Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 7
Express the following decimal numbers in binary form and in binary-coded- decimal form: a \(17 ;\) b. \(8.5 ; c ^{*}9.75 ;\) d. \(73.03125\) \(e .67 .375\).
Short Answer
Expert verified
17: binary \(10001_2\), BCD \(0001\,0111\); 8.5: binary \(1000.1_2\), BCD \(1000\,0101\); 9.75: binary \(1001.11_2\), BCD \(1001\,0111\,0101\); 73.03125: binary \(1001001.00001_2\), BCD \(0111\,0011\,0000\,0011\,0001\,0010\,0101\); 67.375: binary \(1000011.011_2\), BCD \(0110\,0111\,0011\).
Step by step solution
01
Understanding Binary Representation
A binary number consists only of 0s and 1s and is based on powers of 2. To convert a decimal number to binary, we need to express the integer and the fractional parts separately as a sum of powers of 2.
02
Convert 17 to Binary
For the integer 17:1. Find the highest power of 2 less than or equal to 17, which is 16 (i.e., \(2^4\)).2. Subtracting 16 from 17 gives 1, which corresponds to \(2^0\).3. Therefore, 17 in binary is \(10001_2\).
03
Convert 8.5 to Binary
Convert the integer part 8:1. The highest power of 2 less than or equal to 8 is 8 itself (i.e., \(2^3\)).2. No subtraction necessary, so 8 in binary is \(1000_2\).Convert the fractional part 0.5:1. 0.5 equals \(2^{-1}\).2. Therefore, 8.5 in binary is \(1000.1_2\).
04
Convert 9.75 to Binary
Convert the integer part 9:1. 9 can be expressed as \(8 + 1\), which is \(2^3 + 2^0\).2. Hence, 9 in binary is \(1001_2\).Convert the fractional part 0.75:1. 0.75 can be expressed as 0.5 + 0.25 which are \(2^{-1}\) and \(2^{-2}\).2. Therefore, 0.75 in binary is \(.11_2\).Thus, 9.75 in binary is \(1001.11_2\).
05
Convert 73.03125 to Binary
Convert the integer part 73:1. 73 can be expressed as \(64 + 8 + 1\), which is \(2^6 + 2^3 + 2^0\).2. Hence, 73 in binary is \(1001001_2\).Convert the fractional part 0.03125:1. The fraction 0.03125 equals \(2^{-5}\).2. Therefore, 0.03125 in binary is \(.00001_2\).Thus, 73.03125 in binary is \(1001001.00001_2\).
06
Convert 67.375 to Binary
Convert the integer part 67:1. 67 can be expressed as \(64 + 3\), which is \(2^6 + 2^1 + 2^0\).2. Hence, 67 in binary is \(1000011_2\).Convert the fractional part 0.375:1. The fraction 0.375 can be expressed as 0.25 + 0.125 which are \(2^{-2}\) and \(2^{-3}\).2. Therefore, 0.375 in binary is \(.011_2\).Thus, 67.375 in binary is \(1000011.011_2\).
07
Understanding Binary-Coded Decimal (BCD)
In Binary-Coded Decimal, each decimal digit is represented as a four-bit binary number. For example, the decimal digit 2 is represented as \(0010\) in BCD.
08
Convert Decimal to BCD for 17
The decimal number 17 has digits 1 and 7.1. In BCD, 1 is \(0001\) and 7 is \(0111\).2. Therefore, 17 in BCD is \(0001\ 0111\).
09
Convert Decimal to BCD for 8.5
The decimal number 8.5 has digits 8 and 5.1. In BCD, 8 is \(1000\) and 5 is \(0101\).2. Therefore, 8.5 in BCD is \(1000\,0101\).
10
Convert Decimal to BCD for 9.75
The decimal number 9.75 has digits 9, 7, and 5.1. In BCD, 9 is \(1001\), 7 is \(0111\), and 5 is \(0101\).2. Therefore, 9.75 in BCD is \(1001\,0111\,0101\).
11
Convert Decimal to BCD for 73.03125
The decimal number 73.03125 has digits 7, 3, 0, 3, 1, 2, and 5.1. In BCD, 7 is \(0111\), 3 is \(0011\), 0 is \(0000\), 1 is \(0001\), 2 is \(0010\), and 5 is \(0101\).2. Therefore, 73.03125 in BCD is \(0111\,0011\,0000\,0011\,0001\,0010\,0101\).
12
Convert Decimal to BCD for 67.375
The decimal number 67.375 has digits 6, 7, 3, and 7.1. In BCD, 6 is \(0110\), 7 is \(0111\), and 3 is \(0011\).2. Therefore, 67.375 in BCD is \(0110\,0111\,0011\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
decimal to binary
Converting a decimal number to binary involves expressing the number in terms of powers of 2, using only the digits 0 and 1. It requires handling both the integer and fractional parts separately. Here's a simple breakdown:
- For the **integer part**, identify the largest power of 2 less than or equal to the number. Subtract it, and repeat this process with the remainder until you reach zero.
- For example, with the integer 17, the largest power of 2 is 16, or \(2^4\). Subtracting 16 from 17 leaves you with 1, which is \(2^0\). Therefore, 17 is represented as \(10001_2\) in binary.
- For the **fractional part**, multiply by 2 and observe the integer portion. Record each step and continue the process with the remaining fraction.
- As an illustration, converting 8.5 involves writing 8 as \(2^3\), resulting in \(1000_2\), and the fractional .5 is \(2^{-1}\), resulting in .1. Combined, 8.5 becomes \(1000.1_2\).
binary-coded decimal
Binary-Coded Decimal, or BCD, provides a method to represent decimal numbers where each digit is separately portrayed in binary.
- Each decimal digit transforms into a 4-bit binary equivalent.
- For instance, the digit 2 translates into \(0010\) in BCD format.
- A decimal number like 17 is split into its digits 1 and 7, which become \(0001\) and \(0111\) respectively in BCD. Hence, 17 is represented as \(0001\,0111\) in BCD.
- For numbers with a fractional component, such as 8.5, treat each digit separately again: 8 becomes \(1000\) and 5 becomes \(0101\), so in BCD it is \(1000\,0101\).
fractional binary conversion
Converting a fractional decimal to binary requires a separate approach where the fractional part is handled distinctly from the integer part. Here鈥檚 how to manage fractional conversions:
- The fractional component is converted by multiplying by 2 and observing the whole number part of the result, which becomes the next digit.
- For example, to convert 0.75, multiply by 2 to get 1.5. The integer part, 1, becomes the next binary digit. Continue with the remaining .5; multiply by 2 again to achieve 1.0, stopping there since there's no remainder.
- So, 0.75 in binary is represented as \(.11_2\).
- By using this technique, you can convert decimals with complex fractional parts to binary, such as seen in the conversion of .375, where the steps lead to \(.011_2\).
