How do I multiply numbers infinitely many times?

Algebra Level 2

0. 3 3 3 3 × 0. 6 6 6 6 0. D D D D \begin{array}{cccccccc} && & 0. & 3 & 3 & 3&3\ \ldots \\ \times && & 0. & 6 & 6 & 6&6\ \ldots \\ \hline && & 0. & D& D &D&D\ \ldots \\ \end{array}

The above is a long multiplication with only the first 4 decimal places of each number shown.

What is D ? D?

1 2 3 4 5

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2 solutions

Jordan Cahn
Dec 3, 2018

0. 3333 = 1 3 0.\overline{3333} = \dfrac{1}{3} and 0. 6666 = 2 3 0.\overline{6666}=\dfrac{2}{3} .

1 3 × 2 3 = 2 9 = 0. 2222 \frac{1}{3}\times\frac{2}{3} = \frac{2}{9} = 0.\overline{2222}

Munem Shahriar
Dec 9, 2018

0. 3 = 0.33333333..... = 0.3 + 0.03 + 0.003 + . . . . = 0.3 1 0.1 [ r = 0.03 0.3 = 0.1 ] = 0.3 0.9 = 1 3 \begin{aligned} 0.\overline 3 & = 0.33333333..... \\ & = 0.3 + 0.03+0.003+.... \\ & = \dfrac{0.3}{1-0.1} ~~~~~ [r = \dfrac{0.03}{0.3} = 0.1] \\ & = \dfrac{0.3}{0.9} \\ & = \dfrac 13\\ \end{aligned}

0. 6 = 0.66666666..... = 0.6 + 0.06 + 0.006 + . . . . = 0.6 1 0.1 [ r = 0.06 0.6 = 0.1 ] = 0.6 0.9 = 6 9 = 2 3 \begin{aligned} 0.\overline 6 & = 0.66666666..... \\ & = 0.6 + 0.06+0.006+.... \\ & = \dfrac{0.6}{1-0.1} ~~~~~ [r = \dfrac{0.06}{0.6} = 0.1] \\ & = \dfrac{0.6}{0.9} \\ & = \dfrac 69 = \dfrac 23 \\ \end{aligned}

So 1 3 × 2 3 = 2 9 = 0.2222222...... \dfrac 13 \times \dfrac 23 = \dfrac 29 = 0.2222222......

Hence D = 2 D = 2

Unnessesary

Leafy Li - 2 years, 3 months ago

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