Whole Numbers, Decimals and Fractions

Welcome to this detailed session on Whole Numbers, Decimals, and Fractions. These are fundamental concepts in mathematics, and a strong understanding of them is crucial not just for competitive exams like the SSC CGL, but for everyday life as well. We'll break down each concept, explore their properties, and look at how they are used in problem-solving.

Whole Numbers

Whole numbers are the simplest form of numbers. They include all the natural numbers (1, 2, 3, ...) and zero (0). So, the set of whole numbers is {0, 1, 2, 3, 4, ...}.

Properties of Whole Numbers

Understanding the properties of whole numbers helps in simplifying calculations and solving problems efficiently.

• Closure Property:
  • Addition: The sum of any two whole numbers is always a whole number. (e.g., 5 + 7 = 12)
  • Multiplication: The product of any two whole numbers is always a whole number. (e.g., 6 × 4 = 24)
  • Subtraction: The difference between two whole numbers is not always a whole number. (e.g., 3 - 5 = -2, which is not a whole number). So, whole numbers are not closed under subtraction.
  • Division: The quotient of two whole numbers is not always a whole number. (e.g., 7 ÷ 3 = 2.33..., which is not a whole number). So, whole numbers are not closed under division.
• Commutative Property:
  • Addition: The order of addition does not affect the sum. (a + b = b + a). (e.g., 9 + 2 = 11, and 2 + 9 = 11)
  • Multiplication: The order of multiplication does not affect the product. (a × b = b × a). (e.g., 3 × 8 = 24, and 8 × 3 = 24)
  • Subtraction: Subtraction is not commutative. (a - b ≠ b - a). (e.g., 10 - 4 = 6, but 4 - 10 = -6)
  • Division: Division is not commutative. (a ÷ b ≠ b ÷ a). (e.g., 12 ÷ 3 = 4, but 3 ÷ 12 = 0.25)
• Associative Property:
  • Addition: The grouping of numbers in addition does not affect the sum. (a + (b + c) = (a + b) + c). (e.g., 2 + (3 + 4) = 2 + 7 = 9, and (2 + 3) + 4 = 5 + 4 = 9)
  • Multiplication: The grouping of numbers in multiplication does not affect the product. (a × (b × c) = (a × b) × c). (e.g., 2 × (3 × 4) = 2 × 12 = 24, and (2 × 3) × 4 = 6 × 4 = 24)
  • Subtraction: Subtraction is not associative. (a - (b - c) ≠ (a - b) - c). (e.g., 10 - (5 - 2) = 10 - 3 = 7, but (10 - 5) - 2 = 5 - 2 = 3)
  • Division: Division is not associative. (a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c). (e.g., 24 ÷ (6 ÷ 2) = 24 ÷ 3 = 8, but (24 ÷ 6) ÷ 2 = 4 ÷ 2 = 2)
• Distributive Property: Multiplication distributes over addition. (a × (b + c) = (a × b) + (a × c)). (e.g., 5 × (3 + 4) = 5 × 7 = 35, and (5 × 3) + (5 × 4) = 15 + 20 = 35). This property is very useful for simplifying calculations.
• Identity Element:
  • Additive Identity: 0 is the additive identity for whole numbers, as a + 0 = a for any whole number 'a'.
  • Multiplicative Identity: 1 is the multiplicative identity for whole numbers, as a × 1 = a for any whole number 'a'.
• Role of Zero:
  • When any whole number is added to 0, the result is the number itself (a + 0 = a).
  • When any whole number is multiplied by 0, the result is 0 (a × 0 = 0).
  • Division by zero is undefined.

Operations on Whole Numbers

We perform addition, subtraction, multiplication, and division on whole numbers. The results of addition and multiplication are always whole numbers, but subtraction and division may yield results that are not whole numbers.

Decimals

Decimals are a way of representing numbers that are less than one or have fractional parts, using a decimal point. A decimal number has two parts: the whole number part and the fractional part, separated by a decimal point. For example, in 15.75, 15 is the whole number part and .75 is the fractional part.

Understanding Place Value in Decimals

The digits to the right of the decimal point represent fractions with denominators that are powers of 10.

• The first digit to the right of the decimal point is in the tenths place (1/10).
• The second digit is in the hundredths place (1/100).
• The third digit is in the thousandths place (1/1000), and so on.

Example: 0.123 means 1/10 + 2/100 + 3/1000.

Converting Fractions to Decimals

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

• Example: 3/4 = 3 ÷ 4 = 0.75
• Example: 1/2 = 1 ÷ 2 = 0.5
• Example: 1/3 = 1 ÷ 3 = 0.333... (This is a recurring decimal)

Converting Decimals to Fractions

To convert a decimal to a fraction:

1. Write the decimal number 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 if possible.

• Example: 0.75 = 75/100. Divide both numerator and denominator by 25 to get 3/4.
• Example: 0.125 = 125/1000. Divide both by 125 to get 1/8.
• Example: 2.5 = 25/10 = 5/2.

Operations on Decimals

• Addition and Subtraction: Align the decimal points vertically and add or subtract as usual.
  • Example: 12.34 + 5.678 = 18.018
  • Example: 25.1 - 7.05 = 18.05
• Multiplication: Multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and place the decimal point in the result so that it has that many decimal places.
  • Example: 2.5 × 1.2 = 3.00 (25 × 12 = 300. There is 1 decimal place in 2.5 and 1 in 1.2, so 1+1=2 decimal places in the result). We write it as 3.00 or simply 3.
  • Example: 0.05 × 0.3 = 0.015 (5 × 3 = 15. There are 2 decimal places in 0.05 and 1 in 0.3, so 2+1=3 decimal places in the result).
• Division:
  • Dividing by a whole number: Perform long division. Place the decimal point in the quotient directly above the decimal point in the dividend.
    • Example: 15.6 ÷ 4 = 3.9
  • Dividing by a decimal: Make the divisor a whole number by multiplying both the divisor and the dividend by a power of 10 (e.g., 10, 100, 1000). Then, perform the division as above.
    • Example: 7.5 ÷ 0.5 = 75 ÷ 5 = 15
    • Example: 1.44 ÷ 1.2 = 14.4 ÷ 12 = 1.2

Fractions

A fraction represents a part of a whole. It is written in the form a/b, where 'a' is the numerator and 'b' is the denominator. The denominator 'b' tells us how many equal parts the whole is divided into, and the numerator 'a' tells us how many of those parts we have. The denominator can never be zero (b ≠ 0).

Types of Fractions

• Proper Fraction: The numerator is smaller than the denominator (e.g., 1/2, 3/4, 7/10). The value of a proper fraction is always less than 1.
• Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 5/3, 7/7, 10/4). The value of an improper fraction is 1 or greater than 1.
• Mixed Fraction: It consists of a whole number and a proper fraction (e.g., 1 1/2, 3 2/5). It represents a value greater than 1.
• Equivalent Fractions: Fractions that represent the same value, even though they have different numerators and denominators. They can be obtained by multiplying or dividing the numerator and denominator by the same non-zero number. (e.g., 1/2 = 2/4 = 3/6).
• Unit Fraction: A fraction where the numerator is 1 (e.g., 1/3, 1/5, 1/10).

Converting Between Improper Fractions and Mixed Fractions

• Improper Fraction to Mixed Fraction: Divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same.
  • Example: Convert 17/5 to a mixed fraction. 17 ÷ 5 = 3 with a remainder of 2. So, 17/5 = 3 2/5.
• Mixed Fraction to Improper Fraction: Multiply the whole number by the denominator, add the numerator to this product, and use this sum as the new numerator. The denominator remains the same.
  • Example: Convert 3 2/5 to an improper fraction. (3 × 5) + 2 = 15 + 2 = 17. So, 3 2/5 = 17/5.

Operations on Fractions

• Addition and Subtraction:
  • When denominators are the same: Add or subtract the numerators and keep the denominator the same.
    • Example: 2/5 + 1/5 = 3/5
    • Example: 7/8 - 3/8 = 4/8 = 1/2
  • When denominators are different: Find a common denominator (usually the Least Common Multiple or LCM of the denominators). Convert each fraction to an equivalent fraction with the common denominator, then add or subtract the numerators.
    • Example: 1/2 + 1/3. LCM of 2 and 3 is 6. So, 1/2 = 3/6 and 1/3 = 2/6. Then, 3/6 + 2/6 = 5/6.
    • Example: 5/6 - 2/9. LCM of 6 and 9 is 18. So, 5/6 = 15/18 and 2/9 = 4/18. Then, 15/18 - 4/18 = 11/18.
• Multiplication: Multiply the numerators together and multiply the denominators together. Simplify the resulting fraction if possible.
  • Example: 2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15.
  • Example: 3/4 × 8/9. You can simplify before multiplying: (3/4) × (8/9) = (3/9) × (8/4) = (1/3) × (2/1) = 2/3. Or, (3 × 8) / (4 × 9) = 24/36, which simplifies to 2/3.
• Division: To divide one fraction by another, invert the second fraction (find its reciprocal) and multiply.
  • Example: 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3.
  • Example: 5/8 ÷ 3/4 = 5/8 × 4/3 = (5 × 4) / (8 × 3) = 20/24. Simplify this to 5/6.

Simplifying Fractions

To simplify a fraction, divide both the numerator and the denominator by their Greatest Common Divisor (GCD). This is also called reducing the fraction to its lowest terms.

• Example: Simplify 18/24. The GCD of 18 and 24 is 6. So, 18 ÷ 6 = 3 and 24 ÷ 6 = 4. The simplified fraction is 3/4.

Interconversion between Whole Numbers, Decimals, and Fractions

Understanding how to convert between these forms is essential for problem-solving.

• Whole Number to Fraction: Any whole number 'n' can be written as a fraction n/1. (e.g., 7 = 7/1)
• Whole Number to Decimal: Any whole number 'n' can be written as a decimal n.0. (e.g., 5 = 5.0)
• Decimal to Fraction: As explained earlier, write the decimal as the numerator and a power of 10 as the denominator, then simplify. (e.g., 0.2 = 2/10 = 1/5)
• Fraction to Decimal: Divide the numerator by the denominator. (e.g., 3/5 = 3 ÷ 5 = 0.6)
• Decimal to Whole Number: If the decimal part is .0, it's a whole number. Otherwise, if the decimal is less than 0.5, round down to the nearest whole number. If it's 0.5 or greater, round up. (e.g., 5.0 is 5, 5.2 rounds to 5, 5.7 rounds to 6). Note: For exact conversions, we typically convert to fractions first if needed.
• Fraction to Whole Number: A fraction can be converted to a whole number only if the denominator divides the numerator exactly. (e.g., 10/2 = 5, which is a whole number). Otherwise, it results in a decimal or a mixed number.

Example Problem Solving

Let's solve a problem that combines these concepts:

Question: Simplify: (2.5 + 3/4) × (1/2 - 0.1)

Solution:

1. Convert all numbers to the same format. Let's use fractions.
   • 2.5 = 25/10 = 5/2
   • 3/4 remains 3/4
   • 1/2 remains 1/2
   • 0.1 = 1/10
2. Substitute these into the expression:
   • (5/2 + 3/4) × (1/2 - 1/10)
3. Solve the first bracket (addition):
   • Find LCM of 2 and 4, which is 4.
   • 5/2 = 10/4.
   • So, (10/4 + 3/4) = 13/4.
4. Solve the second bracket (subtraction):
   • Find LCM of 2 and 10, which is 10.
   • 1/2 = 5/10.
   • So, (5/10 - 1/10) = 4/10 = 2/5.
5. Multiply the results of the brackets:
   • (13/4) × (2/5) = (13 × 2) / (4 × 5) = 26/20.
6. Simplify the final fraction:
   • 26/20 = 13/10.
7. Convert to decimal if required:
   • 13/10 = 1.3

Therefore, the simplified value is 13/10 or 1.3.

Mastering whole numbers, decimals, and fractions provides a solid foundation for more advanced mathematical topics. Practice converting between them and performing operations diligently to build speed and accuracy for your exams.