Answer:
a) 3₁₀
b) 6₁₀
c) 7₁₀
f) 19₁₀
i) 181₁₀
Step-by-step explanation:
Binary to Decimal Conversion (Positional Notation Method)
- Multiply each digit by the base (2) raised to the power dependent upon the position of that digit in the binary number.
- Sum all the values obtained for each digit.
- Express the number as a decimal number by placing subscript 10 after it.
For a binary number with 'n' digits:
- The right-most digit is multiplied by 2⁰
- The left-most digit is multiplied by [tex]\sf 2^{n-1}[/tex]
For example, to convert the binary number 111001₂ into a decimal:
[tex]\begin{array}{ c c c c c c}1 & 1 & 1 & 0 & 0 & 1\\\downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0\\\end{array}[/tex]
Multiply each digit by the base (2) raised to the power as indicated above and sum them:
[tex]=(1 \times 2^5)+(1 \times 2^4)+(1 \times 2^3)+(0 \times 2^2)+(0 \times 2^1)+(1 \times 2^0)[/tex]
[tex]= 32+16+8+0+0+1[/tex]
[tex]= 57[/tex]
Finally, express as a decimal number ⇒ 111001₂ = 57₁₀
Question (a)
[tex]\begin{aligned}\implies 11_2 & = (1 \times 2^1)+(1 \times 2^0)\\& = 2+1\\& = 3\end{aligned}[/tex]
Therefore, 11₂ = 3₁₀
Question (b)
[tex]\begin{aligned}\implies 110_2 & = (1 \times 2^2)+(1 \times 2^1)+(0 \times 2^0)\\& = 4+2+0\\& = 6\end{aligned}[/tex]
Therefore, 110₂ = 6₁₀
Question (c)
[tex]\begin{aligned}\implies 111_2 & = (1 \times 2^2) +(1 \times 2^1)+(1 \times 2^0)\\& =4+2+1\\& = 7\end{aligned}[/tex]
Therefore, 111₂ = 7₁₀
Question (f)
[tex]\begin{aligned}\implies 10011_2 & =(1 \times 2^4)+(0 \times 2^3)+ (0 \times 2^2) +(1 \times 2^1)+(1 \times 2^0)\\& =16+0+0+2+1\\& = 19\end{aligned}[/tex]
Therefore, 10011₂ = 19₁₀
Question (i)
[tex]\phantom{)))}10110101_2 \\\\=(1 \times 2^7)+(0 \times 2^6)+(1 \times 2^5)+(1 \times 2^4)+(0 \times 2^3)+ (1 \times 2^2) +(0 \times 2^1)+(1 \times 2^0)\\\\=128+0+32+16+0+4+0+1\\\\= 181[/tex]
Therefore, 10110101₂ = 181₁₀
