Q1.
Math Talk: Complete the pattern — 1/2 = 0.5, 1/(2×2) = 0.25, 1/(2×2×2) = 0.125, 1/(2×2×2×2) = 0.0625, 1/(2×2×2×2×2) = ? and 1/5 = 0.2, 1/(5×5) = 0.04, 1/(5×5×5) = 0.008, 1/(5×5×5×5) = 0.0016, 1/(5×5×5×5×5) = ? What pattern do you observe? Why are 2 and 5 related in this way?
Answer
The two missing values are 0.03125 and 0.00032.
| Powers of 2 | Decimal | Powers of 5 | Decimal |
|---|---|---|---|
| 1/2 | 0.5 | 1/5 | 0.2 |
| 1/(2×2) | 0.25 | 1/(5×5) | 0.04 |
| 1/(2×2×2) | 0.125 | 1/(5×5×5) | 0.008 |
| 1/(2×2×2×2) | 0.0625 | 1/(5×5×5×5) | 0.0016 |
| 1/(2×2×2×2×2) | 0.03125 | 1/(5×5×5×5×5) | 0.00032 |
The patterns:
- Every one of these decimals ends — none of them goes on for ever.
- At each step down the list the decimal is halved on the left and divided by 5 on the right.
- Both columns need one more decimal place at each step: 1, 2, 3, 4, 5 places.
- The digits in each row of the two columns multiply to a power of ten: 5 × 2 = 10, 25 × 4 = 100, 125 × 8 = 1000, 625 × 16 = 10000, 3125 × 32 = 100000.
1/32 = 1×3125 / 32×3125 = 3125/100000 = 0.03125
1/3125 = 1×32 / 3125×32 = 32/100000 = 0.00032
1/3125 = 1×32 / 3125×32 = 32/100000 = 0.00032
Why 2 and 5 are related this way: because 10 = 2 × 5. Every power of ten is made of exactly as many 2s as 5s: 100000 = 2⁵ × 5⁵. So a denominator built only from 2s can always be completed into a power of ten by supplying the missing 5s, and a denominator built only from 5s by supplying the missing 2s. That is why fractions with denominators built only from 2s and 5s always give decimals that stop — and why every other denominator, like 3 or 7, gives a decimal that never stops.
Tip: the two decimals in a row are a matching pair. 0.03125 and 0.00032 both have 5 decimal places, and 3125 × 32 = 100000 shows exactly why.