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Base Conversion Guide
UCR · Math 135A
Decimal Binary Octal Hexidecimal (Base 10) (Base 2) (Base 8) (Base 16) 0 0000 00 00 1 0001 01 01 2 0010 02 02 3 0011 03 03 4 0100 04 04 5 0101 05 05 6 0110 06 06 7 0111 07 07 8 1000 10 08 9 1001 11 09 10 1010 12 0A 11 1011 13 0B 12 1100 14 0C 13 1101 15 0D 14 1110 16 0E 15 1111 17 0F
- Convert from base β to base 10. Integer Part:
(anan− 1 · · · a 1 a 0 )β = an ∗ βn^ + an− 1 ∗ βn−^1 + ... + a 1 ∗ β^1 + a 0 ∗ β^0 = (x) 10
Fraction Part:
(0.b 1 b 2 b 3 · · · )β = b 1 ∗ β−^1 + b 2 ∗ β−^2 + b 3 ∗ β−^3 + ... = (x) 10
Example: Convert 21.112 in base 3 to base 10
(21.112) 3 = 2 ∗ 31 + 1 ∗ 30 + 1 ∗ 3 −^1 + 1 ∗ 3 −^2 + 2 ∗ 3 −^3 = 2 ∗ 3 + 1 ∗ 1 + 1 ∗ .333 + 1 ∗ .111 + 2 ∗. 037 = 6 + 1 + .333 + .111 +. 074 = (7. 518 ...) 10
- Convert from base 10 to base β. Integer Part:
(a) Divide the number by β and record the remainder. (b) Divide the resulting quotient by β and record the remainder. (c) Repeat (d) The number in base β is the remainders written in backwards order.
Fraction Part:
(a) Multiply the number by β and record the integer. (b) Multiply the resulting number (ignoring the integer) by β and record the integer. (c) Repeat (d) The number in base β is the integers written in forwards order.
Example: Convert 15.4375 in decimal (base 10) to binary (base 2)
- First convert the integer part: 15 ÷ 2 = 7 R 1 7 ÷ 2 = 3 R 1 3 ÷ 2 = 1 R 1 1 ÷ 2 = 0 R 1 Hence (15) 10 = (1111) 2.
- Second convert the fraction part:
4375 ∗ 2 = 0. 8750
875 ∗ 2 = 1. 750
75 ∗ 2 = 1. 50
5 ∗ 2 = 1. 0 Hence (0.4375) 10 = (0.0111) 2
- Final Answer: (15.4375) 10 = (1111.0111) 2
- Convert from base α to base β.
- First convert from base α to base 10.
- Second convert from base 10 to base β.
- This implies (52.234375) 10 = (110100.001111) 2 = (1.10100001111) 2 × 25
- The mantissa with added zeros is then 10100001111000000000000. (c) Identify the exponent:
- First we solve for c: c − 127 = 5 implies c = (132) 10
- Second we convert 132 from base 10 to binary.
132 ÷ 8 = 16 R 4
16 ÷ 8 = 2 R 0 2 ÷ 8 = 0 R 2 So (132) 10 = (204) 8 = (010 000 100) 2 = (10 000 100) 2
- The exponent is 10000100.
(d) Put it all together: First list the number from step (a), then step (c), then step (b):
(11000010010100001111000000000000) 2
(e) Convert to hexidecimal:
- First split into numbers of length 4:
(1100 0010 0101 0000 1111 0000 0000 0000) 2
- Second convert each 4 digit number to hexidecimal to get the answer:
(C 250 F 000) 16
- Convert from single precision machine representation to decimal. Steps:
(a) Convert the machine representation in hexidecimal to binary. (b) Identify the sign, exponent, mantissa
- sign: first digit
- exponent: next 8 digits
- mantissa: last 23 digits (can ignore trailing zeros) (c) Convert the sign
- 0 represents +
- 1 represents - (d) Convert the exponent
- Convert the exponent number from binary to decimal, call the number in base 10 c.
- Calculate c − 127, call this number n. (e) Convert the mantissa
- Suppose your mantissa is b 1 b 2 · · · bm, then write
± 1 .b 1 b 2 · · · bm × 2 n
where the sign is determined by step (c).
- Convert this number to standard form (i.e. not scientific notation).
- Convert this number to base 8, then to base 10, i.e. decimal. This is your final answer.
Example: Convert the single precision machine representation (BA390000) 16 to decimal.
(a) Convert the machine representation to binary:
(BA390000) 16 = (1011 1010 0011 1001 0000 0000 0000 0000) 2
(b) Identify the sign, exponent, mantissa:
- sign: 1
- exponent: 01110100
- mantissa (without trailing zeros): 0111001 (c) Convert the sign: 1 represents - (negative) (d) Convert the exponent:
- Convert from base 2 to base 10 (01 110 100) 2 = (164) 8 = 1 ∗ 82 + 6 ∗ 81 + 4 ∗ 80 = 64 + 48 + 4 = (116) 10
