Before this page: know what each digit is worth in integers and decimals, since every comparison is made one place value at a time.
Putting numbers in order
0.8 is greater than 0.75: 0.8 = 0.80, and 8 tenths is more than 7 tenths, so the hundredths column is never reached.
For positive numbers, line up the place values and compare from left to right. The first place where the digits differ decides the order. On a number line, further right is larger.
Ascending and descending
Ordering numbers means arranging them by size:
ascending order — from smallest to largest
descending order — from largest to smallest
−3, 0, 2, 4 is in ascending order: it reads left to right along the number line. “Write these numbers in order of size, starting with the smallest” asks for ascending order.
Comparing whole numbers
Example: put 4,582, 4,528 and 4,805 in ascending order
- 4,582 sits between 4,500 and 4,600.
- 4,528 is a little further left, so it is smaller than 4,582.
- 4,805 is beyond 4,800, making it the largest.
All three numbers have 4 thousands, so compare their hundreds.
4,805 has 8 hundreds. The other two have 5 hundreds, so 4,805 is the largest.
4,528 and 4,582 first differ in the tens place. Since 2 tens is less than 8 tens, 4,528 comes first.
Ascending order: 4,528, 4,582, 4,805
Comparing decimals
To compare 3.7, 3.17, 3.705 and 3.07, line up the decimal points so that digits with the same place value sit underneath each other.
Placeholder zeros added to the end of a decimal change nothing, and they give each decimal the same count of places:
3.7 = 3.70 = 3.700
Example: put 3.7, 3.17, 3.705 and 3.07 in ascending order
- 3.7 sits seven tenths past 3.
- 3.17, or 3.170, is much closer to 3 than 3.7 is.
- 3.705 is a little to the right of 3.700.
- 3.07, or 3.070, is the closest of the four to 3.
| number | ones | decimal point | tenths | hundredths | thousandths |
|---|---|---|---|---|---|
| 3.7 | 3 | point | 7 | 0 | 0 |
| 3.17 | 3 | point | 1 | 7 | 0 |
| 3.705 | 3 | point | 7 | 0 | 5 |
| 3.07 | 3 | point | 0 | 7 | 0 |
The tenths digits put 3.070 first, 3.170 second, and the two numbers beginning 3.70 last.
3.700 and 3.705 first differ in the thousandths place, so 3.700 is smaller.
Ascending order: 3.07, 3.17, 3.7, 3.705
Ordering negative numbers
Every negative number is smaller than zero, and so smaller than every positive number.
Among negative numbers, the value closer to zero is larger, so a temperature of −3°C is higher than one of −8°C.
Example: put 5, −1, −6, 0 and 2 in descending order
- 5 is the furthest value to the right.
- −1 sits one step left of zero, halfway to −2.
- −6 is the furthest value to the left.
- 0 separates the negative and positive values.
- 2 sits between 0 and 5.
The positive numbers come first in descending order: 5, then 2.
Zero comes after the positive numbers and before the negative numbers.
Of the negatives, −1 is closer to zero than −6, so −1 is larger.
Descending order: 5, 2, 0, −1, −6
Ordering a mixture
In −2.4, 0.6, −2.04, 1 and 0.06, the two negatives are smaller than the three positives, and each group is then put in order by place value.
Example: put −2.4, 0.6, −2.04, 1 and 0.06 in ascending order
- −2.4 lies between −2.5 and −2, left of zero.
- 0.6 sits between 0.5 and 1.
- −2.04 is to the right of −2.4, so it is larger.
- 1 is the largest of the five.
- 0.06 is positive, a little to the right of zero.
Write the negative decimals as −2.40 and −2.04: 2.40 is further from zero than 2.04, so −2.40 is the smaller.
Write the positive values as 0.06, 0.60 and 1.00. Their place values put them in that order.
Ascending order: −2.4, −2.04, 0.06, 0.6, 1
Key points
- Compare place values from left to right, starting with the greatest place.
- Padding a decimal with zeros does not change its value: 3.7 = 3.700.
- For negative numbers, the number nearer zero is greater.
- Ascending means smallest to greatest; descending reverses that order.