Question

Question #0d788

Answer 1

Technically, `1.bar 6 =1 2/3`.
`1.6=1 6/10 or 1 3/5`.

Explanation:

Use your calculator or long division to divide 2 by 3. You get `2/3=.6666... or .bar 6`
I think maybe your confusion comes from the notation using the bar over the repeating number.
`.6` is not the same as `.bar6`; when you see that bar over a number following a decimal, it means the number keeps repeating.

Also, you might know that `1/3=.3333... = .bar3`
And `1/3+1/3=2/3`, and

certainly `.333...+.333...=.666...`, or, more succinctly,
`.bar3+.bar3=.bar6`

Does that answer your question?

PS. As a math teacher, I would like to sincerely thank you for working on building conceptual understanding and facility with fractions!! Good work!

Answer 2

`1.6 = 1 6/10 = 1 3/5`

`1.66666666...` (recurring decimal) `= 1.bar6 = 1 2/3`

Explanation:

There are two types of rational decimal fractions.

• terminating decimals - (those which stop)
  `" "0.8," " 3.875, " " 4.5, " "0.1378`

• recurring decimals - those which repeat to infinity.
  `" "0.44444....," "3.151515...." " 5.653653653....`

These are indicated with a bar or dot on the recurring digits:
`0.bar4," "3.bar(15)," "5.bar(653)" "or 0.dot4," "3.dot1dot5," "5.dot6 5dot3" "`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`1.6` (terminating decimal) is NOT actually equal to `1 2/3`

`1.6 = 1 6/10 = 1 3/5`

But, `1.66666666...` (recurring decimal) `= 1.bar6 = 1 2/3`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This arises because `1/3` means `1div3` and when you divide `1` by `3`, the answer keeps recurring,

Try it yourself with a calculator or by division:

(There is always a remainder of `1`)

`1 div 3 = 3|ul(1.0000000...)`
`color(white)(xxxxxxxx)0.3333333...`

The same happens with `2/3` which means `2div3`

When you divide `2` by `3`, the answer keeps recurring.

(There is always a remainder of `2`)

`2 div 3 = 3|ul(2.0000000...)`
`color(white)(xxxxxxxx)0.6666666...`

If you have a whole number and a fraction, this does not affect the decimal at all:

`1 2/3 = 1.bar6`

`2 2/3 = 2.bar6`

`5 2/3 = 5.bar6`