Chapter 1, Problem 35CRP

a.

Explanation of Solution

Express $5\frac{3}{4}$ in binary notation:

The process of conversion of base ten representation to binary representation is,

• Divide the number by 2 until the quotient is zero to convert the integer part to its binary equivalent.
• Multiply the fractional part to the powers of $\frac{1}{2}$ from left to right.
• Convert number 5 into binary equivalent.

$\begin{array}{ccc}2& 5& \\ 2& 2& 1\\ 2& 1& 0\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(5\right)}_{10}={\left(101\right)}_2$

• Convert the number $\frac{3}{4}$  into binary representation

b.

Explanation of Solution

Express $15\frac{15}{16}$ in binary notation:

The process of conversion of base ten representations to binary representation is,

• Divide the number by 2 until the quotient is zero to convert the integer part to its binary equivalent.
• Multiply the fractional part to the powers of $\frac{1}{2}$ from left to right.
• Convert the number 15 into binary equivalent.

$\begin{array}{ccc}2& 15& \\ 2& 7& 1\\ 2& 3& 1\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(15\right)}_{10}={\left(1111\right)}_2$

• Convert the number $\frac{15}{16}$  into binary representation

c.

Explanation of Solution

Express $5\frac{3}{8}$ in binary notation:

The process of conversion of base ten representation to binary representation is,

• Divide the number by 2 until the quotient is zero to convert the integer part to its binary equivalent.
• Multiply the fractional part to the powers of $\frac{1}{2}$ from left to right.
• Convert the number 5 into binary equivalent.

$\begin{array}{ccc}2& 5& \\ 2& 2& 1\\ 2& 1& 0\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(5\right)}_{10}={\left(101\right)}_2$

• Convert the number $\frac{3}{8}$  into binary representation

d.

Explanation of Solution

Express $1\frac{1}{4}$ in binary notation:

The process of conversion of base ten representation to binary representation is,

• Divide the number by 2 until the quotient is zero to convert the integer part to its binary equivalent.
• Multiply the fractional part to the powers of $\frac{1}{2}$ from left to right.
• Convert the number 1 into binary equivalent.

$\begin{array}{ccc}2& 1& \\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(1\right)}_{10}={\left(1\right)}_2$

• Convert the number $\frac{1}{4}$  into binary representation

e.

Explanation of Solution

Express $6\frac{5}{8}$ in binary notation:

The process of conversion of base ten representation to binary representation is,

• Divide the number by 2 until the quotient is zero to convert the integer part to its binary equivalent.
• Multiply the fractional part to the powers of $\frac{1}{2}$ from left to right.
• Convert the number 6 into binary equivalent.

$\begin{array}{ccc}2& 6& \\ 2& 3& 0\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(6\right)}_{10}={\left(110\right)}_2$

• Convert the number $\frac{5}{8}$  into binary representation