We deal with fractional numbers every day. From currency to measuring materials at work, our lives depend on fractions. However, when we need to calculate measurements or prices we need decimal numbers. Decimal numbers make math easier, but they are not as intuitive as fractions. Decimals can tell you how much pizza everyone should get, but we need fractions to know how divide up the pie. Fortunately, decimals and fractions refer the same numbers, and we can usually use them interchangeably. Fractions can be converted to decimals, and a decimal can be changed to a fraction with just a few simple steps.

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# What are Fractions and Decimals?

Fractions are ratios of two whole numbers. The first number is called the numerator and it represents the number of pieces we have. The second number is called the denominator, and it represents how many pieces make up the whole. We use them to represent values that cannot represent as whole numbers such as half (1/2) and quarter (1/4). We can also combine them with whole numbers to create mixed number such as 346¾.

We can use mixed numbers to represent any value in the world, but they have their limitations. Fractions take up a lot of space to write, and fraction math is quite complicated. You probably already know how difficult it is to write fractions on a computer, tablet, or smartphone. Instead of using a horizontal bar, we end up using a backslash (/). True, word processors such as Word offer ways to properly write fractions such as 346¾, but you cannot use them online or in text messages. Fortunately, we can use decimal numbers instead.

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# Decimals are Fractions

Decimal numbers let us save space while writing fractions. Decimal numbers use the same math rules we use with whole numbers streamlining and simplifying any and all calculations involving them. Decimals are written as a single number preceded by a period. A zero is usually placed in front of the period unless we have a whole number instead. Technically, we do not have mixed numbers when dealing with decimals. Instead, we have what are known as rational numbers. Rational numbers are numbers we can write as fractions. We write 346¾ as a rational number as 346.75. Irrational numbers are numbers we cannot express as fractions. Before you can convert your decimal to a fraction, you will need to know the difference.

Rational and irrational numbers are written using the decimal number system that represents numbers using ten symbols called digits. Each digit represents a number itself. These digits are the familiar symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We create larger numbers by using combinations of these digits such as grouping 5 and 7 to create 57. Each digit in a number represents a different power of ten that we must multiply that digit to put it in that place. The exponent of this power of ten represents how far away the digit is from the decimal point. Whole numbers use positive exponents while decimals use negative exponents.

For example:

3598.726

3 is in the +3 position from the decimal point as its power of ten is 10^{3} or 1000.

5 is +2 with 10^{2} or 100 as its power of ten.

9 is +1 with 10^{1} or 10 as its power of ten.

8 is in the 0 position with 10^{0} or 1 as the power of ten.

7 is in the -1 position with the power of ten 1/10 or 10^{-1}.

2 is in the -2 position with the power of ten 1/100 or 10^{-2}.

6 is in the -3 position with the power of ten 1/1000 or 10^{-3}.

These powers of ten let us change a decimal to a fraction.

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https://www.udemy.com/an-introduction-to-real-numbers/

# Change Decimals into Fractions and Back Again

Decimals are fractions where the denominator is a power of 10. For example, we can express 8/10, 83/100, 83/1000, and 8/10000 as 0.8, 0.83, 0.083, and 0.0008. We use these powers of 10, as negative powers of 10, to represent decimal digit places in numbers.

We use negative exponents to write factions when the numerator is 1. For instance 1/100 is 100^{-1}. Since 100 is 10^{2}, we can also write 1/100 as 10^{-2}. We change decimals to fractions by inspecting the power of ten for the rightmost digit.

To change a decimal into a fraction, you just create write down a fraction with the decimal without the decimal point as the numerator. You then include the associated power of ten as the denominator. That’s it. You now changed a decimal into a fraction. You can continue further to reduce the fraction as you see fit, but the fraction you just created is the decimal number in faction form.

For example, let’s change the decimal 0.1250 into a fraction. First, 0.1250 ends with a 0. This 0 is four places from the decimal point. That means it is in the 10^{-4} digit place. 10^{-4} is the fraction 1/10^{4}, so the power of ten we have is 10^{4} or 10,000. Now, to transform our decimal 0.1250 into a fraction, we create a fraction with our number without the leading 0 and the decimal point as the numerator, and the power or ten as a denomination to create 1250/10000. We now have our decimal as a fraction. We can further reduce it to 125/1000 or 1/8 is we want.

We can even do this when your decimal number contains a whole number. You just include the whole number with the decimal in the numerator of your fraction without the decimal point. We then continue procedure as we did before without the whole number. For example 2.125 becomes 2125/1000 or 17/8 or 2 1/8.

We can get our decimal back by converting our fraction to a fraction with a power of ten as a denominator, and then just write the numerator by itself with the rightmost digit in the position dictated by the power of ten we used. We place the decimal point after the digit in the ones place and before the digit in the tenths place. If our number does not contain any whole number, we add a leading 0 to finish it off. After that, you will have your original decimal number back.