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Quiz 2
1. Compute $\dfrac{21}{4} \cdot \dfrac{15}{3}$.
Solution: We multiply across and get $$\dfrac{21}{4} \cdot \dfrac{15}{3} = \dfrac{21 \cdot 15}{4 \cdot 3} = \dfrac{315}{12}=\dfrac{105}{4}.$$ 2. Compute $\dfrac{1}{3} + \dfrac{3}{4}$.
Solution: Find a common denominator. In this case the common denominator is 12. Now rewrite each fraction in terms of that common denominator (recall that $\dfrac{a}{a}=1$ and that $1 \cdot x = x$ for any $x$). $$\dfrac{1}{3} = \dfrac{1}{3} \cdot 1 = \dfrac{1}{3} \cdot \dfrac{4}{4} = \dfrac{4}{3 \cdot 4} = \dfrac{4}{12}$$ and $$\dfrac{3}{4} = \dfrac{3}{4} \cdot 1 = \dfrac{3}{4} \cdot \dfrac{3}{3} = \dfrac{9}{12}.$$ Now we may add and get $$\dfrac{1}{3} + \dfrac{3}{4} = \dfrac{4}{12} + \dfrac{9}{12} = \dfrac{4+9}{12} = \dfrac{13}{12}.$$ 3. Is $\pi$ a rational number?
Solution: No. We said this in class. It should have appeared in your notes.
4. Is $\dfrac{1}{2}$ an integer?
Solution: No. The integers are the numbers $\{\ldots,-2,-1,0,1,2,\ldots\}$. The number $\dfrac{1}{2}$ does not appear in that list.
5. Is $\dfrac{3}{7}$ a real number?
Solution: Yes. The real numbers are the whole real line. We did an example in class where we placed fractions on the number line -- that showed that rational numbers are real numbers.