Introduction to Decimals

Welcome to the exciting journey into the world of decimals! Decimals are an essential part of mathematics, and understanding them will open new doors in your math knowledge. Decimals represent parts of a whole, just like fractions do, but in a slightly different way. In this chapter, we will explore what decimals are, how they relate to fractions, and how you can use them in various examples and problems.

What are decimals?

Decimals are a way to express numbers that are between whole numbers. When you think about decimal numbers, imagine dividing a whole number into smaller parts. Suppose we have the number 1, and we want to divide it into ten equal parts. Each part will be a decimal fraction of 1, represented as 0.1.

This is what the decimal 0.1 looks like on the number line:

0------0.1-------0.2------0.3-----------0.4-------0.5-----------0.6-----------0.7------0.8----------0.9------1

This row divides 0 through 1 into 10 equal parts. Each part is 0.1, making it easy to see how decimals break down whole numbers.

Decimals and place value

Decimals also have place value like whole numbers. Let's take a look at the number 3.25 and understand its place value:

Whole Number Division: 3
Decimal part: .25

The number 3.25 has two parts: the whole number part (3) and the decimal part (.25). The decimal part is read "twenty-five hundredths" because it is 25 parts out of 100.

This is how place values work for decimals:

3.25
,
| | +---> Hundredth place (5)
| +-----> 10th position (2)
+--------> units place (3)

Understanding place value will help you read and write decimal numbers correctly.

Converting fractions to decimals

Decimals and fractions are closely related because both represent parts of a whole. Let's convert an ordinary fraction to a decimal and see how they are connected.

The fraction 1/2 can be converted to a decimal. To do this, we divide the numerator (1) by the denominator (2).

Calculate: 1 ÷ 2 = 0.5

So, the fraction 1/2 is equal to the decimal 0.5.

Visually, it looks like this:

Fraction: 1/2

Division: 1 ÷ 2 = 0.5

Decimal: 0.5

This calculation means that 1/2 of something is equal to 0.5 of that same thing.

Converting decimals to fractions

It is also possible to convert a decimal back to a fraction. Let's take the decimal 0.75 and turn it into a fraction.

The number 0.75 means "seventy-five hundredths", or 75 parts of 100. Thus, the fraction is:

0.75 = 75/100

To simplify a fraction, you divide both the numerator and denominator by their greatest common divisor (GCD). For 75 and 100, this number is 25.

Simplification:

75 ÷ 25 = 3
100 ÷ 25 = 4

75/100 = 3/4

So the decimal 0.75 is equivalent to the simplified fraction 3/4.

Comparing decimals

Understanding how to compare decimals is important when you want to know which decimal is bigger or smaller. Let's take the decimals 0.3 and 0.29. Which is bigger?

0.3 = 30/100
0.29 = 29/100

Since the number 30/100 is greater than 29/100, the decimal 0.3 is greater than 0.29.

You can compare decimals visually as follows:

0.3: 0------0.1-------0.2------[0.3]-------0.4
0.29: 0------0.1------0.2-----0.29

As you can see, 0.3 is right after 0.29 on the number line.

Adding and subtracting decimals

Adding and subtracting decimals is similar to adding and subtracting whole numbers. To do this, make sure the decimal points line up.

For example, add 0.4 and 0.35:

  0.40
+ 0.35
,
  0.75

This result means 0.4 + 0.35 = 0.75.

Let's try subtracting 0.6 from 0.95:

  0.95
- 0.60
,
  0.35

Here, 0.95 - 0.6 equals 0.35.

Multiplying decimals

Multiplying decimals is simple. First multiply them as if they were whole numbers, and then count and place the decimal point.

Multiply 0.3 by 0.4:

0.3 (3/10) × 0.4 (4/10) = 12/100 = 0.12

When you multiply 0.3 and 0.4 the answer is 0.12.

Dividing decimals

Dividing decimals is a little more complicated. Change the divisor to a whole number by moving the decimal point to the right. Then, apply the same steps to the dividend and divide as usual.

Divide 1.2 by 0.4:

Change 0.4 -> 4 by moving the decimal one place to the right
Do the same for 1.2 -> 12

12 ÷ 4 = 3

So dividing 1.2 by 0.4 will give 3.

Use of decimals in real life

Understanding how decimals work is useful in everyday life. You can use decimals when handling money, measuring length, or calculating weight.

Imagine you bought a pencil for $0.75 and an eraser for $0.25. How much did you spend? Add the decimals:

  0.75
+ 0.25
,
  1.00

You spent a total of $1.00.

Practicing with decimals

Strengthen your understanding of decimals by practising converting fractions to decimals, comparing them, and calculating with them. Here are some exercises you can try yourself:

1. Convert the fraction 3/5 to a decimal.
2. Compare these decimals and find out which is larger: 0.48 and 0.52.
3. Add 0.7 and 0.25.
4. Multiply 0.2 by 0.6.
5. Divide 2.4 by 0.3.

Each time you work with decimals, visualize the number line and look at the place values to help you understand the size of the numbers. Remember, practicing regularly will make you more confident in using decimals every day.

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