Understanding Decimals

Decimals are an essential part of math, and they are used in a variety of tasks every day. From handling money to measuring objects, decimals make dealing with fractions easier and more understandable. In this detailed guide, we'll learn everything about decimals, including what they are, how they work, and how you can use them in various mathematical problems.

What are decimals?

Decimals are numbers that have a decimal point (.) that represents a part of a whole number. They're an alternative way to represent fractions and are part of the base-10 number system, which is the same system we use for whole numbers.

For example, the decimal number 0.5 is equal to the fraction 1/2. The number 3.75 represents three and seventy-five hundredths. In this number system, each place value to the right of the decimal point represents a power of tenth, such as tenths, hundredths, thousandths, and so on.

Place value system in decimal

Just like whole numbers have place values like ones, tens, hundreds, etc., decimals also have place values, but they extend to the right of the decimal point as well. Let's look at this step by step.

  Place Value Chart
Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths Example: 2 3 6 7 . 8 9 5

In the number 2367.895: - 2 is in the thousands place - 3 is in the hundreds place - 6 is in the tens place - 7 is in the units place - 8 is in the tenths place - 9 is in the hundredths place - 5 is in the thousandths place

Visual representation:

Reading and writing decimals

Reading and writing decimal numbers involves understanding place values. Let's look at an example:

Decimal: 45.768

- First, read the whole number part: "forty-five" - Then, the decimal point is read as "and" - Finally, read the digits individually and mention the fractional place value: "seven hundred sixty-eight thousandths".

Writing in words:

The number 45.768 is written "forty-five and seven hundred sixty-eight thousandths."

Comparing and ordering decimals

To compare decimals, imagine lining them up to the decimal point and then comparing the digits from left to right. An easy way to do this is:

  Compare 0.45 and 0.450
1. Line them up:
  0.450
  0.450
2. Compare digits from left of the decimal to right:
  0 - same
  . - same
  4 - same
  5 - same
  0 - same (adds no value)
The numbers are equal.

When ordering decimal places, follow the same procedure:

  Numbers: 0.3, 0.44, 0.405, 0.475
1. Line them up by the point:
  0.300
  0.440
  0.405
  0.475
2. Compare digits from left to right:
Order: 0.3, 0.405, 0.44, 0.475

Adding and subtracting decimals

Addition and subtraction with decimals is the same as for whole numbers, but we align the numbers with the decimal point:

Example: Add 3.56 and 4.78

  3.56
+ 4.78
------
  8.34

Example: Subtract 8.23 from 9.54

  9.54
- 8.23
------
  1.31

To begin, line up the decimals, then add or subtract as you would with whole numbers. Start with the rightmost digit and work your way to the left.

Multiplying and dividing decimals

Multiplying decimals:

Multiplying decimals involves the following steps:

  Example: Multiply 2.5 by 1.3
Step 1: Ignore the decimals and multiply as whole numbers:
  25 x 13
------
  75  (5 * 15)
+ 250 (2 * 25)
------
  325
Step 2: Count total decimal places in both original numbers (1+1=2)
Step 3: Place the decimal in the result, so it has 2 decimal places: 3.25

In this example, 2.5 and 1.3 have one decimal place each. The product 3.25 also has two decimal places.

Dividing decimals:

Dividing decimals requires careful attention to the place of the decimal place.

  Example: Divide 3.75 by 1.5
Step 1: Multiply both numbers by 10 to make them whole:
  37.5 ÷ 15
Step 2: Divide as with whole numbers:
  37.5 ÷ 15 = 2.5
Result: 2.5

Division is simplified by adjusting the divisor and dividend to make the divisor a whole number.

Converting decimals to fractions

To convert a decimal to a fraction, follow these steps:

  Convert: 0.75
Step 1: Write it as a fraction of over 100 or 10, etc., based on how many decimal places:
  0.75 = 75/100
Step 2: Simplify the fraction:
  75/100 = 3/4

Similar to:

  Convert: 0.5
  0.5 = 5/10 = 1/2

Converting fractions to decimals

Converting fractions to decimals is usually done by division: divide the numerator by the denominator.

  Convert: 3/8
Step 1: Divide 3 by 8 to get
  3 ÷ 8 = 0.375
Result: 0.375

This process involves simple division, resulting in a decimal.

Decimal rounding

Rounding off decimals ensures that numbers are easier to estimate or approximate. To round off decimals, follow the following rules:

Example: Round 2.678 to the nearest hundred:

  1. Identify place value to the right to round.
2. If the digit to the right is 5 or more, increase the targeted digit by 1.
  2.678 rounds to 2.68

Practice this by identifying other decimal places like nearest decimal, thousandth, etc.

Use of decimals in real-world situations

Decimal numerals appear often in everyday life: decimal numerals are used in currency (for example, $5.75), in measurements (for example, 2.4 m, 3.6 kg), etc.

Example in money:

Bob buys a coffee for $2.45 and a bagel for $1.75. How much does he spend in total?

  2.45
+ 1.75
------
  4.20

Bob spends a total of $4.20.

Example in measurement:

One piece of wood is 3.5 m long. The other piece is 2.8 m long. What is the total length?

  3.5
+ 2.8
------
6.3 meters

In woodworking, shopping for goods, traveling distances, and many other everyday applications, decimals help with precision and detail that fractions cannot efficiently provide without complicated conversions.

Conclusion

Understanding decimals is important for mathematical literacy, making arithmetic tasks more manageable and practical. Their versatility in representing whole and fractional quantities ensures their widespread use in science, finance, engineering, and everyday transactions. Practicing operations and understanding the relationship between decimals and fractions increases proficiency in dealing with decimals.

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