2.5 Number Sense: Rounding, Comparing, and Ordering Rational and Irrational Numbers

Number Sense: Rounding, Comparing, and Ordering Rational and Irrational Numbers

Quick Answer: The TSIA2 Quantitative Reasoning strand asks you to compare magnitudes of rational and irrational numbers on both the CRC and the Diagnostic. The Diagnostic also tests rounding to a given place value (including 10, 100, and 1,000) and comparing and ordering whole numbers, decimals, fractions, and percents, including on a number line. The dependable method is to convert every value to the same form and estimate irrational values between known benchmarks before you compare.

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1. Place Value and Rounding

Every digit's value depends on its place:

The Rounding Procedure

1. Underline the target place (the place you are rounding to).
2. Look only at the digit immediately to its right (the decider digit).
3. If the decider is 5 or more, add 1 to the target digit. If it is 4 or less, keep the target digit.
4. Replace every digit to the right of the target with zeros (whole numbers) or drop them (decimals).

Rounding Traps

• Only the next digit decides. When 2.4449 is rounded to the nearest tenth, the decider is 4, so the result is 2.4. Rounding one step at a time (2.4449 → 2.445 → 2.45 → 2.5) is a classic error.
• Keep full precision until the end. If a problem multiplies 3.46 × 1.8, compute 6.228 first and round at the end (6.23). Do not round 3.46 to 3.5 first, which gives 6.30.
• Money is rounded to the nearest cent (hundredth) unless the question says otherwise.

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2. Comparing and Ordering Numbers in Different Forms

When a list mixes fractions, decimals, and percents, convert everything to decimals (to three or four places) and compare place by place from the left.

Order from least to greatest:  0.58,  5/8,  57%,  0.6,  7/12

Convert:
  0.58  = 0.5800
  5/8   = 0.6250
  57%   = 0.5700
  0.6   = 0.6000
  7/12  = 0.5833...

Ordered: 57% < 0.58 < 7/12 < 0.6 < 5/8

Shortcuts for Fractions

• Cross-multiplication test: To compare a/b and c/d (with positive denominators), compare a × d with b × c. For 7/9 versus 11/14: 7 × 14 = 98 and 9 × 11 = 99, so 7/9 < 11/14.
• Benchmark fractions: Compare each fraction to 0, 1/2, or 1. For example, 5/11 is just below 1/2 because 5.5/11 = 1/2, while 6/11 is just above it.
• Same numerator: With equal numerators, the larger denominator gives the smaller fraction (3/8 < 3/7).

Negative Numbers and the Number Line

On a number line, values increase to the right. For negatives, the number with the larger absolute value is smaller:

• −0.75 < −0.7, because −0.75 lies farther left.
• −3/4 < −2/3, because 0.75 > 0.667 in absolute value.

Percents Greater Than 100% or Less Than 1%

• 125% = 1.25, which is greater than 1.
• 0.4% = 0.004, which is much less than 4% (0.04).

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3. Rational vs. Irrational Magnitudes

A rational number can be written as a ratio of integers. Its decimal either terminates or repeats. An irrational number has a non-terminating, non-repeating decimal: √2, √15, π. On the TSIA2, you usually need to estimate irrational numbers, not compute them exactly.

Estimating Square Roots with Perfect-Square Benchmarks

• √50 lies between √49 = 7 and √64 = 8, and very close to 7 because 50 is close to 49. √50 ≈ 7.07.
• √30 lies between 5 and 6. Because 30 is about halfway from 25 to 36, √30 ≈ 5.5 (more precisely 5.48).
• Useful values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, π ≈ 3.1416.

Comparing Radical Expressions by Squaring

For positive numbers, squaring preserves order. To compare 2√5 and 3√2:

(2√5)² = 4 × 5 = 20
(3√2)² = 9 × 2 = 18
20 > 18, so 2√5 > 3√2

The same trick shows that 4√3 = √48 is less than 7 = √49.

π and Its Fraction Approximations

• π ≈ 3.14159...
• 22/7 ≈ 3.142857..., slightly greater than π.
• 3.14 is slightly less than π.

So 3.14 < π < 22/7.

Worked Ordering Problem

Order from least to greatest:  √10,  π,  3.2,  10/3

Estimate:
  √10  ≈ 3.162  (between √9 = 3 and √16 = 4, very close to 3)
  π    ≈ 3.142
  3.2  = 3.200
  10/3 ≈ 3.333

Ordered: π < √10 < 3.2 < 10/3

Operations That Keep or Remove Irrationality

• Rational + irrational is always irrational (3 + √2).
• Nonzero rational × irrational is always irrational (5π).
• Irrational × irrational may be rational: √2 × √8 = √16 = 4, and √3 × √3 = 3.
• A square root is rational only when the number under it is a perfect square (√49 = 7 is rational; √48 is not).

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4. Scientific Notation Comparisons

Large and small quantities in context problems may use scientific notation, a × 10ⁿ with 1 ≤ a < 10. Compare the exponents first, then the leading factors:

• 3.2 × 10⁵ > 9.8 × 10⁴, because 10⁵ > 10⁴.
• 4.1 × 10⁻³ (0.0041) < 2.5 × 10⁻² (0.025).

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5. TSIA2 Number-Sense Traps

1. Comparing digits instead of values. 0.9 > 0.456 even though 456 has more digits. Line up decimal places (0.900 versus 0.456).
2. Treating 22/7 or 3.14 as exactly π. Answer choices may separate "π" from "22/7" in an ordering question.
3. Squaring negative comparisons. Squaring preserves order only for non-negative numbers: −3 < 2, but (−3)² = 9 > 4 = 2².
4. Rounding too early in multi-step problems, which pushes the final answer onto a distractor.