What pattern do you observe? Why are 2 and 5 related in this way?

Question

Homi · Accepted Answer

We are looking at patterns when 1 is divided by powers of 2 and powers of 5.

Step 1 — Observing the patterns

Let us look at the left side first. We are dividing 1 by powers of 2. Each time, the denominator is multiplied by another 2. The decimal value is divided by 2 each time. The number of decimal places increases by one.

Let us look at the right side now. We are dividing 1 by powers of 5. Each time, the denominator is multiplied by another 5. The decimal value is divided by 5 each time. The number of decimal places increases by one.

Step 2 — Calculating the missing values

Let us find the missing value on the left side. The expression is $1/(2 \times 2 \times 2 \times 2 \times 2)$ . This is $1/2^5$ . The previous value was $1/2^4 = 0.0625$ . We divide this value by 2.

$0.0625 \div 2$

$= 0.03125$

$\boxed{0.03125}$

Let us find the missing value on the right side. The expression is $1/(5 \times 5 \times 5 \times 5 \times 5)$ . This is $1/5^5$ . The previous value was $1/5^4 = 0.0016$ . We divide this value by 5.

$0.0016 \div 5$

$= 0.00032$

$\boxed{0.00032}$

Step 3 — Explaining the relationship between 2 and 5

Our number system is based on 10. The prime factors of 10 are 2 and 5. When a fraction has a denominator that is only made of factors of 2 or 5, it can be written as a terminating decimal. This means the decimal will end. We can multiply the numerator and denominator by the missing factors to make the denominator a power of 10. For example, for $1/2^n$ , we multiply by $5^n/5^n$ .

$\frac{1}{2^n} = \frac{1 \times 5^n}{2^n \times 5^n}$

$= \frac{5^n}{(2 \times 5)^n}$

$= \frac{5^n}{10^n}$

For $1/5^n$ , we multiply by $2^n/2^n$ .

$\frac{1}{5^n} = \frac{1 \times 2^n}{5^n \times 2^n}$

$= \frac{2^n}{(5 \times 2)^n}$

$= \frac{2^n}{10^n}$

This makes the conversion to decimal form very direct. The number of decimal places will be equal to $n$ .

Answer

(i) The pattern is that for $1/2^n$ , the decimal value is halved at each step, and the number of decimal places increases by one. For $1/5^n$ , the decimal value is divided by five at each step, and the number of decimal places increases by one. (ii) The numbers 2 and 5 are the prime factors of 10. This allows fractions with denominators that are powers of 2 or 5 to be easily converted into terminating decimals by making the denominator a power of 10. (iii) The missing value for $1/(2 \times 2 \times 2 \times 2 \times 2)$ is 0.03125. The missing value for $1/(5 \times 5 \times 5 \times 5 \times 5)$ is 0.00032.

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