Understanding Decimals

Decimals are an important part of math that you use in your daily life. You see them on price tags, when calculating distance, time, and even in measurements in science. But what exactly are decimals? Let's understand what decimals are and how they work.

What is a decimal?

Decimals are a way to write fractions. Instead of writing the fraction as 1/2, you can write it as 0.5. Decimal numbers are based on 10. Numbers can have parts less than a whole, and decimals are used to show those parts. They use a point called a "decimal point" to separate whole numbers from parts of a whole.

Place value in decimal

Just as whole numbers have place values, so do decimals. The decimal point helps you understand these place values. Here is an example of place values in the decimal system:

    thousands hundreds tens units tenths hundredths thousandths
       1 5 3 2. 4 7 6

In the number 1532.476, the digit 1 is in the thousandths place, 5 is in the hundreds place, and so on. To the right of the decimal point, we have the tenths, hundredths, and thousandths places.

Reading decimals

When you read decimals, you speak the number before the decimal point as a whole number. Then, you speak "point" for the decimal point, followed by reading each digit after the decimal separately. For example, 45.67 is read as "forty-five point six seven." If there are final zeros, they are usually not read, such that 45.6700 is still "forty-five point six seven."

Writing decimals

Writing decimals involves understanding their value according to the place value chart. Let's look at some examples:

• The decimal point 0.8 is the same as eight-tenths.
• The decimal place 0.25 is twenty-five hundredths.
• The decimal 0.375 is three hundred seventy-five thousandths.

Viewing decimals

Visual aids can really help in understanding decimals. Let's use some examples:

In this example, the entire rectangle represents a whole, and the shaded portion represents 0.5 or half of that whole.

Here 0.75 means three-fourth of the whole. The shaded area represents the part that makes up 0.75.

Comparing decimals

Comparing decimals is a simple task once you understand place value. Here are the steps to compare decimals:

1. Align the decimal points of the numbers you want to compare.
2. Start from the left and compare digits until you find a different digit.
3. The number which has the larger first fractional digit will be the larger decimal place.

Consider the 0.76 and 0.8 comparison:

    Number 1: 0. 7. 6
    Number 2: 0. 8 0

Since the digits in the tens place are different (7 vs. 8), 0.8 is greater than 0.76.

Adding decimals

To add decimals, follow these steps:

1. Line up the decimal points of the numbers you're adding.
2. Start at the right and add each column of digits, just like with whole numbers.
3. If the sum in a column is greater than 10, carry it over to the next column to the left.

Let's add 2.75 and 3.45:

      2.75
    + 3.45
    ,
      6.20

By aligning the decimal points and adding, we get that 2.75 + 3.45 = 6.2.

Subtracting decimals

Subtraction is similar to addition, except there are a few key steps:

1. Line up the decimal points of the included numbers.
2. Subtract each column as you would with whole numbers.
3. If a smaller digit is subtracted from a larger digit, you may need to borrow a digit from the column to the left.

Example: Subtract 1.32 from 2.45:

      2.45
    - 1.32
    ,
      1.13

So, 2.45 - 1.32 = 1.13.

Multiplying decimals

When multiplying decimals, follow these steps:

1. Multiply the decimal points and multiply the numbers as if they were whole numbers.
2. Count the total number of decimal places in both factors (the numbers you're multiplying).
3. Place the decimal point in the product (result) with the total number of places.

For example, multiplying 0.6 and 0.7 gives:

      0.6
    x 0.7
    ,
      42 (Ignore decimals and multiply as whole numbers: 6 x 7 = 42)
      
    Result: 0.42 (since there are two decimal places in total, we place the decimal point in the product with two places)

The result of 0.6 x 0.7 is 0.42.

Dividing decimals

You can divide decimals as follows:

1. Move the decimal point in the divisor (the number you're dividing by) to the right until it becomes a whole number.
2. Move the decimal point in the dividend (the number you want to divide by) the same number of places to the right.
3. Then, divide the new dividend by the new divisor as you normally would.

Example: Divide 4.5 by 0.5:

    4.5 ÷ 0.5
    = 45 ÷ 5 (shift the decimal point one place in both numbers)

    Result: 9

Here, 4.5 ÷ 0.5 equals 9.

Converting decimals to fractions

Decimals can be easily converted into fractions, and here's how:

1. Write the decimal after dividing by 1.
2. Multiply the numerator (top number) and the denominator (bottom number) by 10 for each digit after the decimal point.
3. Simplify the fraction, if possible.

Example: Convert 0.75 to a fraction:

    0.75 = 75 / 100 (Multiply both numerator and denominator by 100)
    Simplified: 3 / 4

Thus, 0.75 is 3/4 in its simplest form.

Practice problems

Here are some practice problems to test your understanding of decimals:

• Convert 0.5 to a fraction.
• What is the sum of 7.95 and 2.45?
• Subtract 5.2 from 8.6.
• Multiply 0.9 by 0.3.
• Divide 9.6 by 0.4.

Conclusion

Decimals are very important to understand because they are widely used in our everyday lives and many areas of study. From reading and writing to performing operations such as addition, subtraction, multiplication and division, decimals help us understand numbers less than an integer in a precise way. Practice often to get more comfortable with decimals and you will find that they become a handy tool for measuring, calculating and comparing numbers.

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