1.1 Real Numbers
Ideas in this lesson
- Recognize real-number types and decimal representations
- Use commutative, associative, and distributive properties
- Interpret negatives, subtraction, and additive inverses
- Calculate with fractions and justify restrictions
- Read, locate, and compare numbers on the real line
- Describe sets and combine intervals
- Use absolute value to measure magnitude and distance
- Prove real-number and absolute-value claims
- Interpret measurement data and real-world number models
Recognize real-number types and decimal representations · Worked example
A student says, “Every expression with a fraction bar is rational.” Test this claim using \(\dfrac{\sqrt{2}}{1}\), \(\dfrac{12}{4}\), and \(\dfrac{5}{0}\). Give a corrected rule.
Answer: The claim is false: \(\dfrac{\sqrt{2}}{1}\) is irrational, \(\dfrac{12}{4}=3\) is rational, and \(\dfrac{5}{0}\) is undefined. A ratio of integers with nonzero denominator is rational.
\(\dfrac{\sqrt{2}}{1}=\sqrt{2}\), which is irrational. Writing a number over \(1\) does not change its value or classification.
\(\dfrac{12}{4}=3\) is natural, integer, rational, and real.
\(\dfrac{5}{0}\) is undefined; it is not a real number.
A ratio of two integers with a nonzero denominator is rational. A noninteger numerator alone does not prove irrationality: for example, \(\dfrac{(\dfrac{1}{2})}{1}=\dfrac{1}{2}\) is rational.
\(\dfrac{12}{4}=3\) is natural, integer, rational, and real.
\(\dfrac{5}{0}\) is undefined; it is not a real number.
A ratio of two integers with a nonzero denominator is rational. A noninteger numerator alone does not prove irrationality: for example, \(\dfrac{(\dfrac{1}{2})}{1}=\dfrac{1}{2}\) is rational.