Math Problem Statement

Convert binary numbers 110111₂ and 11100011₂ to decimal.

Solution

The notebook page shows the conversion of binary numbers to decimal numbers. Let's break down each step as shown in the image:

(a) Conversion of 110111₂ to decimal:

  1. The binary number 110111₂ is converted to decimal by expanding it using powers of 2:

    1101112=(1×25)+(1×24)+(0×23)+(1×22)+(1×21)+(1×20)110111_2 = (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)

  2. Calculating the powers of 2:

    =32+16+0+4+2+1= 32 + 16 + 0 + 4 + 2 + 1

  3. Summing all the values:

    =58= 58

So, 1101112=5810110111_2 = 58_{10}.

(b) Conversion of 11100011₂ to decimal:

  1. The binary number 11100011₂ is converted to decimal by expanding it using powers of 2:

    111000112=(1×27)+(1×26)+(1×25)+(0×24)+(0×23)+(0×22)+(1×21)+(1×20)11100011_2 = (1 \times 2^7) + (1 \times 2^6) + (1 \times 2^5) + (0 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)

  2. Calculating the powers of 2:

    =128+64+32+0+0+0+2+1= 128 + 64 + 32 + 0 + 0 + 0 + 2 + 1

  3. Summing all the values:

    =227= 227

So, 111000112=2271011100011_2 = 227_{10}.

Let me know if you want further details or if you have any questions!

Related Questions:

  1. How do you convert a decimal number to binary?
  2. What is the significance of each digit in a binary number?
  3. How can binary numbers be used in computer systems?
  4. What is the process for converting binary numbers to hexadecimal?
  5. How can you check if a binary number is correct after conversion?

Tip: Always double-check each step when converting between number systems to ensure accuracy, especially when dealing with larger binary numbers.

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Math Problem Analysis

Mathematical Concepts

Number Systems
Binary to Decimal Conversion

Formulas

Decimal value of binary number = Sum of (binary digit × 2^position)

Theorems

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Suitable Grade Level

Grades 6-8