# Question #0d788

Dec 13, 2017

Technically, $1. \overline{6} = 1 \frac{2}{3}$.
$1.6 = 1 \frac{6}{10} \mathmr{and} 1 \frac{3}{5}$.

#### Explanation:

Use your calculator or long division to divide 2 by 3. You get $\frac{2}{3} = .6666 \ldots \mathmr{and} . \overline{6}$
I think maybe your confusion comes from the notation using the bar over the repeating number.
$.6$ is not the same as $. \overline{6}$; when you see that bar over a number following a decimal, it means the number keeps repeating.

Also, you might know that $\frac{1}{3} = .3333 \ldots = . \overline{3}$
And $\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$, and

certainly $.333 \ldots + .333 \ldots = .666 \ldots$, or, more succinctly,
$. \overline{3} + . \overline{3} = . \overline{6}$

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

Dec 13, 2017

$1.6 = 1 \frac{6}{10} = 1 \frac{3}{5}$

$1.66666666 \ldots$ (recurring decimal) $= 1. \overline{6} = 1 \frac{2}{3}$

#### Explanation:

There are two types of rational decimal fractions.

• terminating decimals - (those which stop)
$\text{ "0.8," " 3.875, " " 4.5, " } 0.1378$

• recurring decimals - those which repeat to infinity.
$\text{ "0.44444....," "3.151515...." } 5.653653653 \ldots .$

These are indicated with a bar or dot on the recurring digits:
$0. \overline{4} , \text{ "3.bar(15)," "5.bar(653)" "or 0.dot4," "3.dot1dot5," "5.dot6 5dot3" }$

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$1.6$ (terminating decimal) is NOT actually equal to $1 \frac{2}{3}$

$1.6 = 1 \frac{6}{10} = 1 \frac{3}{5}$

But, $1.66666666 \ldots$ (recurring decimal) $= 1. \overline{6} = 1 \frac{2}{3}$

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

This arises because $\frac{1}{3}$ means $1 \div 3$ 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 $1$)

$1 \div 3 = 3 | \underline{1.0000000 \ldots}$
$\textcolor{w h i t e}{\times \times \times \times} 0.3333333 \ldots$

The same happens with $\frac{2}{3}$ which means $2 \div 3$

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

(There is always a remainder of $2$)

$2 \div 3 = 3 | \underline{2.0000000 \ldots}$
$\textcolor{w h i t e}{\times \times \times \times} 0.6666666 \ldots$

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

$1 \frac{2}{3} = 1. \overline{6}$

$2 \frac{2}{3} = 2. \overline{6}$

$5 \frac{2}{3} = 5. \overline{6}$