# Decimals

**Decimals** are numbers with a fractional part, which is written after a **decimal mark**. The mark separates any whole part from the fractional part. Some examples of a decimal numbers are: 10.2, 0.23, and 19.999.

For "Decimal," the base-10 number system, see: Decimal Numbers (base-10)

Digits in a decimal number have a value that is determined by their placement around the decimal mark. Since our number system is based on ten, the first digit to the right of the decimal mark has a value of \( \frac{1}{10} \) of a unit. The second digit after the decimal point is equal to \( \frac{1}{100} \) part of a unit, and in general the \(n\)-th digit to the right of the decimal point is equal to \( \frac{1}{10^n} \)-th part of a unit.

## Adding Decimals

To add two decimal numbers together, we must ensure that we add up all the corresponding place values, just as we have to be careful when adding \( 201 + 35 \) to ensure that we add the hundreds to the hundreds, the tens to the tens, etc.. A simple trick is to line up the decimal points in both numbers, which ensures that the place values are now lined up.

## What is \( 5.12 + 1.034 \) ?

Lining up the decimal points carefully, we get: \[ \begin{align} & 5.& 1& \quad 2 & \\ + & 1.& 0 &\quad 3 & 4 \\ \hline & 6.& 1 &\quad 5 & 4 \end{align} \]

Thus the answer is \( 6.154 \). \( _\square \)

## What is \( 3.6 + 7.12 ?\)

Lining up the decimal points carefully, we get: \[ \begin{align} & 3.& 6 & \\ + & 7.& 1 &\quad 2 \\ \hline & 10.& 7 &\quad 2 \end{align} \]

Thus the answer is \( 10.72 .\) \( _\square \)

## What is \( 0.332 + 0.83 ?\)

Lining up the decimal points carefully, we get: \[ \begin{align} & 0.& 3& \quad 3 & 2 \\ + & 0.& 8 &\quad 3 & \\ \hline & 1.& 1 &\quad 6 & 2 \end{align} \]

Thus the answer is \( 1.162 .\) \( _\square \)

## What is \( 0.043 + 1.028 ?\)

Lining up the decimal points carefully, we get: \[ \begin{align} & 0.& 0& \quad 4 & 3 \\ + & 1.& 0 &\quad 2 & 8 \\ \hline & 1.& 0 &\quad 7 & 1 \end{align} \]

Thus the answer is \( 1.071 .\) \( _\square \)

## Multiplying Decimals

Multiplying decimals can be done most simply by ignoring the decimal point during the multiplication and then dealing with it after obtaining the product. For example, if the problem is to multiply \( 2.3 \times 5.1 \) we can easily find that \( 23 \times 51 = 1173 \). Then, noting that we had two digits after the decimal place, the result is 11.73.

Note that this approach is possible because \( 2.3 = \frac{1}{10} \times 23 \) and \( 5.1 = \frac{1}{10} \times 51 \), thus \( 2.3 \times 5.1 = 23 \times 51 \times \frac{1}{100} \).

## What is \( 4.1 \times 2 \) ?

We have \( 41 \times 2 = 82 \), and counting digits to the right of the decimal place in our original question, we see that the answer should be multiplied by \( \frac{1}{10} \) or moved 1 decimal place. Thus, the answer is \( 8.2 \). \( _\square \)

## What is \( 3.2 \times 0.5 \) ?

We have \( 32 \times 5 = 160 \), and counting digits to the right of the decimal place in our original question, we see that the answer should be multiplied by \( \frac{1}{100} \) or moved 2 decimal places. Thus, the answer is \( 1.60 \).

Another way to solve this problem is to notice that \(0.5 = \frac{1}{2} \) and therefore, we would like to find \( (3.2) \cdot \frac{1}{2} = 1.6. _\square \)

## What is \( 0.005 \times 0.06 \) ?

We have \( 5 \times 6 = 30 \), and counting digits to the right of the decimal place in our original question, we see that the answer should be multiplied by \( \frac{1}{10^5} \) or moved 5 decimal places. Thus, the answer is \( 0.0003 \). \( _\square\)

## What is \( 6.12 \times .15 \) ?

We can see that \( 612 \times 15 = 9180 \), and counting digits to the right of the decimal place in our original question, we see that the answer should be multiplied by \( \frac{1}{10000} \) or moved 4 decimal places. Thus the answer is \( 0.9180 \). \( _\square \)

## Converting to Fractions

Main page: Converting Decimals and Fractions

An **integer** can simply be written as fraction by making the numerator the number itself and the denominator one. For example the number \(9\) can be written in fractional form as \(\frac{9}{1}\).

A **terminating decimal** can be written as fraction by writing it the way you say it. For example the decimal \(1.5\) is **one and five tenths** or \(1\frac{5}{10}=1\frac{1}{2}\). Adding the terms \(1+\frac{1}{2}=\frac{2}{2}+\frac{1}{2}=\frac{3}{2}\).

A **repeating decimal** can be written as fraction using an algebraic method.

## Converting to Percentages

Main page: Converting Decimals and Percentages

To convert a number written as a percent to decimal, simply divide the number by 100. Thus, \( 56\% = 56 \div 100 = 0.56 \).

To convert a decimal to a percent, just multiply the decimal number by 100. Thus, \( 0.23 = \left(.23 \times 100\right) \% = 23 \% \).