Decimal Representation of Real Numbers

The concept of decimal representation is an essential part of understanding real numbers. Real numbers include all the numbers you would normally find on a number line. These include rational numbers (such as fractions and integers) and irrational numbers (such as square roots of non-perfect squares and π). Decimal representation helps us express these numbers in a readable and understandable format.

Decimal numbers are a way of representing numbers using the base-10 numeral system, which is also the standard system for representing integers and non-integer numbers. In this system, each digit after the decimal point represents a fraction whose denominator is a power of ten.

What is a decimal number?

Before going deep into the topic, let us understand what a decimal number is. A decimal number has two parts:

• The whole number part, which is the number before the decimal point.
• The fractional part, which is the number after the decimal point.

For example, consider the number 45.678. Here, 45 is the whole number part, and 678 is the fractional part.

Mathematically, 45.678 can also be written as

45 + 0.678

Where 0.678 is a decimal fraction. To expand 0.678:

0.678 = 6 * 0.1 + 7 * 0.01 + 8 * 0.001
      = 6/10 + 7/100 + 8/1000

Thus, each digit has its own place value, determined by its position relative to the decimal point.

Visualization of decimal numbers

Let's look at the number 45.678 using the number line.

Note that 45.678 falls between 45 and 46 on the number line. The fractional part (0.678) means that the value is greater than 45 but less than 46, which shows how decimals determine the exact location on the number line.

Types of decimal representation

The decimal representation can be divided into two types:

1. Ending decimals: These decimals come at the end. For example, the decimal representation of the fraction 1/2 is 0.5, and for 1/4 it is 0.25.
2. Non-terminating decimals: These decimals continue forever. Non-terminating decimals are further classified into:
   • Repeating decimals: A sequence of digits is repeated an infinite number of times. For example, the decimal representation of 1/3 is 0.3333..., which can be written as 0.(3).
   • Non-repeating decimals: The sequence of digits never repeats. These are usually irrational numbers, such as π.

Ending decimal example

Let's work through a simple example of a terminating decimal:

Consider the fraction 1/8. To convert it to a decimal, divide 1 by 8:

            0.125
          ,
        8 | 1.000
           - 8
          ,
            20
           - 16
          ,
             40
            - 40
          ,
              0

So 1/8 is equal to 0.125, which is a terminating decimal.

Repeating decimal examples

Consider the fraction 2/3 for a recurring decimal:

            0.6666...
          ,
        3 | 2.000
           - 18
          ,
             20
           - 18
          ,
              20
            - 18
          ,
               20 ...

The digit '6' continues indefinitely, so its decimal is 0.(6), which shows that '6' is a repeating sequence.

Non-recurring decimal examples

An example of a non-recurring, non-terminating decimal is the number π, which is approximately equal to 3.141592653589793..., and the digits continue without repeating.

Mixed number

Mixed numbers have both integer and fractional parts, such as 7.85, which is the sum of the whole number 7 and the decimal fraction 0.85.

It is very easy to convert a mixed number to a decimal. For example, the number 7 + 43/100 has a decimal equivalent of 7.43.

Operations on decimal numbers

It is very important to understand how mathematical operations are performed on decimal numbers. Let's learn about the basic operations:

Add

To add decimals, align the decimal points and proceed as for whole numbers:

Example: Add 12.56 and 7.4.

   12.56
+ 7.40
,
   19.96

Subtraction

Subtract decimals by aligning the decimal points and subtracting like whole numbers:

Example: Subtract 4.38 from 9.15.

   9.15
- 4.38
,
   4.77

Multiplication

Multiply the decimals as whole numbers, then place the decimal point in the answer by counting the total decimal places in both numbers:

Example: Multiply 6.3 by 0.5.

       6.3
  × 0.5
,
      315 (ignore first decimal)
   + 000
,
    3.15 (one decimal in each multiplier, so two in total)

Division

Dividing decimals involves moving the decimal place to make the divisor a whole number and adjusting the dividend accordingly:

Example: Divide 4.5 by 0.3.

   Moving the decimal point forward one place, 4.5 becomes 45, and 0.3 becomes 3.

   45 ÷ 3 = 15

Converting fractions to decimals

Conversion from fractions to decimals is achieved by division. If the division process terminates, the decimal is terminating. If it is repeated, the decimal is repeating. Let's look at examples of this process:

Example of ending decimals

Convert 7/8 to decimal:

   0.875
  ,
8 | 7.000
   - 64
  ,
     60
    - 56
  ,
      40
    - 40
  ,
       0

The answer is 0.875, which is a terminating decimal.

Example of a repeating decimal

Convert 1/9 to decimal:

   0.1111...
  ,
9 | 1.0000
   -0.9
  ,
    100
   - 90
  ,
     100
   ,

The digit 1 is repeated indefinitely, so the answer is 0.(1).

Converting decimals to fractions

To convert a decimal to a fraction, write the decimal number above its place value. Reduce the resulting fraction to its simplest form.

Example of ending decimals

Convert 0.75 to a fraction:

0.75 = 75/100 = 3/4 (after simplification)

Example of a repeating decimal

Consider converting the repeating decimal 0.6666... to a fraction. Let x = 0.666....

1. Multiply both sides by 10:
   10x = 6.666...
   
2. Subtract the first equation from the second:
   10x - x = 6.666... - 0.666...

3. Solve for x:
   9x = 6
   x = 6/9
   Simplification:
   x = 2/3

Decimal rounding

Rounding is the process of rounding off decimals to a specific precision.

Rounding off rules

• If the digit on the right is less than 5, round it down.
• If the digit on the right is 5 or more, round it up.

Example of rounding

Round off 4.736 to two decimal places:

Check the third decimal place (6) - it is ≥5, so increase the second decimal place by 1: 4.74.

Conclusion

In conclusion, mastering the decimal representation is crucial to understanding real numbers. They serve as a bridge that allows us to move seamlessly between the exact values of fractions and the immediacy of arithmetic operations on the number line. From terminating to non-terminating decimals, each offers a unique insight into the relationships and behaviors of numbers. Practice with a variety of examples to strengthen your knowledge, ensuring that you can confidently navigate through mathematical problems and solutions. Real numbers, with their infinite potential, continue to play a vital role in mathematics and the world around us.

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