Comparing and Ordering Numbers
This topic covers the fundamental skill of comparing and ordering different types of numbers, including integers, decimals, and fractions, using relational symbols. It's a foundational concept for interpreting data and solving multi-step problems across engineering and science.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Numbers on a number line increase in value from left to right. This means any positive number is greater than any negative number.
- For negative numbers, the number with the larger magnitude (absolute value) is actually smaller. For example, -10 is less than -2.
- To compare fractions, convert them to equivalent fractions with a common denominator. The fraction with the larger numerator is the larger number.
- To compare a mix of fractions, decimals, and integers, the most reliable method is to convert all numbers into the same format, usually decimals.
Definitions
- =
- is equal to
- ≠
- is not equal to
- <
- is less than
- >
- is greater than
- ≤
- is less than or equal to
- ≥
- is greater than or equal to
Worked example
Arrange the following numbers in ascending order (smallest to largest): 2/3, -0.7, 7/11, -3/4, 0.65
- 1
Convert all numbers to decimals to ensure a consistent format for comparison.
- 2
2/3 ≈ 0.667
- 3
-0.7 remains -0.7
- 4
7/11 ≈ 0.636 (since 1/11 is approx 0.09, 7/11 is approx 0.63)
- 5 -3/4 = -0.75
- 6
0.65 remains 0.65
- 7
The list in decimal form is:
0.667, -0.7, 0.636, -0.75, 0.65.
- 8
Order the negative numbers first.
-0.75 is smaller than -0.7.
- 9
Order the positive numbers:
0.636 < 0.65 < 0.667.
- 10
Combine the ordered lists and revert to the original forms:
-3/4, -0.7, 7/11, 0.65, 2/3.
Answer: -3/4, -0.7, 7/11, 0.65, 2/3
Common mistakes
- ×Incorrectly ordering negative numbers, for example thinking -100 is larger than -10 because 100 is larger than 10.
- ×Making arithmetic errors when finding a common denominator for fractions, especially with large or awkward numbers.
- ×Misinterpreting ≤ and ≥; for example, stating that x ≤ 5 is false if x = 5.
- ×Rounding decimals too early or inaccurately during conversion, which can lead to incorrect ordering.
No-calculator tips
- ✓To quickly compare two fractions a/b and c/d, use cross-multiplication. Compare the products a*d and c*b. If a*d > c*b, then a/b > c/d.
- ✓When comparing decimals, align the decimal points and pad with zeros to make them the same length. For example, to compare 0.52 and 0.518, compare 0.520 and 0.518.
- ✓Benchmark common fractions to their decimal equivalents (e.g., 1/4=0.25, 1/3≈0.33, 1/8=0.125) to help estimate the size of more complex fractions.