Demystifying - Ordering integers, decimals and negatives

Introduction

Before this page: know what each digit is worth in integers and decimals, since every comparison here is made one place value at a time.

Numbers do not always look like their size. 0.75 has more digits than 0.8 but is smaller. −7 looks larger than −2 and is not. 9,999 and 10,000 sit next to each other despite having different lengths. Sorting by eye works until one of these appears, and then it fails quietly: the answer looks reasonable and is wrong.

Ordering is the habit that takes the guesswork out. Compare one place value at a time, or read the positions along a number line, and the size of a number stops depending on how it happens to be written.

What does it mean to order numbers?

Ordering numbers means arranging them according to their value:

  • ascending order — from smallest to largest

  • descending order — from largest to smallest

A number line makes the order visible. Numbers further to the left are smaller, and numbers further to the right are larger.

A number line from negative four to four The values increase from negative four on the left to four on the right. An arrow above the line labels the direction from smaller to larger. values increase −4−3 −2−1 01 234 smaller larger

For example, −3, 0, 2, 4 is in ascending order: it reads from left to right along the number line.

Ordering whole numbers

Place value tells us which of two positive whole numbers is larger.

First compare the digits in the largest place. If they are equal, move one place to the right and compare again. Stop at the first place where the digits are different.

Example: put 4,582, 4,528 and 4,805 in ascending order

  1. 4,582 sits between 4,500 and 4,600.
  2. 4,528 is a little further left, so it is smaller than 4,582.
  3. 4,805 is beyond 4,800, making it the largest.

  1. All three numbers have 4 thousands, so compare their hundreds.

  2. 4,805 has 8 hundreds. The other two have 5 hundreds, so 4,805 is the largest.

  3. 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

Ordering decimal numbers

Line up the decimal points so that digits with the same place value sit underneath each other.

You can add zeros to the end of a decimal without changing its value. This makes decimal numbers the same length and easier to compare:

3.7 = 3.70 = 3.700

Example: put 3.7, 3.17, 3.705 and 3.07 in ascending order

  1. 3.7 can be written as 3.700.
  2. 3.17, or 3.170, is much closer to 3 than 3.7 is.
  3. 3.705 is just to the right of 3.700.
  4. 3.07, or 3.070, is the closest number to 3.

  1. Add placeholder zeros: 3.700, 3.170, 3.705, 3.070.

  2. The tenths digits put 3.070 first, 3.170 second, and the two numbers beginning 3.70 last.

  3. 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 every positive number.

Among negative numbers, the value closer to zero is larger. This means −2 is larger than −7, even though 7 is larger than 2.

Example: put 5, −1, −6, 0 and 2 in descending order

  1. 5 is the furthest value to the right.
  2. −1 belongs just to the left of zero.
  3. −6 is the furthest value to the left.
  4. 0 separates the negative and positive values.
  5. 2 sits between 0 and 5.

  1. The positive numbers come first in descending order: 5, then 2.

  2. Zero comes after the positive numbers and before the negative numbers.

  3. Of the negatives, −1 is closer to zero than −6, so −1 is larger.

Descending order: 5, 2, 0, −1, −6

Using temperature

Temperature is a useful context for negative numbers. A temperature of −8°C is colder, and therefore lower, than −3°C. If the temperature rises from −8°C to −3°C, its value has increased because it has moved closer to zero.

Ordering a mixture of numbers

Separate negative values from positive values, then compare place values as before.

Example: put −2.4, 0.6, −2.04, 1 and 0.06 in ascending order

  1. −2.4 lies close to the left end of the line.
  2. 0.6 sits between 0.5 and 1.
  3. −2.04 is to the right of −2.4, so it is larger.
  4. 1 is the largest value here.
  5. 0.06 is positive, but only just to the right of zero.

  1. Write the negative decimals as −2.40 and −2.04. Since −2.40 lies further left, it is smaller.

  2. Write the positive values as 0.06, 0.60 and 1.00. Their place values put them in that order.

  3. Every negative value comes before every positive value in ascending order.

Ascending order: −2.4, −2.04, 0.06, 0.6, 1

Common mistakes

  • Treating negatives like positives−9 is smaller than −4 because it lies further left on the number line.
  • Counting decimal digits0.625 is not larger than 0.7. Write 0.7 as 0.700, then compare place values.
  • Adding zeros in the wrong placeZeros may be added to the end of a decimal: 2.5 = 2.50. Writing 2.05 changes the value.
  • Ignoring the requested directionCheck whether the question asks for ascending (smallest first) or descending (largest first).

Practice and continue

Practice

Ordering integers, decimals and negatives

Ordering numbers, including negatives and decimals of different lengths.

 Next lesson

The inequality symbols

The less-than, greater-than, at-most, at-least and not-equal signs.