71-72 : Express each repeating decimal as a fraction. (See...

$71-72$ : Express each repeating decimal as a fraction. (See the margin note on page $11 . )$
$$
\begin{array}{llll}{\text { (a) } 0 . \overline{7}} & {\text { (b) } 0.2 \overline{8}} & {\text { (c) } 0 . \overline{57}}\end{array}
$$

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers. Since repeating decimals can always be written as fractions, they are a subset of rational numbers. Understanding this concept helps in identifying and working with various types of numbers in mathematics.

Algebraic Manipulation

Algebraic manipulation refers to the set of techniques used to rearrange and simplify equations. In the context of repeating decimals, it is used to set up equations based on shifting digits (using powers of 10) and then subtracting to solve for the variable representing the decimal, thereby converting the repeating decimal to a simple fraction.

Repeating Decimals

A repeating decimal is a decimal number in which a digit or a group of digits repeats infinitely. Recognizing repeating decimals is important because they have an equivalent representation as fractions, making them a type of rational number.

Conversion of Repeating Decimals to Fractions

This concept involves transforming a repeating decimal into a fraction using algebraic techniques. Typically, a variable is set equal to the repeating decimal, then multiplied by an appropriate power of 10 to shift the decimal point such that subtracting the original variable eliminates the repeating portion, allowing for the construction of an equation that can be solved to yield the fraction.

Transcript

00:01 Okay, so we want to write this as a decimal.

00:05 So let's start by noting that 0 .77.

00:09 Well, let's multiply this by, let's see, if we multiply 100 by 0 .77, we're going to get, so let's write that as 100x is equal to 77 .77 and so on.

00:28 And then we can multiply this by 10x, where x is equal to our repeating term.

00:35 So 10 times that, that's going to give us 7 .77 and so on.

00:41 So let's notice that if we subtract these two numbers, we're going to end up with a 90x is equal to 77 minus 7, which is 70.

00:56 And then these repeating decimals, they cancel out.

00:59 Okay, so we see that x.

01:01 To well 70 over 90 that's equal to 7 over 9 we can reduce that okay and now let's move on to our next portion so here we have two repeating numbers so i'm going to multiply this by um let's see i want let's try 1 000 times 0 .28 .28 .2 8 .2 8.

01:32 Okay um but i wanted to start with 28 to 8 right after our decimal.

01:39 So what about 10 ,000? so let's see, 0 .28 .28.

01:55 Okay, so let's go with 1000 times x, where this is our x.

02:03 So that's going to give us 2828 times 28 and so on.

02:09 And now i want our next term multiplied by x to have 28 and then 0 .28 .28.

02:16 And so on.

02:18 So if i multiply this by 100, that's what i get.

02:22 So 100x is equal to the following...