Math Problem Statement
Q2: Binary Operations and Number SystemsConvert the binary number ( (1101.011)_2 ) to decimal.Perform the binary multiplication: [ (110)_2 \times (101)_2 ]
Solution
Part 1: Convert the binary number to decimal
In a binary number, each digit represents a power of 2, with the decimal point separating the positive and negative powers. Let's break down the conversion:
Step 1: Convert the integer part
The binary number is converted to decimal by multiplying each bit by , where is the position of the bit from the right (starting at 0):
This simplifies to:
Step 2: Convert the fractional part
The fractional part is converted by multiplying each bit by , where is the position of the bit from the left of the decimal point (starting at 1):
This simplifies to:
Step 3: Combine integer and fractional parts
The decimal value is:
So, in decimal is .
Part 2: Perform the binary multiplication
Let's multiply the two binary numbers step by step.
Step 1: Write the binary numbers
We will perform the multiplication manually in binary:
Step 2: Multiply each bit
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Multiply (shift by 0 positions):
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Multiply (shift by 1 position):
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Multiply (shift by 2 positions):
Step 3: Add the results together:
Thus, .
Step 4: Convert the result to decimal
Thus, .
Final Results:
Would you like more details on any step, or do you have any questions?
Related Questions:
- How do you convert decimal numbers to binary?
- What is the process of dividing two binary numbers?
- How does binary addition differ from decimal addition?
- How would you convert a binary number to hexadecimal?
- How are fractional binary numbers handled in multiplication?
Tip:
When multiplying binary numbers, think of it like long multiplication in decimal, with shifts instead of carrying.
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Math Problem Analysis
Mathematical Concepts
Binary to Decimal Conversion
Binary Multiplication
Number Systems
Formulas
Binary to Decimal Conversion: integer part - sum(bit * 2^position), fractional part - sum(bit * 2^(-position))
Binary Multiplication
Theorems
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Suitable Grade Level
Grades 8-10