place).
  4. 274 is two hundred seventy-four. 830 is eight hundred thirty.
  5. Compare hundreds first: 158 and 185 have 1 hundred, 518 and 581 have 5. Then compare tens in each pair: 158, 185, 518, 581.
  6. 10 more: 395 has 9 tens; one more ten makes 10 tens, which is a new hundred. So it becomes 4 hundreds, 0 tens and 5 ones: 405. 100 more: 3 hundreds become 4 hundreds: 495.
  7. 13 ones is 1 ten and 3 ones. So the number is 6 hundreds, 1 ten and 3 ones: 613. (Writing "6013" would be wrong: each place can only hold one digit, from 0 to 9.)
  8. Biggest: put the biggest digit in the biggest place: 940. Smallest: the hundreds digit cannot be 0 (then it would not be a three-digit number), so use 4 in the hundreds, then 0, then 9: 409.

Common mistake

Writing what you hear. Some people hear "three hundred five", write 300, and then put the 5 after it: "3005". But the number has only 3 hundreds, 0 tens and 5 ones, so it has only three digits: 305. Fill the places, do not glue the words together.

Comparing by the last digit. 299 is not bigger than 301 just because 9 is bigger than 1. Always start comparing from the left, the biggest place.

Where it leads