Chapter 3, Problem 1P

Convert the following base-2 numbers to base-10: (a) 101101, (b) 101.011, and (c) 0.01101.

To determine

To calculate: The number in base-10 if the number in base-2 is $101101$.

Answer to Problem 1P

Solution:

The number in base-10 is $45$.

Explanation of Solution

Given:

The number in base-2 is $101101$.

Formula used:

Conversion formula from base-2 to base-10,

$N={\displaystyle \sum _i{b}_i{2}^i}$

Where, N is real positive number, b is the digit and I is the integer.

Calculation:

Consider the Base-2 Number, $101101$ .

Convert this number to Base-10 number by multiplying the place values to the digits and adding them together.

The place value of the digits are $2^{+5}$, $2^{+4}$, $2^{+3}$, $2^{+2}$, $2^{+1}$, $2^0$ from left to right.

Therefore,

$\begin{array}{c}{\left(101101\right)}_2=1\times 2^{+5}+0\times 2^{+4}+1\times 2^{+3}+1\times 2^{+2}+0\times 2^{+1}+1\times 2^0\\ =32+0+8+4+0+1\\ ={\left(45\right)}_{10}\end{array}$

Hence, the converted base-10 value of the base-2 number is $45$ .

To determine

To calculate: The number in base-10 if the number in base-2 is $101.011$.

Answer to Problem 1P

Solution:

The number in base-10 is $5.375$.

Explanation of Solution

Given:

The number in base-2 is $101.011$.

Formula used:

Conversion formula from base-2 to base-10,

$N={\displaystyle \sum _i{b}_i{2}^i}$

Where, N is real positive number, b is the digit and I is the integer.

Calculation:

Consider the Base-2 Number, $101.011$ .

Convert this number to Base-10 number by multiplying the place values to the digits and adding them together.

In this case the binary digits after the point has place values $2^{-1}$, $2^{-2}$, $2^{-3}$ from left to right and $2^{+2}$, $2^{+1}$, $2^0$ from left to right before the point. Therefore,

$\begin{array}{c}{\left(101.011\right)}_2=1\times 2^{+2}+0\times 2^{+1}+1\times 2^0+0\times 2^{-1}+1\times 2^{-2}+1\times 2^{-3}\\ =4+0+1+0+0.25+0.125\\ ={\left(5.375\right)}_{10}\end{array}$

Hence, the converted base-10 value of the base-2 number is $5.375$ .

To determine

To calculate: The number in base-10 if the number in base-2 is $0.01101$.

Answer to Problem 1P

Solution:

The number in base-10 is $0.40625$.

Explanation of Solution

Given:

The number in base-2 is $0.01101$.

Formula used:

Conversion formula from base-2 to base-10,

$N={\displaystyle \sum _i{b}_i{2}^i}$

Where, N is real positive number, b is the digit and I is the integer.

Calculation:

Consider the Base-2 Number, $0.01101$ .

In this case the binary digits after the point has place values $2^{-1}$, $2^{-2}$, $2^{-3}$, $2^{-4}$, $2^{-5}$, from left to right. So, multiply the place values with the digits of the base-2 number and add them to get the base-10 number.

$\begin{array}{c}{\left(0.01101\right)}_2=0\times 2^0+0\times 2^{-1}+1\times 2^{-2}+1\times 2^{-3}+0\times 2^{-4}+1\times 2^{-5}\\ =0+0+0.25+0.125+0+0.03125\\ ={\left(0.40625\right)}_{10}\end{array}$

Hence, the converted base-10 value of the base-2 number is $0.40625$.