SAT Math - Types of Numbers

Express 374 in base six.

$1422_{\textrm{six}}$

$1522_{\textrm{six}}$

$1502_{\textrm{six}}$

$1402_{\textrm{six}}$

Explanation

To convert a base ten number to base six, divide the number by six, with the remainder being the digit in the units place; continue, dividing each successive quotient by 6 and putting the remainder in the next position to the left until the final quotient is less than 6.

$374 \div 6 = 62\textrm{ R }\underline{2}$ - the rightmost digit is 2

$62 \div 6 = 10 \textrm{ R }\underline{2}$ - the next digit to the left is 2

$10 \div 6 = \underline{1} \textrm{ R }\underline{4}$ - the next digit to the left is 4, and the digit to the left of that is 1.

$374 = 1422_{\textrm{six}}$

Express 73 as a binary (base two) number.

$1001001_{\textrm{two}}$

$1000101_{\textrm{two}}$

$1010001_{\textrm{two}}$

$1000111_{\textrm{two}}$

$1010111_{\textrm{two}}$

Explanation

73 can be uniquely expressed as the sum of distinct powers of two as follows:

$73 = 64 + 8 + 1 = 2^{6} + 2^{3}+ 2^{0}$

Therefore, there is a "1" in the seventh, fourth, and first positions from the right, and a "0" everywhere else. So,

$73 = 1001001_{\textrm{two}}$

Express $13241 _{\textrm{five}}$ in base ten.

Explanation

Place values in the base five system are powers of five rather than powers of ten. $13241 _{\textrm{five}}$ is equal to

$1 \times 5^{4} + 3 \times 5 ^{3} + 2 \times 5^{2} + 4 \times 5 + 1$

$= 1 \times 625 + 3 \times 125 + 2 \times 25 + 4 \times 5 + 1$

Which of the following numbers is an imaginary number?

$\sqrt{7}$

$\frac{9}{8}$

Explanation

By definition, an imaginary number is a number with the term "i" in it. For this problem 7i is the only answer with the imaginary part.

Express $1011101_{\textrm{two}}$ as a base ten number.

Explanation

Place values in the base two system are powers of two rather than powers of ten.

$1011101_{\textrm{two}}$ is equal to:

$1\times 2^{6}+0\times 2^{5}+ 1\times 2^{4}+1 \times 2^{3}+ 1 \times 2^{2}+ 0 \times 2^{1} +1$

$= 1\times 64+0\times32+ 1\times 16+1 \times 8+ 1 \times 4+ 0 \times2+1$

Express 68 as a base two (binary) number.

$1000100_{\textrm{two}}$

$100010_{\textrm{two}}$

$100 100_{\textrm{two}}$

$100 1000_{\textrm{two}}$

$110 000_{\textrm{two}}$

Explanation

68 can be expressed as the sum of distinct powers of two as follows:

$68 = 64 + 4 = 2^{6}+ 2^{2}$

Therefore, there are 1's in the seventh and third place from the right, and 0's everywhere else, so

$68 = 1000100_{\textrm{two}}$

Which of the following is a rational number?

$\frac{1}{2},\sqrt{5},2i,3.658$

$\frac{1}{2}$

$\sqrt{5}$

None

Explanation

A rational number is any number that can be expressed as an integer p or a fraction p/q. In the set of number given only $\tiny \frac{1}{2}$ fits into that category.