Question

Marcy performed a calculation and the result was 4.8333333..., is it rational or irrational?

Answer 1

If the `3`'s continue endlessly, it a sign of a ratio.

Explanation:

To find out what the ratio is, you could multiply by `3`, as a series of `3`'s will become an endless series of `9`'s.

`4.8333...xx3=14.4999...=14.5`

So `14.5/3=4.8333...` The `14.5` is not a whole number, but we can double both the denominator and the numerator:

`4.8333...=29/6` which is certainly a ratio.

Answer 2

All recurring decimals are rational numbers.

Explanation:

All recurring decimals are rational numbers.

They are formed when one rational number is divided by another rational number.

Any divisor which has a factor that cannot be divided exactly into a power of 10 produces a recurring decimal.

For example: `20div3, " "15 div 6," "25 div 11`

These include division by `3, 6, 7, 9, 11, 13, 15, 18` and so on.

It is to your advantage to learn the common recurring decimals and their fraction equivalents:

`0.bar3 = 1/3 " "0.bar6 = 2/3`

`0.1bar6 = 1/6" " 0.8bar3 = 5/6`

`0.bar1 = 1/9 " "0.bar2 = 2/9" "0.bar4 = 4/9`

`0.bar5 = 5/9" "0.bar7 = 7/9" "0.bar8 = 8/9`

Answer 3

`4.8bar(3) = 29/6` is a rational number.

Explanation:

First, in case you have not met it, let me introduce you to bar notation for repeating decimals...

When you have a decimal representation with a repeating pattern, draw a bar over the digit or sequence of digits which repeat.

`4.8333... = 4.8bar(3)`

Next, allow me to demonstrate a method that allows you to convert any repeating decimal into a fraction.

Given:

`4.8bar(3)`

Separate the number into a sum of whole number and fractional part:

`4.8bar(3) = color(blue)(4) + 0.8bar(3)`

Take the reciprocal of the resulting fractional part `0.8bar(3)` to find:

`color(white)(4.8bar(3)) = color(blue)(4) + 1/1.2`

Repeat with the resulting decimal, dividing it into whole number and fractional parts:

`color(white)(4.8bar(3)) = color(blue)(4) + 1/(color(blue)1+0.2)`

Take the reciprocal of the fractional part to find:

`color(white)(4.8bar(3)) = color(blue)(4) + 1/(color(blue)1+1/color(blue)(5))`

Having arrived at a whole number (`color(blue)(5)`), we can simplify the resulting expression:

`color(white)(4.8bar(3)) = 4 + 1/(5/5+1/5)`

`color(white)(4.8bar(3)) = 4 + 1/(6/5)`

`color(white)(4.8bar(3)) = 4 + 5/6`

`color(white)(4.8bar(3)) = 24/6 + 5/6`

`color(white)(4.8bar(3)) = 29/6`

`color(white)()`
Footnote

If you try doing this with the decimal representation of an irrational number, then you will not get a whole number, but you may find useful rational numbers along the way.

Let me show you what I mean:

`pi ~~ 3.14159265359`

We find:

`3.14159265359 = 3+0.14159265359`

`color(white)(3.14159265359) ~~ 3+1/7.06251330592`

`color(white)(3.14159265359) ~~ 3+1/(7+1/15.9965944)`

`color(white)(3.14159265359) ~~ 3+1/(7+1/(15+1/1.00341724))`

`color(white)(3.14159265359) ~~ 3+1/(7+1/(15+1/(1+1/292.63382)))`

Stopping here, notice the large denominator `292.63382`.

That means that we can get a good approximation to `pi` by truncating just before this:

`pi ~~ 3+1/(7+1/(15+1/1)) = 3+1/(7+1/16) = 3+1/(113/16) = 3+16/113 = 355/113`