Solved on Dec 14, 2023

Determine if $0.393993999399993999993\ldots$ is rational or irrational. The pattern of digits repeating indefinitely indicates the number is rational.

STEP 1

Assumptions
1. The given number is $0.393993999399993999993\ldots$
2. A number is rational if it can be expressed as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
3. A number is irrational if it cannot be expressed as a fraction of two integers.
4. A repeating decimal is a rational number.

STEP 2

Identify the pattern in the decimal part of the number.
The given number has a pattern where the number of 9s after the digit 3 increases by one each time: 39, 399, 3999, etc.

STEP 3

Express the number as a sum of its individual parts to see if it can be written as a repeating decimal.
The number can be expressed as: $0.393993999399993999993\ldots = 0.3 + 0.09 + 0.0009 + 0.000009 + \ldots$

STEP 4

Recognize that this is a geometric series where the first term $a$ is $0.09$ and the common ratio $r$ is $0.01$ .

STEP 5

Use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$ , where $a$ is the first term and $r$ is the common ratio.

STEP 6

Check if the common ratio $r$ is between $-1$ and $1$ (excluding $-1$ and $1$ ), because only then the infinite geometric series will converge to a sum.
Since $r = 0.01$ , we have $-1 < r < 1$ , so the series converges.

STEP 7

Calculate the sum of the infinite geometric series using the formula.
$S = \frac{a}{1-r}$

STEP 8

Substitute the values of $a$ and $r$ into the formula.
$S = \frac{0.09}{1-0.01}$

STEP 9

Simplify the denominator.
$1 - 0.01 = 0.99$

STEP 10

Calculate the sum $S$ .
$S = \frac{0.09}{0.99}$

STEP 11

Simplify the fraction by dividing both the numerator and the denominator by $0.09$ .
$S = \frac{0.09 / 0.09}{0.99 / 0.09}$

STEP 12

Perform the division.
$S = \frac{1}{11}$

STEP 13

Now, add the initial $0.3$ (which is $3/10$ ) to the sum $S$ .
$Total = \frac{3}{10} + \frac{1}{11}$

STEP 14

Find a common denominator for the two fractions.
The common denominator is $10 \times 11 = 110$ .

STEP 15

Express both fractions with the common denominator.
$Total = \frac{3 \times 11}{110} + \frac{1 \times 10}{110}$

STEP 16

Perform the multiplication.
$Total = \frac{33}{110} + \frac{10}{110}$

STEP 17

Add the two fractions.
$Total = \frac{33 + 10}{110}$

STEP 18

Simplify the sum.
$Total = \frac{43}{110}$

STEP 19

Since we have expressed the original number as a fraction of two integers, the number is rational.
The number $0.393993999399993999993\ldots$ is rational because it can be expressed as the fraction $\frac{43}{110}$ .

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Determine if $0.393993999399993999993\ldots$ is rational or irrational. The pattern of digits repeating indefinitely indicates the number is rational.

STEP 1

STEP 2

Identify the pattern in the decimal part of the number.
The given number has a pattern where the number of 9s after the digit 3 increases by one each time: 39, 399, 3999, etc.

STEP 3

Express the number as a sum of its individual parts to see if it can be written as a repeating decimal.
The number can be expressed as: $0.393993999399993999993\ldots = 0.3 + 0.09 + 0.0009 + 0.000009 + \ldots$

STEP 4

Recognize that this is a geometric series where the first term $a$ is $0.09$ and the common ratio $r$ is $0.01$ .

STEP 5

Use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$ , where $a$ is the first term and $r$ is the common ratio.

STEP 6

Check if the common ratio $r$ is between $-1$ and $1$ (excluding $-1$ and $1$ ), because only then the infinite geometric series will converge to a sum.
Since $r = 0.01$ , we have $-1 < r < 1$ , so the series converges.

STEP 7

Calculate the sum of the infinite geometric series using the formula.
$S = \frac{a}{1-r}$

STEP 8

Substitute the values of $a$ and $r$ into the formula.
$S = \frac{0.09}{1-0.01}$

STEP 9

Simplify the denominator.
$1 - 0.01 = 0.99$

STEP 10

Calculate the sum $S$ .
$S = \frac{0.09}{0.99}$

STEP 11

Simplify the fraction by dividing both the numerator and the denominator by $0.09$ .
$S = \frac{0.09 / 0.09}{0.99 / 0.09}$

STEP 12

Perform the division.
$S = \frac{1}{11}$

STEP 13

Now, add the initial $0.3$ (which is $3/10$ ) to the sum $S$ .
$Total = \frac{3}{10} + \frac{1}{11}$

STEP 14

Find a common denominator for the two fractions.
The common denominator is $10 \times 11 = 110$ .

STEP 15

Express both fractions with the common denominator.
$Total = \frac{3 \times 11}{110} + \frac{1 \times 10}{110}$

STEP 16

Perform the multiplication.
$Total = \frac{33}{110} + \frac{10}{110}$

STEP 17

Add the two fractions.
$Total = \frac{33 + 10}{110}$

STEP 18

Simplify the sum.
$Total = \frac{43}{110}$

STEP 19

Since we have expressed the original number as a fraction of two integers, the number is rational.
The number $0.393993999399993999993\ldots$ is rational because it can be expressed as the fraction $\frac{43}{110}$ .

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Determine if 0.393993999399993999993…0.393993999399993999993\ldots0.393993999399993999993… is rational or irrational. The pattern of digits repeating indefinitely indicates the number is rational.

STEP 1

STEP 2

Identify the pattern in the decimal part of the number.The given number has a pattern where the number of 9s after the digit 3 increases by one each time: 39, 399, 3999, etc.

STEP 3

STEP 4

Recognize that this is a geometric series where the first term aaa is 0.090.090.09 and the common ratio rrr is 0.010.010.01.

STEP 5

Use the formula for the sum of an infinite geometric series, which is a1−r\frac{a}{1-r}1−ra​, where aaa is the first term and rrr is the common ratio.

STEP 6

Check if the common ratio rrr is between −1-1−1 and 111 (excluding −1-1−1 and 111), because only then the infinite geometric series will converge to a sum.Since r=0.01r = 0.01r=0.01, we have −1<r<1-1 < r < 1−1<r<1, so the series converges.

STEP 7

Calculate the sum of the infinite geometric series using the formula.S=a1−rS = \frac{a}{1-r}S=1−ra​

STEP 8

Substitute the values of aaa and rrr into the formula.S=0.091−0.01S = \frac{0.09}{1-0.01}S=1−0.010.09​

STEP 9

Simplify the denominator.1−0.01=0.991 - 0.01 = 0.991−0.01=0.99

STEP 10

Calculate the sum SSS.S=0.090.99S = \frac{0.09}{0.99}S=0.990.09​

STEP 11

Simplify the fraction by dividing both the numerator and the denominator by 0.090.090.09.S=0.09/0.090.99/0.09S = \frac{0.09 / 0.09}{0.99 / 0.09}S=0.99/0.090.09/0.09​

STEP 12

Perform the division.S=111S = \frac{1}{11}S=111​

STEP 13

Now, add the initial 0.30.30.3 (which is 3/103/103/10) to the sum SSS.Total=310+111Total = \frac{3}{10} + \frac{1}{11}Total=103​+111​

STEP 14

Find a common denominator for the two fractions.The common denominator is 10×11=11010 \times 11 = 11010×11=110.

STEP 15

Express both fractions with the common denominator.Total=3×11110+1×10110Total = \frac{3 \times 11}{110} + \frac{1 \times 10}{110}Total=1103×11​+1101×10​

STEP 16

Perform the multiplication.Total=33110+10110Total = \frac{33}{110} + \frac{10}{110}Total=11033​+11010​

STEP 17

Add the two fractions.Total=33+10110Total = \frac{33 + 10}{110}Total=11033+10​

STEP 18

Simplify the sum.Total=43110Total = \frac{43}{110}Total=11043​

STEP 19

Since we have expressed the original number as a fraction of two integers, the number is rational.The number 0.393993999399993999993…0.393993999399993999993\ldots0.393993999399993999993… is rational because it can be expressed as the fraction 43110\frac{43}{110}11043​.

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Determine if $0.393993999399993999993\ldots$ is rational or irrational. The pattern of digits repeating indefinitely indicates the number is rational.

Identify the pattern in the decimal part of the number.
The given number has a pattern where the number of 9s after the digit 3 increases by one each time: 39, 399, 3999, etc.

Recognize that this is a geometric series where the first term $a$ is $0.09$ and the common ratio $r$ is $0.01$ .

Use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$ , where $a$ is the first term and $r$ is the common ratio.

Check if the common ratio $r$ is between $-1$ and $1$ (excluding $-1$ and $1$ ), because only then the infinite geometric series will converge to a sum.
Since $r = 0.01$ , we have $-1 < r < 1$ , so the series converges.

Calculate the sum of the infinite geometric series using the formula.
$S = \frac{a}{1-r}$

Substitute the values of $a$ and $r$ into the formula.
$S = \frac{0.09}{1-0.01}$

Simplify the denominator.
$1 - 0.01 = 0.99$

Calculate the sum $S$ .
$S = \frac{0.09}{0.99}$

Simplify the fraction by dividing both the numerator and the denominator by $0.09$ .
$S = \frac{0.09 / 0.09}{0.99 / 0.09}$

Perform the division.
$S = \frac{1}{11}$

Now, add the initial $0.3$ (which is $3/10$ ) to the sum $S$ .
$Total = \frac{3}{10} + \frac{1}{11}$

Find a common denominator for the two fractions.
The common denominator is $10 \times 11 = 110$ .

Express both fractions with the common denominator.
$Total = \frac{3 \times 11}{110} + \frac{1 \times 10}{110}$

Perform the multiplication.
$Total = \frac{33}{110} + \frac{10}{110}$

Add the two fractions.
$Total = \frac{33 + 10}{110}$

Simplify the sum.
$Total = \frac{43}{110}$

Since we have expressed the original number as a fraction of two integers, the number is rational.
The number $0.393993999399993999993\ldots$ is rational because it can be expressed as the fraction $\frac{43}{110}$ .