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 (1101.011)2(1101.011)_2 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 110121101_2

The binary number 110121101_2 is converted to decimal by multiplying each bit by 2n2^n, where nn is the position of the bit from the right (starting at 0):

11012=1×23+1×22+0×21+1×201101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0

This simplifies to:

=1×8+1×4+0×2+1×1=8+4+0+1=13= 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 = 8 + 4 + 0 + 1 = 13

Step 2: Convert the fractional part .0112.011_2

The fractional part is converted by multiplying each bit by 2n2^{-n}, where nn is the position of the bit from the left of the decimal point (starting at 1):

.0112=0×21+1×22+1×23.011_2 = 0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3}

This simplifies to:

=0+14+18=14+18=28+18=38=0.375= 0 + \frac{1}{4} + \frac{1}{8} = \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} = 0.375

Step 3: Combine integer and fractional parts

The decimal value is:

1101.0112=13+0.375=13.375101101.011_2 = 13 + 0.375 = 13.375_{10}

So, (1101.011)2(1101.011)_2 in decimal is 13.37513.375.


Part 2: Perform the binary multiplication (110)2×(101)2(110)_2 \times (101)_2

Let's multiply the two binary numbers step by step.

Step 1: Write the binary numbers

1102=610,1012=510110_2 = 6_{10}, \quad 101_2 = 5_{10}

We will perform the multiplication manually in binary:

     110  (This is 6 in binary)\ \ \ \ \ 110 \ \ (\text{This is 6 in binary}) ×  101  (This is 5 in binary)\times \ \ 101 \ \ (\text{This is 5 in binary})

Step 2: Multiply each bit

  • Multiply 110×1110 \times 1 (shift by 0 positions): 110110

  • Multiply 110×0110 \times 0 (shift by 1 position): 000000

  • Multiply 110×1110 \times 1 (shift by 2 positions): 1100011000

Step 3: Add the results together:

110+0000+11000=1001102110 + 0000 + 11000 = 100110_2

Thus, (110)2×(101)2=1001102(110)_2 \times (101)_2 = 100110_2.

Step 4: Convert the result to decimal

1001102=1×25+0×24+0×23+1×22+1×21+0×20100110_2 = 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 =32+0+0+4+2+0=38= 32 + 0 + 0 + 4 + 2 + 0 = 38

Thus, 1001102=3810100110_2 = 38_{10}.


Final Results:

  1. (1101.011)2=13.37510(1101.011)_2 = 13.375_{10}
  2. (110)2×(101)2=1001102=3810(110)_2 \times (101)_2 = 100110_2 = 38_{10}

Would you like more details on any step, or do you have any questions?

Related Questions:

  1. How do you convert decimal numbers to binary?
  2. What is the process of dividing two binary numbers?
  3. How does binary addition differ from decimal addition?
  4. How would you convert a binary number to hexadecimal?
  5. 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