수와 연산 전체 문제

ⓐ $\frac{-19}{1}$ ⓑ $\frac{841}{100}$

ⓐ $0.29,0.81\overset{-}{6}$ ⓑ $2.515115111\ldots$

ⓐ $2.6\overset{-}{3},0.125$ ⓑ $0.418302\ldots$

• ⓐ Recognize that 36 is a perfect square, since ${6}^{2}=36.$ So $\sqrt{36}=6,$ therefore $\sqrt{36}$ is rational.
• ⓑ Remember that ${6}^{2}=36$ and ${7}^{2}=49,$ so 44 is not a perfect square. Therefore, the decimal form of $\sqrt{44}$ will never repeat and never stop, so $\sqrt{44}$ is irrational.

ⓐ rational ⓑ irrational

ⓐ irrational ⓑ rational

• ⓐ There is no real number whose square is $-169.$ Therefore, $\sqrt{-169}$ is not a real number.
• ⓑ Since the negative is in front of the radical, $-\sqrt{64}$ is $-8,$ Since $-8$ is a real number, $-\sqrt{64}$ is a real number.

ⓐ not a real number ⓑ real number

ⓐ real number ⓑ not a real number

• ⓐ Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
• ⓑ The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so $-\sqrt{64}=-8.$ So the integers are $-7,8,-\sqrt{64}.$
• ⓒ Since all integers are rational, then $-7,8,-\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}\,\text{and}\,5.9$ are rational. So the list of rational numbers is $-7,\frac{14}{5},8,5.9,-\sqrt{64}.$
• ⓓ Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.
• ⓔ All the numbers listed are real numbers.

ⓐ $4,\sqrt{49}$ ⓑ $-3,4,\sqrt{49}$ ⓒ $-3,0.\overset{-}{3},\frac{9}{5},4,\sqrt{49}$ ⓓ $-\sqrt{2}$ ⓔ $-3,-\sqrt{2},0.\overset{-}{3},\frac{9}{5},4,\sqrt{49}$

ⓐ $6,\sqrt{121}$ ⓑ $-\sqrt{25},-1,6,\sqrt{121}$ ⓒ $-\sqrt{25},-\,\frac{3}{8},-1,6,\sqrt{121}$ ⓓ $2.041975\ldots$ ⓔ $-\sqrt{25},-\,\frac{3}{8},-1,6,\sqrt{121},2.041975\ldots$

Locate and plot the integers, $4,-3.$

Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot $\frac{3}{4}.$ Similarly plot $-\,\frac{1}{4}.$

Now locate the improper fractions $\frac{6}{5},-\,\frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\,\frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

ⓐ > ⓑ > ⓒ < ⓓ <

ⓐ < ⓑ < ⓒ > ⓓ <

A proper fraction has value less than one. The decimal number 0.4 is equivalent to $\frac{4}{10},$ a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See (the referenced figure).