O so many 0's

Number Theory Level 2

What is 0.1 × 0.01 × 0.001 × 0.0001 × ⋯ × 0.0000000001 0.1 \times 0.01 \times 0.001 \times 0.0001 \times \cdots \times 0.0000000001 ?

Note : The expression above is a product of 10 decimal numbers.

0 0 1 0 55 10^{55} 1 0 − 11 10^{-11} 1 0 − 55 10^{-55} 1 0 − 45 10^{-45}

Margaret Zheng by Margaret Zheng , 20, USA

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2 solutions

Margaret Zheng
May 22, 2016

Relevant wiki: Decimals

Observe that this expression could be written as 1 0 − 1 × 1 0 − 2 × 1 0 − 3 × . . . × 1 0 − 10 10^{-1} \times 10^{-2} \times 10^{-3} \times ... \times 10^{-10} .

Using rules of exponents , we get

1 0 − 1 × 1 0 − 2 × 1 0 − 3 × . . . × 1 0 − 10 10^{-1} \times 10^{-2} \times 10^{-3} \times ... \times 10^{-10} = 1 0 − ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ) 10^{-(1+2+3+4+5+6+7+8+9+10)} = 1 0 − 55 \boxed{10^{-55}} .

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Ashish Menon
May 23, 2016

The expression above would have ∑ n = 1 10 n = 10 × 11 2 = 55 \displaystyle \sum_{n = 1}^{10} n = \dfrac{10 × 11}{2} = 55 numbers after the decimal point. So the answer is 10 − 55 \boxed{{10}^{-55}} .

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