Rational Numbers

[sbsbsbsbsb]

How do you change a rational number into a fraction?

Im not talking about a simple number, im thinking of a more complex number like 1.621 or 1.517 or something like that. Can anyone help?

 

[mmm4444bot]

1.621 = 1621/1000

1.517 = 1517/1000

Wanna try some repeating decimals ?

 

[sbsbsbsbsb]

Oh wow thats a lot easier than i thought!! thanks for the reply.
yeah that would be great

 

[mmm4444bot]

Let's express the Rational number 1.621111… as an improper fraction.

The basic strategy is to multiply by powers of ten, so that the repeating part subtracts away.

Start by letting x = 1.621111…

          1000x = 1621.11111...
     _     100x =  162.11111...
     ----------------------------
           900x = 1459.00000...


          x = 1459/900

 

[sbsbsbsbsb]

ok i think i have it. let me know if this is right...
im going to use 1.51777.... as x
1000x=1517.777....
- 100x=0151.7777....
---------------------------------------------------
900x=1366
x= 1366/900 =683/450

is that right?

 

[mmm4444bot]

x = 1366/900 = 683/450

is that right? Yes

As a check, you can divide 683 by 450 using a regular calculator; it will show you something like 1.517777778.

1.1688311688311688311688311688311688311688312

 

[BigGlenntheHeavy]

\(\displaystyle Another \ way, \ to \ wit:\)

\(\displaystyle 1.\overline{168831} \ = \ 1+.168831+.000000168831+000000000000168831+...\)

\(\displaystyle Hence, \ after \ 1, \ we \ have \ a \ geometric \ progression \ with \ a \ = \ .168831 \ and \ r \ = \ .000001.\)

\(\displaystyle Ergo, \ 1.\overline{168831} \ = \ 1+\frac{.168831}{1-.000001} \ = \ 1 \ + \ \frac{.168831}{.999999} \ =1 \ + \ \ \frac{168831}{999999} \ = \ \frac{90}{77}.\)

\(\displaystyle Now, \ try \ one, \ for \ example: \ N \ = \ 1.03\overline{287}.\)

 

[mmm4444bot]

Clever, Glenn.

1.03 287 287 287…

(1/100)(103.287 287 287…)

(1/100)(103 + 0.287 + 0.000287 + 0.000000287 + …)

a1 = 0.287

r = 10^(-3)

(1/100)(103 + 0.287/[1 - 0.001])

(1/100)(103 + 287/999)

103/100 + 297/99900

25796/24975

 

[BigGlenntheHeavy]

\(\displaystyle N \ = \ 1.03\overline{287} \ = \ 1+.03+.00287+.00000287+.00000000287+...\)

\(\displaystyle N \ = \ 1+\frac{3}{100}+\frac{.00287}{1-.001}, \ a \ = \ .00287, \ r \ = \ .001.\)

\(\displaystyle N \ = \ 1+\frac{3}{100}+\frac{.00287}{.999} \ = \ 1+\frac{3}{100}+\frac{287}{99900} \ = \ \frac{25796}{24975}.\)