Decimals

Decimals are a way of representing fractions where the denominator is a power of 10. They are an extension of the place value system used for whole numbers. Understanding decimals is crucial for various calculations in everyday life and in competitive exams.

Understanding Place Value in Decimals

Just like whole numbers have place values for ones, tens, hundreds, etc., decimals have place values to the right of the decimal point. The first digit to the right of the decimal point represents tenths (1/10), the second represents hundredths (1/100), the third represents thousandths (1/1000), and so on.

For example, in the number 123.456:

• 1 is in the hundreds place.
• 2 is in the tens place.
• 3 is in the ones place.
• . is the decimal point.
• 4 is in the tenths place (meaning 4/10).
• 5 is in the hundredths place (meaning 5/100).
• 6 is in the thousandths place (meaning 6/1000).

This can be written as: 123.456 = 100 + 20 + 3 + 4/10 + 5/100 + 6/1000

Types of Decimals

Decimals can be broadly classified into two types: terminating and non-terminating.

Terminating Decimals

A terminating decimal is a decimal that has a finite number of digits after the decimal point. This happens when the denominator of the fraction (in its simplest form) has only prime factors of 2 and 5.

Examples:

• 1/2 = 0.5
• 3/4 = 0.75
• 7/8 = 0.875
• 1/10 = 0.1

Non-Terminating Decimals

A non-terminating decimal is a decimal that has an infinite number of digits after the decimal point. These can be further divided into repeating and non-repeating decimals. For competitive exams, we primarily focus on repeating decimals, which represent rational numbers.

Repeating Decimals (Recurring Decimals)

In a repeating decimal, a digit or a group of digits repeats infinitely after the decimal point. A bar is often placed over the repeating part to indicate this.

Examples:

• 1/3 = 0.333... = 0.̄3
• 2/3 = 0.666... = 0.̄6
• 1/6 = 0.1666... = 0.1̄6
• 5/11 = 0.454545... = 0.̄45

The repeating part is called the 'repetend'.

Non-Repeating Decimals (Irrational Numbers)

These decimals have an infinite number of digits after the decimal point, and no digit or group of digits repeats in a pattern. These represent irrational numbers.

Examples:

• √2 = 1.41421356...
• π (Pi) = 3.14159265...

Converting Fractions to Decimals

To convert a fraction to a decimal, you divide the numerator by the denominator.

Example: Convert 5/8 to a decimal.

Divide 5 by 8: 5 ÷ 8 = 0.625

Example: Convert 2/3 to a decimal.

Divide 2 by 3: 2 ÷ 3 = 0.666... = 0.̄6

Converting Decimals to Fractions

To convert a terminating decimal to a fraction:

1. Write the decimal number without the decimal point as the numerator.
2. The denominator will be 1 followed by as many zeros as there are digits after the decimal point.
3. Simplify the fraction.

Example: Convert 0.75 to a fraction.

Numerator: 75 Denominator: 100 (since there are two digits after the decimal point) Fraction: 75/100 Simplified fraction: 3/4

Example: Convert 0.125 to a fraction.

Numerator: 125 Denominator: 1000 Fraction: 125/1000 Simplified fraction: 1/8

Converting Repeating Decimals to Fractions

This is a key skill for competitive exams.

Case 1: Purely Repeating Decimal (e.g., 0.̄3, 0.̄45)

1. Let the decimal be 'x'.
2. Write the repeating part as the numerator.
3. The denominator will be as many 9s as there are digits in the repeating part.

Example: Convert 0.̄3 to a fraction.

Repeating part is 3 (one digit). Fraction = 3/9 = 1/3

Example: Convert 0.̄45 to a fraction.

Repeating part is 45 (two digits). Fraction = 45/99 = 5/11

Case 2: Mixed Repeating Decimal (e.g., 0.1̄6, 0.23̄45)

1. Let the decimal be 'x'.
2. Write the number without the decimal point and the bar as the numerator.
3. Subtract the non-repeating part (the digits before the repeating block) from the numerator.
4. The denominator will consist of as many 9s as there are digits in the repeating part, followed by as many 0s as there are digits in the non-repeating part.
5. Simplify the fraction.

Example: Convert 0.1̄6 to a fraction.

Number without bar: 16 Non-repeating part: 1 Numerator: 16 - 1 = 15 Repeating part has 1 digit (6) → one 9. Non-repeating part has 1 digit (1) → one 0. Denominator: 90 Fraction: 15/90 = 1/6

Example: Convert 0.23̄45 to a fraction.

Number without bar: 2345 Non-repeating part: 23 Numerator: 2345 - 23 = 2322 Repeating part has 2 digits (45) → two 9s. Non-repeating part has 2 digits (23) → two 0s. Denominator: 9900 Fraction: 2322/9900 = 1161/4950 = 387/1650 = 129/550

Operations on Decimals

The rules for addition, subtraction, multiplication, and division of decimals are similar to those for whole numbers, with the key being the correct placement of the decimal point.

Addition and Subtraction of Decimals

To add or subtract decimals, align the decimal points vertically. Write zeros in any empty place values to make the numbers have the same number of decimal places. Then, add or subtract as usual, keeping the decimal point in the same position.

Example: Add 12.345 and 5.67

Align the decimal points: ``` 12.345 + 5.670 (added a zero for alignment) --------- 18.015 ```

Example: Subtract 8.25 from 15.1

Align the decimal points: ``` 15.10 (added a zero for alignment) - 8.25 --------- 6.85 ```

Multiplication of Decimals

To multiply decimals:

1. Multiply the numbers as if they were whole numbers, ignoring the decimal points initially.
2. Count the total number of decimal places in all the numbers being multiplied.
3. Place the decimal point in the product so that it has the same number of decimal places as counted in step 2.

Example: Multiply 2.5 by 1.2

Multiply 25 by 12: 25 × 12 = 300 In 2.5, there is 1 decimal place. In 1.2, there is 1 decimal place. Total decimal places = 1 + 1 = 2. Place the decimal point 2 places from the right in 300: 3.00 So, 2.5 × 1.2 = 3.00 or 3.

Example: Multiply 0.34 by 0.5

Multiply 34 by 5: 34 × 5 = 170 In 0.34, there are 2 decimal places. In 0.5, there is 1 decimal place. Total decimal places = 2 + 1 = 3. Place the decimal point 3 places from the right in 170: 0.170 So, 0.34 × 0.5 = 0.170 or 0.17.

Division of Decimals

To divide decimals:

1. Make the divisor a whole number by moving its decimal point to the right.
2. Move the decimal point in the dividend the same number of places to the right as you moved it in the divisor. Add zeros if necessary.
3. Perform the division as if they were whole numbers.
4. Place the decimal point in the quotient directly above the decimal point in the dividend.

Example: Divide 12.5 by 0.5

Divisor is 0.5. Move decimal 1 place right to make it 5. Dividend is 12.5. Move decimal 1 place right to make it 125. Now divide 125 by 5: 125 ÷ 5 = 25. So, 12.5 ÷ 0.5 = 25.

Example: Divide 6.72 by 1.2

Divisor is 1.2. Move decimal 1 place right to make it 12. Dividend is 6.72. Move decimal 1 place right to make it 67.2. Now divide 67.2 by 12. ``` 5.6 _______ 12|67.2 60 --- 7.2 7.2 --- 0 ``` So, 6.72 ÷ 1.2 = 5.6.

Comparing Decimals

To compare decimals, start from the leftmost digit and compare them place by place.

1. If the whole number parts are different, the one with the larger whole number part is greater.
2. If the whole number parts are the same, move to the tenths place and compare. The decimal with the larger digit in the tenths place is greater.
3. If the tenths digits are the same, move to the hundredths place and compare, and so on.
4. If one decimal runs out of digits while comparing, you can add trailing zeros to it to make the comparison easier.

Example: Compare 3.14 and 3.141

• Whole number parts are the same (3).
• Tenths digits are the same (1).
• Hundredths digits are the same (4).
• The first number (3.14) has no further digits. The second number (3.141) has a digit (1) in the thousandths place.
• We can write 3.14 as 3.140.
• Comparing 3.140 and 3.141, the thousandths digit 0 is less than 1.
• Therefore, 3.141 is greater than 3.14.

Example: Compare 0.5 and 0.499

• Whole number parts are the same (0).
• Tenths digit in 0.5 is 5. Tenths digit in 0.499 is 4.
• Since 5 is greater than 4, 0.5 is greater than 0.499.

Approximation and Rounding

Often in calculations, we need to approximate or round decimals to a certain number of decimal places.

Rounding Rules:

1. Identify the digit in the place value to which you want to round.
2. Look at the digit immediately to its right.
3. If the digit to the right is 5 or greater, round up the digit in the desired place value (add 1 to it).
4. If the digit to the right is less than 5, keep the digit in the desired place value as it is.
5. Drop all digits to the right of the rounded digit.

Example: Round 15.783 to one decimal place.

The digit in the first decimal place is 7. The digit to its right is 8. Since 8 is greater than or equal to 5, we round up the 7. 15.783 rounded to one decimal place is 15.8.

Example: Round 2.3456 to three decimal places.

The digit in the third decimal place is 5. The digit to its right is 6. Since 6 is greater than or equal to 5, we round up the 5. 2.3456 rounded to three decimal places is 2.346.

Example: Round 10.1234 to two decimal places.

The digit in the second decimal place is 2. The digit to its right is 3. Since 3 is less than 5, we keep the 2 as it is. 10.1234 rounded to two decimal places is 10.12.

Recurring Decimal Identities

There are some useful identities involving recurring decimals that can save time in exams.

• 0.̄a = a/9
• 0.̄ab = ab/99
• 0.̄abc = abc/999
• 0.āb = (ab - a)/90
• 0.ab̄c = (abc - ab)/900
• 0.ābc = (abc - a)/990

Example: What is the value of 0.̄3 + 0.̄6?

0.̄3 = 3/9 = 1/3 0.̄6 = 6/9 = 2/3 Sum = 1/3 + 2/3 = 3/3 = 1.

Example: What is the value of 0.1̄6 + 0.2̄7?

0.1̄6 = (16 - 1)/90 = 15/90 = 1/6 0.2̄7 = (27 - 2)/90 = 25/90 = 5/18 Sum = 1/6 + 5/18 Find a common denominator (18): (3/18) + (5/18) = 8/18 = 4/9. As a decimal, 4/9 = 0.444... = 0.̄4.

Decimal Representation of Rational Numbers

Every rational number (a number that can be expressed as p/q, where p and q are integers and q ≠ 0) can be represented as either a terminating decimal or a repeating decimal.

A rational number p/q (in simplest form) results in a terminating decimal if and only if the prime factorization of the denominator q contains only powers of 2 and 5.

If the denominator q has prime factors other than 2 and 5, the decimal representation will be a repeating decimal.

Example: 7/20 = 7/(2² × 5). Denominator has only 2s and 5s. So, it's a terminating decimal. 7/20 = 0.35.

Example: 5/12 = 5/(2² × 3). Denominator has a factor of 3. So, it's a repeating decimal. 5/12 = 0.41666... = 0.41̄6.

Decimal Representation of Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction p/q. Their decimal representation is non-terminating and non-repeating.

Examples include √2, √3, π, e. Their decimal expansions go on forever without any repeating pattern.

Practical Applications

Decimals are used extensively in:

• Currency: Representing money (e.g., $10.50).
• Measurements: Length, weight, volume (e.g., 2.5 meters, 1.75 kg).
• Science and Engineering: Calculations involving precise values.
• Finance: Interest rates, stock prices.
• Statistics: Averages, probabilities.