Express the repeating decimal as a ratio of two integers.
2.6231313131....
2.62 + .0031 + .000031 + .00000031 ....
= 2.62 + .0031/(1`-.01)
= 2.62 + .0031/.99
= 262/100 + 31/9900
= 262*99/9900 +31/9900
=(25938 +31)/9900
= 25969/9900
=2,6231313131........
= 262048/99900
Ignore that last line, early error
To express a repeating decimal as a ratio of two integers, we can use the fact that if a decimal repeats, it can be expressed as a fraction over 9s.
Let's consider the repeating part of the decimal, which is "1313" in this case. The number of repeating digits is 4, so we write the repeating decimal as:
2.6231313131... = 2.623 + 0.1313 / (10^4 - 1)
Now, simplify the fraction:
0.1313 / (10^4 - 1)
To find the value of the denominator, calculate 10^4 - 1:
10^4 - 1 = 10000 - 1 = 9999
The fraction becomes:
0.1313 / 9999
To express this fraction as a ratio of two integers, multiply numerator and denominator by 10000 (since there are four digits after the decimal point in the repeating part):
0.1313 * 10000 / 9999
= 1313 / 9999
So, the repeating decimal 2.6231313131... can be expressed as the ratio 1313/9999.