Answer
a. \(5\frac{3}{4} = 101.11_2 \)
b. \(15\frac{15}{16} = 1111.1111_2 \)
c. \( 5\frac{3}{8} = 101.011_2 \)
d. \( 1\frac{1}{4} = 1.01_2 \)
e. \( 6\frac{5}{8} = 110.101_2 \)
Work Step by Step
To express each of the given values in binary notation, let's first convert the integer and fractional parts separately.
a. \( 5\frac{3}{4} \)
Integer part: \(5_{10} = 101_2\)
Fractional part: \( \left( \frac{3}{4} \right)_{10} = \left( 0.11 \right)_2 \)
Combining them, we get \( 5\frac{3}{4} = 101.11_2 \).
b. \( 15\frac{15}{16} \)
Integer part: \(15_{10} = 1111_2\)
Fractional part: \( \left( \frac{15}{16} \right)_{10} = \left( 0.1111 \right)_2 \)
Combining them, we get \( 15\frac{15}{16} = 1111.1111_2 \).
c. \( 5\frac{3}{8} \)
This is a mixed number, so let's convert it into an improper fraction first: \( 5\frac{3}{8} = \frac{5 \times 8 + 3}{8} = \frac{43}{8} \).
Integer part: \(5_{10} = 101_2\)
Fractional part: \( \left( \frac{3}{8} \right)_{10} = \left( 0.011 \right)_2 \)
Combining them, we get \( 5\frac{3}{8} = 101.011_2 \).
d. \( 1\frac{1}{4} \)
This mixed number can be converted to the improper fraction \( \frac{5}{4} \).
Integer part: \(1_{10} = 1_2\)
Fractional part: \( \left( \frac{1}{4} \right)_{10} = \left( 0.01 \right)_2 \)
Combining them, we get \( 1\frac{1}{4} = 1.01_2 \).
e. \( 6\frac{5}{8} \)
This mixed number can be converted to the improper fraction \( \frac{53}{8} \).
Integer part: \(6_{10} = 110_2\)
Fractional part: \( \left( \frac{5}{8} \right)_{10} = \left( 0.101 \right)_2 \)
Combining them, we get \( 6\frac{5}{8} = 110.101_2 \).
