A universal trick: get to a common scale
Comparing two numbers is easy when they look alike and hard when they don't. The whole strategy is to make them comparable. Two reliable ways: convert every number to a decimal (just divide), or put every fraction over a common denominator and compare numerators. Once two numbers share a scale, the order relation reads straight off, exactly as on the number line.
Which is larger, 5/8 or 3/5 ? Method A — common denominator (LCD = 40): 5/8 = 25/40 3/5 = 24/40 25 > 24 → 5/8 > 3/5 Method B — decimals: 5/8 = 0.625 3/5 = 0.600 0.625 > 0.600 → 5/8 > 3/5 ✓ same answer
Negatives and roots, handled with care
Two pitfalls deserve a second look. First, with negatives the order reverses your gut: among negatives, the one closer to zero is larger, so −1/2 > −3/4 (since 0.5 < 0.75 in size, but they're on the negative side). Second, to compare an irrational like sqrt(2) with a rational, use a decimal approximation: sqrt(2) ≈ 1.414, so sqrt(2) > 1.4 but sqrt(2) < 1.5. Approximations are honest tools here, as long as you carry enough digits to settle the comparison.
Ordering a whole mixed list
- Convert every number to a decimal approximation, carrying enough digits to break ties.
- Place them mentally on the number line: negatives to the left of 0, positives to the right.
- Read off from left to right to list them smallest → largest, then translate each back to its original form.
Order from least to greatest:
−3/2 , 0.8 , −1 , sqrt(2) , 3/4
Step 1 — decimals:
−3/2 = −1.5
0.8 = 0.8
−1 = −1.0
sqrt(2) ≈ 1.414
3/4 = 0.75
Step 2 — on the line:
−1.5 < −1.0 < 0.75 < 0.8 < 1.414
Step 3 — back to originals:
−3/2 < −1 < 3/4 < 0.8 < sqrt(2)