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Rational numbers are ratios of integers with a nonzero denominator; 2 is irrational. ∣x∣ is distance from zero.
Remember |x| cannot be negative.
Worked example 1
01 / A measurement system stores the exact values 0, 81, −2.75, and 2, but displays the last as 1.414. Classify each exact value and explain whether the display changes its type.
02 / 0 is an integer and a rational real number, but is not natural under this section's convention.
03 / 81=9, so this value is natural, integer, rational, and real. A radical symbol alone does not make a number irrational.
04 / −2.75=−411, so it is rational and real but not an integer.
05 / 2 is irrational and real. The stored value remains irrational; 1.414=500707 is a separate rational approximation.
Worked example 2 · Recognize real-number types and decimal representations
A timing calculation gives 0.16 minutes exactly. Convert it to a fraction, then explain whether 0.167 minutes is exact.
Let x=0.1666…. Moving past the nonrepeating digit gives 10x=1.666…. Move one more place to align the repeating tail: 100x=16.666…. Subtract the aligned values: 90x=15. Therefore x=9015=61. Rounding to three decimal places gives x≈0.167. It is not exact: 0.167−61=30001 minute.
Read the graphs
Read, locate, and compare numbers on the real line
A number line has eight equal spaces from −1 to 1. Point P is three spaces to the right of −1. Point Q has coordinate −0.3. Find P and decide which point is farther right. Also find the distance between the two plotted points.Number-line scale and endpoint markings.
The total coordinate change is 1−(−1)=2. Eight equal spaces therefore represent 82=41 unit each. Three spaces add 43 to the starting coordinate: P=−1+43=−41. Write the coordinate as −0.25. Because −0.25>−0.3, P is farther right than Q. Negative coordinates still increase as we move right. The graph-read distance is ∣−41−(−103)∣=∣201∣=201=0.05.
Answer: The total coordinate change is 1−(−1)=2. Eight equal spaces therefore represent 82=41 unit each. Three spaces add 43 to the starting coordinate: P=−1+43=−41. Write the coordinate as −0.25. Because −0.25>−0.3, P is farther right than Q. Negative coordinates still increase as we move right. The graph-read distance is ∣−41−(−103)∣=∣201∣=201=0.05.
Describe sets and combine intervals
The first number line represents test A and the second represents test B. Read the endpoint markings to determine their acceptance intervals. Find the readings accepted by at least one test and by both tests. A technician excludes −1 from the intersection because it is inside, rather than at an endpoint of, the first interval. Explain the mistake and account for all four endpoints. In a written extension, draw each resulting set on a labeled number line. Show included and excluded endpoints, any gaps, and every unbounded direction.Number-line scale and endpoint markings.Number-line scale and endpoint markings.
The union covers every reading from −4 to 8 but neither outer endpoint: A∪B=(−4,8). The shared readings run from −1 through 3, both included: A∩B=[−1,3]. The value −1 lies inside A and is included at the left endpoint of B, so both tests accept it. A value need not be an endpoint of both intervals to be in their intersection. The value 3 is included by A and lies inside B. The outer endpoints −4 and 8 are excluded by their respective intervals and absent from the other interval, so neither belongs to the union.
Answer: Union (−4,8); intersection [−1,3]. Both −1 and 3 belong to both tests.
Interpret measurement data and real-world number models
Read the temperatures for towns A and B from the chart. For every day calculate A minus B and the absolute difference, including units. Which list preserves more information, and why?Read both towns from each labeled pair of bars.
The bar pairs give 18−22=−4, 24−21=3, 20−20=0, and 16−19=−3. Taking absolute values gives 4,3,0,3. The zero means equal temperatures. The signed list also shows which town is warmer: a negative difference means A is cooler, a positive difference means A is warmer. Absolute differences give only the temperature gaps.
Answer: Signed differences: −4,3,0,−3 degrees Celsius; magnitudes: 4,3,0,3 degrees Celsius.
15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.