Number System Questions and Answers

Answer: D

The base of the octal number system is 8.

The octal system uses the eight digits 0 to 7, and each place value is a power of 8. For example, octal 17 means \(1 \times 8 + 7 = 15\) in decimal. Each octal digit corresponds to exactly three binary digits, which is why octal was used as shorthand for binary.

Base 2 is the binary system, base 10 the decimal system, and base 4 the quaternary system, which is rarely used. Base 16, not offered, is hexadecimal.

Answer: B

Correct answer: Multiplying

The integer part of a decimal number is converted by repeated division by 2, taking the remainders. A fraction is converted the other way: multiply repeatedly by 2 and take the integer part that carries out each time. For example \(0.625\times 2 = 1.25\) gives 1, \(0.25\times 2=0.5\) gives 0, \(0.5\times 2=1.0\) gives 1, so \(0.625_{10}=0.101_2\).

Answer: A

Correct answer: (140)10

\[ 214_8 = 2\times 64 + 1\times 8 + 4 = 128+8+4 = 140 \]

Answer: B

Correct answer: 100

\(1+1+1+1 = 4\), and four in binary is 100. The other options are not equal to four: 111 is 7, 1111 is 15 and 1001 is 9.

Answer: A

Correct answer: 491

Each octal digit carries a power of 8:

\[ 753_8 = 7\times 64 + 5\times 8 + 3 = 448+40+3 = 491 \]

Answer: C

Correct answer: (AA)16

Divide by 16 and read the remainders upward:

\[ 170 \div 16 = 10 \ \text{remainder}\ 10 \]

Both the quotient and the remainder are 10, which is A in hexadecimal, so the answer is AA. As a check, \( A\!A_{16} = 10\times 16 + 10 = 170 \).

Answer: C

Correct answer: 421

Each octal digit carries a power of 8:

\[ 645_8 = 6\times 8^2 + 4\times 8^1 + 5\times 8^0 = 384+32+5 = 421 \]

Answer: B

Correct answer: nibble

A nibble is four bits, and four bits are exactly what one hexadecimal digit represents - which is why two hex digits make a byte. That correspondence is the reason hexadecimal is used for binary data at all.

Answer: D

Correct answer: 11

In binary, \(1+1+1 = 3\), and three is written 11 - one two plus one one. In an adder this appears as a sum bit of 1 with a carry of 1.

Answer: A

Correct answer: binary

Digital circuits have two stable states, high and low, so they represent numbers in base 2. Octal and hexadecimal are used by people as shorthand for binary, but the machine itself works in binary.