1.3 Real Numbers & Decimal Expansions

Step 1

Convert each term to a fraction individually.

Step 2

Term 1: $0.6 = \frac{6}{10} = \frac{3}{5}$ .

Step 3

Term 2: Let $x = 0.777\dots$ . $10x = 7.777\dots$ . $9x = 7 \implies x = \frac{7}{9}$ .

Step 4

Term 3: Let $y = 0.4777\dots$ . $10y = 4.777\dots$ . $100y = 47.777\dots$ . $100y - 10y = 43 \implies 90y = 43 \implies y = \frac{43}{90}$ .

Step 5

Sum: $\frac{3}{5} + \frac{7}{9} + \frac{43}{90}$ .

Step 6

LCM is 90. $\frac{54}{90} + \frac{70}{90} + \frac{43}{90} = \frac{167}{90}$ .

Final Answer

$\frac{167}{90}$

Operation

Number

Quick Reference:

N = Natural Numbers: 1, 2, 3, ...

W = Whole Numbers: 0, 1, 2, ...

Z = Integers: ..., -2, -1, 0, 1, 2, ...

Q = Rational Numbers: p/q form (p, q ∈ Z, q ≠ 0)

R = Real Numbers: Q ∪ Irrational

Select an operation and enter values to see detailed step-by-step solutions.