Decimal

What Is a Decimal?

A decimal is a number that includes a decimal point, used to represent values that are parts of a whole number. Think of it as another way to write a fraction. The decimal point is the dot that separates the whole number part on the left from the fractional part on the right. Each digit to the right of the decimal point has a specific place value, just like the digits to the left.

Understanding place value is the key to understanding decimals. As you move to the right of the decimal point, each place value is ten times smaller than the one before it.

The Decimal Place Value Chart

Here is a chart showing the place values around the decimal point:

For the number $245.683$:

• The $2$ is in the hundreds place (value: $200$).
• The $4$ is in the tens place (value: $40$).
• The $5$ is in the ones place (value: $5$).
• The $6$ is in the tenths place (value: $0.6$ or $\frac{6}{10}$).
• The $8$ is in the hundredths place (value: $0.08$ or $\frac{8}{100}$).
• The $3$ is in the thousandths place (value: $0.003$ or $\frac{3}{1000}$).

How Do You Read and Write Decimals?

Reading and writing decimals correctly is essential. There are a few standard ways to express them.

Let's use the number $12.345$ as an example.

1. Standard Form: This is just the number written with digits: $12.345$.
2. Word Form: To read a decimal in word form, you read the whole number part, say "and" for the decimal point, and then read the number to the right of the decimal, followed by the place value of the last digit. For $12.345$, the last digit ($5$) is in the thousandths place. So, you would read it as: "Twelve and three hundred forty-five thousandths."
3. Expanded Form: This method shows the value of each digit. You write the number as a sum of its place values. For $12.345$, the expanded form is: $10+2+0.3+0.04+0.005$. You can also write this using fractions: $10+2+\frac{3}{10}+\frac{4}{100}+\frac{5}{1000}$.

Here's another example with the number $0.78$:

• Word Form: "Seventy-eight hundredths." (Since there's no whole number, we don't say "and").
• Expanded Form: $0.7+0.08$ or $\frac{7}{10}+\frac{8}{100}$.

How Do You Compare and Order Decimals?

Comparing decimals tells you which number is larger or smaller. It's like comparing whole numbers, but you have to pay close attention to the decimal point. Here’s a reliable method:

1. Line Them Up: Write the numbers you want to compare vertically, lining up the decimal points. If a number doesn't have the same number of decimal places, you can add zeros to the end (this doesn't change its value). For example, $5.6$ is the same as $5.60$ and $5.600$.
2. Compare from Left to Right: Start with the digits in the largest place value (the leftmost column). Compare the digits in that column.
3. Find the First Difference: Keep moving to the right, one column at a time, until you find a column where the digits are different. The number with the larger digit in that column is the larger number.

How Do You Add and Subtract Decimals?

Adding and subtracting decimals is very similar to working with whole numbers. There is one golden rule you must always follow.

When you line up the decimal points, you ensure that you are adding or subtracting digits with the same place value (tenths with tenths, hundredths with hundredths, and so on). You can add placeholder zeros to the end of numbers to make them the same length, which can help prevent mistakes.

Adding Decimals

Let's add $23.45+7.8$.

1. Write the numbers vertically, lining up the decimal points.
2. Add placeholder zeros if it helps you keep the columns straight.
3. Add each column, starting from the right, just like with whole numbers. Regroup (carry over) if necessary.
4. Bring the decimal point straight down into your answer.

  008.920
+ 125.400
+ 000.557
----------

   ¹ ¹
  008.920
+ 125.400
+ 000.557
----------
  134.877

Subtracting Decimals

The process for subtraction is the same. Line up the decimal points, add placeholder zeros if needed, and subtract, borrowing when necessary.

How Do You Multiply Decimals?

Multiplying decimals has a different set of rules. You do not need to line up the decimal points. Instead, you follow these two simple steps:

1. Multiply as Whole Numbers: First, ignore the decimal points completely and multiply the numbers as if they were whole numbers.
2. Count and Place the Decimal: Count the total number of digits to the right of the decimal point in both of the original numbers you multiplied. In your answer, start from the right and move the decimal point that many places to the left.

   582
 x  34
------
  2328  (This is 4 x 582)
17460  (This is 30 x 582)
------
19788

How Do You Divide Decimals?

Division with decimals can seem intimidating, but it follows a clear process. The method changes slightly depending on whether you are dividing by a whole number or another decimal.

Case 1: Dividing by a Whole Number

When the divisor (the number you're dividing by) is a whole number, the process is straightforward.

1. Set up the long division problem as you normally would.
2. Place the decimal point in the quotient (the answer) directly above the decimal point in the dividend (the number being divided).
3. Divide as you would with whole numbers.

For example, to solve $15.75÷5$, you would place the decimal point in the answer right above the one in $15.75$ and then divide. The answer would be $3.15$.

Case 2: Dividing by a Decimal

You can't divide by a decimal directly. You must first change the divisor into a whole number.

1. Move the Divisor's Decimal: Move the decimal point in the divisor all the way to the right to make it a whole number. Count how many places you moved it.
2. Move the Dividend's Decimal: Move the decimal point in the dividend the same number of places to the right. You may need to add zeros as placeholders.
3. Divide: Place the decimal point in the quotient directly above the new position in the dividend and divide as usual.

   42.9
  _____
5 | 214.5
   -20
   ---
    14
   -10
   ---
     45
    -45
    ---
      0

How Are Decimals and Fractions Related?

Decimals and fractions are two different ways to represent the same thing: parts of a whole. Being able to convert between them is a very useful skill.

Converting Decimals to Fractions

This is all about place value. Say the decimal aloud, write it down, and simplify.

1. Read It: Read the decimal using its proper place value name. For example, $0.75$ is read as "seventy-five hundredths."
2. Write It: Write what you just said as a fraction. "Seventy-five hundredths" is $\frac{75}{100}$.
3. Simplify It: Reduce the fraction to its simplest form. Both $75$ and $100$ are divisible by $25$. $\frac{75÷25}{100÷25}=\frac{3}{4}$. So, $0.75=\frac{3}{4}$.

Converting Fractions to Decimals

A fraction is just a division problem. The fraction bar means "divide by."

Let's convert $\frac{2}{5}$ to a decimal. This means we need to calculate $2÷5$. You will need to add a decimal point and a zero to the dividend (the $2$).

   0.4
  ____
5 | 2.0
   -0
   --
    2 0
   -2 0
   ----
      0

So, $\frac{2}{5}=0.4$.

Common Mistakes to Avoid with Decimals

Working with decimals can be tricky at first. Here are some common errors to watch out for:

• Misaligning Decimal Points: This is the most frequent mistake in addition and subtraction. Always line up the decimal points vertically before you start. Adding placeholder zeros can help you keep your columns straight.
• Forgetting to Place the Decimal in Multiplication: After multiplying the numbers, it's easy to forget the final step. Always count the total decimal places in the factors and place the decimal in the product accordingly.
• Moving the Decimal Incorrectly in Division: When dividing by a decimal, remember to move the decimal in both the divisor and the dividend. Whatever you do to the outside number, you must do to the inside number.
• Comparing Decimals Like Whole Numbers: A student might see $0.45$ and $0.8$ and think $0.45$ is larger because $45$ is larger than $8$. By lining them up ($0.45$ vs $0.80$), it's clear that $0.8$ (eighty hundredths) is larger than $0.45$ (forty-five hundredths).
• Reading "and" Incorrectly: The word "and" is reserved specifically for the decimal point. The number $125$ is "one hundred twenty-five." The number $100.25$ is "one hundred and twenty-five hundredths."

Decimal Operations: A Quick Summary

Here's a quick reference for the four basic operations with decimals:

• Addition & Subtraction: Line up the decimal points. Add or subtract as you would with whole numbers. Bring the decimal point straight down into the answer.
• Multiplication: Ignore the decimals and multiply. Then, count the total number of decimal places in the original numbers. Place the decimal in the answer so it has that same number of decimal places.
• Division: If the divisor is a decimal, move the decimal point to the right to make it a whole number. Move the decimal in the dividend the same number of places. Then, bring the decimal point straight up into the quotient and divide.