1. Write the pure repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places:
0.000003 ≈ 0.000003000003
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
0.000003000003 =
0.000003000003 × 100/100 =
(0.000003000003 × 100)/100 =
0.0003000003/100 =
0.0003000003% ≈
0%
(rounded off to a maximum of 2 decimal places)
In other words:
Approximate to the desired number of decimal places...
Multiply the number by 100...
... And then add the percent sign, %
0.000003 ≈ 0%
2. Write the pure repeating (recurring) decimal number as a proper fraction.
0.000003 can be written as a proper fraction. The numerator is smaller than the denominator.
Set up the first equation.
Let y equal the decimal number:
y = 0.000003
Set up the second equation.
Number of decimal places repeating: 6
Multiply both sides of the first equation by 106 = 1,000,000
y = 0.000003
1,000,000 × y = 1,000,000 × 0.000003
1,000,000 × y = 3.000003
Subtract the first equation from the second equation.
Having the same number of decimal places ...
The repeating pattern drops off by subtracting the two equations.
1,000,000 × y - y = 3.000003 - 0.000003 ⇒
(1,000,000 - 1) × y = 3.000003 - 0.000003 ⇒
We now have a new equation:
999,999 × y = 3
Solve for y in the new equation.
999,999 × y = 3 ⇒
y = 3/999,999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 0.000003
According to our calculations:
y = 3/999,999
⇒ 0.000003 = 3/999,999
3. Reduce (simplify) the fraction above: 3/999,999
(to the lowest terms, to its simplest equivalent form, irreducible).
To reduce a fraction divide the numerator and the denominator by their greatest (highest) common factor (divisor), GCF.
Factor both the numerator and the denominator (prime factorization).
In exponential notation (an):
3 is a prime number, it cannot be factored into other prime factors
999,999 = 33 × 7 × 11 × 13 × 37
Calculate the greatest (highest) common factor (divisor), GCF.
Multiply all the common prime factors by the lowest exponents.
GCF (3; 33 × 7 × 11 × 13 × 37) = 3
Divide both the numerator and the denominator by their greatest common factor, GCF.
3/999,999 =
3/(33 × 7 × 11 × 13 × 37) =
(3 ÷ 3) / ((33 × 7 × 11 × 13 × 37) ÷ 3) =
1/(32 × 7 × 11 × 13 × 37) =
1/333,333
1/333,333: Equivalent fractions.
The above fraction cannot be reduced.
That is, it has the smallest possible numerator and denominator.
By expanding it we can build up equivalent fractions.
Multiply the numerator & the denominator by the same number.
Example 1. By expanding the fraction by 4:
1/333,333 = (1 × 4)/(333,333 × 4) = 4/1,333,332
Example 2. By expanding the fraction by 8:
1/333,333 = (1 × 8)/(333,333 × 8) = 8/2,666,664
Of course, the above fractions are reducing...
... to the initial fraction: 1/333,333