Place value in whole numbers
We write numbers with only ten digits: 0 to 9. The place of a digit tells us how much it is worth. From the right, the places are ones, tens, hundreds, thousands, ten thousands and so on.
Value of a digit = digit ร value of its place. In 3 452: 3 ร 1000 + 4 ร 100 + 5 ร 10 + 2 ร 1. Writing a number like this is called expanded form.
The digit itself (4) is its face value. What it is worth (400) is its place value.
Zero as a place holder
In 2 705 the 0 shows there are no tens. Without it, 275 would be a very different number.
Big numbers
Many countries group digits in threes: 1 000 000 (one million). In India we also use lakhs and crores: 10,00,000 = 10 lakh = 1 million. The place values are the same; only the commas move.
Decimals: places smaller than one
After the decimal point the places keep getting 10 times smaller: tenths (0.1), hundredths (0.01), thousandths (0.001).
0.36 = 3 tenths + 6 hundredths = 0.3 + 0.06.
Multiplying by 10 moves every digit one place left: 4.27 ร 10 = 42.7. Dividing by 10 moves every digit one place right: 4.27 รท 10 = 0.427.
Comparing and ordering with <, > and =
Symbols: < means "is less than", > means "is greater than", = means "is equal to". The open mouth faces the bigger number.
Method: 1) Line up the numbers at the decimal point (or at the ones). 2) Fill empty decimal places with zeros so they have the same length. 3) Compare digits from the left. The first place where they differ decides.
Example: 0.5 and 0.47 โ 0.50 and 0.47 โ tenths: 5 > 4, so 0.5 > 0.47. A longer decimal is not always bigger.
Ascending order = smallest to largest. Descending = largest to smallest.
Negative numbers and the number line
Integers are the whole numbers, their negatives and zero: โฆ โ3, โ2, โ1, 0, 1, 2, 3 โฆ
On a number line, numbers grow to the right. Every negative number is less than 0 and less than every positive number.
For two negatives, the one further from zero is smaller: โ7 < โ2. Think of temperature or a bank balance: owing โน700 is worse than owing โน200.
We use negatives for temperatures below 0 ยฐC, floors below ground (โ1 for basement), heights below sea level and money owed.
Try it
Take 5 shop bills or price tags at home. Write each amount in a place-value chart (thousands to hundredths). Then order them from cheapest to costliest. In the 3D, type each amount in free play and check the towers.
Key formulas and definitions
- Value of a digit = digit ร place value
- Each place = 10 ร the place on its right
- 1 = 10 tenths = 100 hundredths = 1000 thousandths
- a < b means a is left of b on the number line
Worked examples
1. Write the value of 6 in 46 213.
6 is in the thousands place, so its value is 6 ร 1000 = 6000.
2. Write 7 305.04 in expanded form.
7 ร 1000 + 3 ร 100 + 0 ร 10 + 5 ร 1 + 0 ร 0.1 + 4 ร 0.01 = 7000 + 300 + 5 + 0.04.
3. Which is greater: 0.7 or 0.65?
Write 0.70 and 0.65. Tenths: 7 > 6. So 0.7 > 0.65.
4. Order in ascending order: 3.08, 3.8, 3.008, 3.18.
Make same length: 3.080, 3.800, 3.008, 3.180. Compare tenths then hundredths: 3.008 < 3.08 < 3.18 < 3.8.
5. Put < or >: โ9 ___ โ4.
โ9 is further left of zero than โ4, so โ9 < โ4.
6. Order from smallest to largest: 2, โ5, 0, โ1, 4.
Start from the far left of the number line: โ5, โ1, 0, 2, 4.
Common mistakes
- Thinking a longer decimal is bigger: 0.125 is less than 0.3 because 1 tenth < 3 tenths.
- Mixing face value and place value: in 582 the face value of 8 is 8, its place value is 80.
- Saying โ8 > โ3 because 8 > 3. For negatives, further from zero means smaller.
- Forgetting zero as a place holder: writing three thousand and five as 305 instead of 3005.