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Magazine Chapter 1: Problem 20
Express the following numbers in denary form: (a) \(1111 \cdot 11_{2}\) (b) \(777.7018\) (c) \(3 \Lambda 3.9 \Lambda 1_{12}\) (d) E02 \(F A B_{16}\)
Short Answer
Expert verified
(a) 15.75
(b) 511.880859375
(c) 567.8321759259
(d) 14696167
Step by step solution
01
Convert Binary to Denary (a)
To convert the binary number \(1111.11_2\) to denary, separate the integer part from the fractional part. The integer part \(1111_2\) equals \((1 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 4 + 2 + 1 = 15_{10}\). The fractional part \(.11_2\) equals \((1 \times 2^{-1}) + (1 \times 2^{-2}) = 0.5 + 0.25 = 0.75\). Adding these together gives the denary number: \(15.75_{10}\).
02
Convert Octal to Denary (b)
The octal number \(777.7018\) consists of the integer part \(777_8\) and the fractional part \(.7018_8\). For the integer part: \(777_8 = (7 \times 8^2) + (7 \times 8^1) + (7 \times 8^0) = 448 + 56 + 7 = 511_{10}\).For the fractional part: \(.7018_8 = (7 \times 8^{-1}) + (0 \times 8^{-2}) + (1 \times 8^{-3}) + (8 \times 8^{-4}) = 0.875 + 0 + 0.001953125 + 0.00390625 = 0.880859375\).The combined denary value is: \(511.880859375_{10}\).
03
Convert Duodecimal to Denary (c)
The duodecimal number \(3 \Lambda 3.9 \Lambda 1_{12}\) can be converted by noting that \(\Lambda\) represents 11 in denary.For the integer part:\(3 \Lambda 3_{12} = (3 \times 12^2) + (11 \times 12^1) + (3 \times 12^0) = 432 + 132 + 3 = 567_{10}\).For the fractional part:\(.9 \Lambda 1_{12} = (9 \times 12^{-1}) + (11 \times 12^{-2}) + (1 \times 12^{-3}) = 0.75 + 0.0763888889 + 0.005787037037 = 0.8321759259\).Thus, the denary value is: \(567.8321759259_{10}\).
04
Convert Hexadecimal to Denary (d)
The hexadecimal number \(E02FAB_{16}\) can be converted by recognizing E = 14 and F = 15.Calculating the integer value:\(E02FAB_{16} = (14 \times 16^5) + (0 \times 16^4) + (2 \times 16^3) + (15 \times 16^2) + (10 \times 16^1) + (11 \times 16^0)\).This equals:\( (14 \times 1048576) + (0 \times 65536) + (2 \times 4096) + (15 \times 256) + (10 \times 16) + (11 \times 1) \).Which results in: \(14680064 + 8192 + 3840 + 160 + 11 = 14696167_{10}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binary to Decimal Conversion
The binary number system uses only two digits, 0 and 1, making it fundamental in digital electronics and computing. Converting a binary number to a decimal (also known as denary) involves understanding the weighted sum of its digits. Each digit in a binary number represents a power of 2, with the rightmost digit being the least significant bit.For example, to convert the binary number \(1111.11_2\) to decimal, separate it into its integer part \(1111_2\) and fractional part \(.11_2\):
- The integer part, \(1111_2\), is calculated as: \((1 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 4 + 2 + 1 = 15_{10}\).
- The fractional part, \(.11_2\), is calculated as: \((1 \times 2^{-1}) + (1 \times 2^{-2}) = 0.5 + 0.25 = 0.75\).
Octal to Decimal Conversion
The octal system, or base-8, uses digits 0 through 7 and is often used in computing as a shorthand for binary numbers. To convert an octal number to decimal, treat each digit as a weight determined by powers of 8.Consider the octal number \(777.7018\). Break it down into the integer part \(777_8\) and the fractional part \(.7018_8\):
- The integer part, \(777_8\), is calculated as: \((7 \times 8^2) + (7 \times 8^1) + (7 \times 8^0) = 448 + 56 + 7 = 511_{10}\).
- The fractional part, \(.7018_8\), is calculated as: \((7 \times 8^{-1}) + (0 \times 8^{-2}) + (1 \times 8^{-3}) + (8 \times 8^{-4}) = 0.875 + 0 + 0.001953125 + 0.00390625 = 0.880859375\).
Duodecimal to Decimal Conversion
The duodecimal system, or base-12, is less common but interesting due to its divisibility properties. It uses twelve different digits, often represented by the numbers 0 to 9 and symbols for ten (usually \(A\)) and eleven (often \(\Lambda\)).To convert a duodecimal number like \(3 \Lambda 3.9 \Lambda 1_{12}\) to decimal, interpret each digit as a power of 12:
- The integer part, \(3 \Lambda 3_{12}\), is calculated with \(\Lambda = 11\), as: \((3 \times 12^2) + (11 \times 12^1) + (3 \times 12^0) = 432 + 132 + 3 = 567_{10}\).
- The fractional part, \(.9 \Lambda 1_{12}\), is calculated as: \((9 \times 12^{-1}) + (11 \times 12^{-2}) + (1 \times 12^{-3}) = 0.75 + 0.0763888889 + 0.005787037037 = 0.8321759259\).
Hexadecimal to Decimal Conversion
Hexadecimal, or base-16, is extensively used in computing because of its compatibility with binary. It uses the digits 0 to 9 and the letters A to F for the values 10 to 15. Converting a hexadecimal number to decimal involves interpreting each digit based on powers of 16.For example, take the hexadecimal \(E02FAB_{16}\):
- Recognize that \(E = 14, F = 15,\) and each letter represents a specific value.
- The conversion process is: \((14 \times 16^5) + (0 \times 16^4) + (2 \times 16^3) + (15 \times 16^2) + (10 \times 16^1) + (11 \times 16^0)\).
- Perform the operations: \(14 \times 1048576 + 0 \times 65536 + 2 \times 4096 + 15 \times 256 + 10 \times 16 + 11 \times 1 = 14696167_{10}\).
