Unit I: Vedic Math, Number System, Average - PEA305 Analytical Skills-I | B.Tech CSE Notes PDF | FineNotes4U

Unit I: Vedic Math, Number System, Average

⭐Vedic Math:

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Contents

1. Multiplication of Two 2-Digit Numbers
   1. Steps:
   2. Special Case:
2. Multiplication of Two 3-Digit Numbers
   1. Steps:
3. Squaring Numbers
   1. Different Methods Based on Ranges:
4. Square Root Calculation
   1. Steps:
5. Cube Root Calculation
   1. Steps:
6. Squaring Tricks
   1. Example:
   2. Another Example:
   3. Types of Numbers
7. Face Value and Place Value
8. Divisibility Rules
9. LCM and HCF
10. Factorization and Factors
11. Factorials and Trailing Zeros
12. Unit Digits
13. Formula for Average
14. Important Properties of Average
15. Age and Average
16. Average of Important Series of Numbers
17. Advanced Average Formulas
18. Average Speed
19. 🚨Thanks for visiting finenotes4u✨

Multiplication of Two 2-Digit Numbers

Steps:

1. Write the Numbers: Arrange the numbers vertically.

   Example:

     21
   x 23
   

2. Multiply Ones Place: Multiply the digits in the ones place and place the product under the line.

     1 × 3 = 3
   

3. Cross Multiply and Add:

   • Multiply the tens digit of the first number with the ones digit of the second number.
   • Multiply the ones digit of the first number with the tens digit of the second number.
   • Add the two products.

     (2 × 3) + (1 × 2) = 6 + 2 = 8
   

4. Multiply Tens Place: Multiply the tens digits and place the result to the left.

     2 × 2 = 4
   

Thus, the answer for 21 × 23 is 483.

Special Case:

If both numbers end in 1, the middle term can be simplified by adding the tens digits:

Example:

  41 × 81

Instead of cross multiplying:

  4 × 8 / (8 + 4) / 1 = 3321

Multiplication of Two 3-Digit Numbers

Consider numbers ABC and DEF:

  ABC
  x DEF

Steps:

1. Multiply the last digits: C × F.
2. Cross multiply B × F and C × E; add any carry from step 1.
3. Multiply A × F, B × E, C × D; add any carry from step 2.
4. Multiply A × E and B × D; add any carry from step 3.
5. Multiply A × D; add any carry from step 4.

Example:

  234 × 651
  2×6/6×3+2×5/3×5+6×4+2×1/4×5+3×1/4×1
  Answer: 152334

Squaring Numbers

Different Methods Based on Ranges:

1. Numbers Ending in 5:

   • Remove 5 and multiply the remaining number by its successor.
   • Append 25 at the end.
   • Example: 65² → 6 × 7 = 42 → Answer: 4225

2. Numbers from 25-50:

   • Calculate difference (d) from 50.
   • First part: (25 - d), Second part: d².
   • Example: 46² → (25 - 4) / 4² = 2116

3. Numbers from 50-75:

   • First part: (25 + d), Second part: d².
   • Example: 54² → (25 + 4) / 4² = 2916

4. Numbers from 75-100:

   • First part: (100 - 2d), Second part: d².
   • Example: 94² → (100 - 12) / 6² = 8836

5. Numbers from 100-125:

   • First part: (100 + 2d), Second part: d².
   • Example: 113² → (100 + 26) / 13² = 12769

Square Root Calculation

Steps:

1. Group digits in pairs from right.
2. Find the nearest square to the first group.
3. Determine the unit's digit based on the square endings.
4. Multiply the ten's digit by its next number and compare.

Example:

√784
Grouping: 7 84
Tens digit: 2 (√4)
Unit digit possibilities: 2 or 8
Final Answer: 28

Cube Root Calculation

Steps:

1. Group digits in sets of three from right.
2. Find the nearest cube root.
3. Use the last digit to determine the unit's place using complements.

Example:

Cube root of 54872:
Grouping: 54 872
Tens digit: 3 (since nearest cube root is 27)
Unit digit: 8 (from cube root complements)
Answer: 38

Squaring Tricks

Example:

Number: 306
Sub-base: 300

Step 1: 3 × (306 + 6) = 3 × 312 = 936
Step 2: 6² = 36
Answer: 93636

Another Example:

Number: 480 (sub-base 500)
480 - 20 = 460
5 × 460 = 2300
20² = 400
Answer: 230400

 

⭐Number System:

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1. Natural Numbers
   These are the numbers 1, 2, 3, 4, 5,... that represent counting numbers.
   Example: 1, 2, 3, 4, 5, 6, 7, ...

2. Whole Numbers
   Whole numbers include all natural numbers plus zero.
   Example: 0, 1, 2, 3, 4, 5, 6, 7,...

3. Integers
   Integers include all positive whole numbers, negative whole numbers, and zero.
   Example: -3, -2, -1, 0, 1, 2, 3, 4, ...

4. Rational Numbers
   These are numbers that can be expressed in the form of a fraction $\frac{p}{q}$, where p and q are integers, and $q \neq 0$.
   Example: $\frac{4}{1}, \frac{3}{5}, 5, -\frac{7}{2}$ etc.
   Note: Any number that terminates or repeats after the decimal point is a rational number.
   Example:

   • $\frac{3}{4} = 0.75$ (terminating)
   • $\frac{2}{3} = 0.666...$ (repeating)

5. Irrational Numbers
   These are numbers that cannot be expressed as a fraction and have non-terminating and non-repeating decimal expansions.
   Example:

   • $\sqrt{2}, \sqrt{3}, \pi, e$etc.

6. Real Numbers
   Real numbers are the set of all rational and irrational numbers.
   Example: $-3, 4, 0.5, \sqrt{5}, \pi$, etc.

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Types of Numbers

7. Even Numbers
   Numbers divisible by 2.
   Example: 2, 4, 6, 8, 10, 12, 14, ...

8. Odd Numbers
   Numbers that are not divisible by 2.
   Example: 1, 3, 5, 7, 9, 11, 13, ...

9. Prime Numbers
   Numbers greater than 1 with only two divisors: 1 and itself.
   Example: 2, 3, 5, 7, 11, 13, 17, ...

10. Twin Primes
    A pair of prime numbers that differ by 2.
    Example: (3, 5), (5, 7), (11, 13), (17, 19), ...

11. Co-prime Numbers
    Two numbers are co-prime if their greatest common divisor (GCD) is 1.
    Example:

    • $\text{GCD}(9, 28) = 1$, so (9, 28) are co-prime.
    • $\text{GCD}(12, 25) = 1$, so (12, 25) are co-prime.

12. Composite Numbers
    These are natural numbers with more than two divisors.
    Example: 4, 6, 8, 9, 12, 14, 15, ...

13. Perfect Numbers
    A number is perfect if the sum of its divisors (excluding itself) is equal to the number.
    Example:

    • 6: Divisors are 1, 2, 3. $1 + 2 + 3 = 6$.
    • 28: Divisors are 1, 2, 4, 7, 14. $1 + 2 + 4 + 7 + 14 = 28$.
    • 496, 8128 are also perfect numbers.

14. Complex Numbers
    These are numbers of the form $a+b\mathrm{i,\; where }$$i = \sqrt{-1}$
    Example: $3 + 4i, 5 - 6i$, where $i$ represents the imaginary unit.

Face Value and Place Value

15. Face Value
    The face value of a digit is simply the digit itself.
    Example:

    • In the number 6728, the face value of 6 is 6, the face value of 7 is 7, etc.

16. Place Value
    The place value of a digit is the value it holds based on its position in the number.
    Example:

    • In the number 6729:
      • The place value of 9 = $9 \times 1 = 9$
        
      • The place value of 2 = $2 \times 10 = 20$
        
      • The place value of 7 = $7 \times 100 = 700$
        

Divisibility Rules

• Divisibility by 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
  Example:

  • 130, 128, 234 are divisible by 2.

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  Example:

  • $6561$ → $6 + 5 + 6 + 1 = 18$ (divisible by 3)
  • $17281$ → $1 + 7 + 2 + 8 + 1 = 19$ (not divisible by 3)

• Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5.
  Example: 100, 125, 895.

LCM and HCF

• HCF (Highest Common Factor)
  The greatest number that divides two or more numbers exactly.
  Example:

  • HCF of 12 and 36 is 12.
  • HCF of 15 and 25 is 5.

• LCM (Lowest Common Multiple)
  The smallest number that is divisible by two or more numbers.
  Example:

  • LCM of 4 and 6 is 12.
  • LCM of 5 and 8 is 40.

Factorization and Factors

• Factors: Numbers that divide a given number exactly without leaving a remainder.
  Example:

  • Factors of 12 are 1, 2, 3, 4, 6, 12.

• Prime Factorization: Expressing a number as a product of its prime factors.
  Example:

  • Prime factorization of 60 is $2^2 \times 3 \times 5$.

Factorials and Trailing Zeros

• Factorial
  The product of all positive integers from 1 to n.
  Example:

  • $4! = 4 \times 3 \times 2 \times 1 = 24$
    
  • $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
    

• Trailing Zeros in Factorials
  The number of zeros at the end of a factorial depends on the number of times 10 is a factor in the number.
  Example:

  • Trailing zeros in $102!$: $\frac{102}{5} + \frac{102}{25} = 20 + 4 = 24$.
    So, $102!$ has 24 trailing zeros.

Unit Digits

• Unit Digit of Products
  Multiply the unit digits of the numbers.
  Example:

  • For $121\times 76\times 528\times \mathrm{172,}$$1 \times 6 \times 8 \times 2 = 96$, so the unit digit is 6.

• Unit Digit of Powers
  Use the cyclicity of powers to find the unit digit.
  Example:

• Unit digit of $2^{49}$:
  The unit digits of powers of 2 repeat in a cycle of 4:
  $2, 4, 8, 6$.
  Dividing 49 by 4 gives a remainder of 1, so $2^{49}$ has the unit digit 2.

⭐Average

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The average is a measure of central tendency, representing the sum of a set of quantities divided by the number of quantities. It provides a way to summarize a group of numbers with a single value.

Formula for Average

$$\text{Average} = \frac{\text{Sum of all quantities}}{\text{Number of quantities}}$$

Example:

If the ages of 4 students are 20 years, 22 years, 18 years, and 24 years, the average age can be calculated as:

$$\text{Average Age} = \frac{20 + 22 + 18 + 24}{4} = \frac{84}{4} = 21 \, \text{years}$$

Important Properties of Average

1. Increased by a constant (a):
   If all the numbers in a data set are increased by a constant $a$, the average is also increased by $a$.

   • Example: If all numbers are increased by 3, the new average is the old average + 3.

2. Decreased by a constant (a):
   If all the numbers are decreased by a constant $a$, the average is also decreased by $a$.

   • Example: If all numbers are decreased by 4, the new average is the old average - 4.

3. Multiplied by a constant (a):
   If all the numbers are multiplied by a constant $a$, the average is also multiplied by $a$.

   • Example: If all numbers are multiplied by 2, the new average is the old average × 2.

4. Divided by a constant (a):
   If all the numbers are divided by a constant $a$, the average is also divided by $a$.

   • Example: If all numbers are divided by 5, the new average is the old average ÷ 5.

Age and Average

1. Effect of change in the average age:
   • If the average age of $n$ persons decreases by $x$ years, the total age of $n$ persons decreases by $n \times x$ years.
   • If the average age of $n$ persons increases by $x$ years, the total age of $n$ persons increases by $n \times x$ years.

Example:

The average age of 6 persons is increased by 2 years when one of them, aged 26 years, is replaced by a new person. What is the age of the new person?

• Solution:
  • Increase in total age = $6 \times 2 = 12$ years.
  • Age of new person = 26 + 12 = 38 years.

The new person is 12 years older than the person who was replaced.

Average of Important Series of Numbers

Here are some useful formulas for calculating the average of different types of number series:

1. Average of first $n$ natural numbers:

   $$\text{Average} = \frac{n + 1}{2}$$
   • Example: For $n = 5$, the average is $\frac{5 + 1}{2} = 3$.

2. Average of first $n$ consecutive even numbers:

   $$\text{Average} = \frac{n + 1}{2}$$
   • Example: For the first 4 even numbers (2, 4, 6, 8), the average is $\frac{4 + 1}{2} = 5$.

3. Average of first $n$ consecutive odd numbers:

   $$\text{Average} = n$$
   
   • Example: For the first 4 odd numbers (1, 3, 5, 7), the average is $4$.

4. Average of consecutive numbers:

   $$\text{Average} = \frac{\text{First Number} + \text{Last Number}}{2}$$
   • Example: For the series 1, 2, 3, 4, 5, the average is $\frac{1 + 5}{2} = 3$.

5. Average of squares of natural numbers till $n$:

   $$\text{Average} = \frac{(n + 1)(2n + 1)}{6}$$
   • Example: For $n = 3$, the average of squares of first 3 natural numbers (1² + 2² + 3²) is $\frac{(3 + 1)(2 \times 3 + 1)}{6} = \frac{4 \times 7}{6} = \frac{28}{6} = 4.67$.

6. Average of cubes of natural numbers till $n$:

   $$\text{Average} = \frac{n(n + 1)^2}{4}$$
   • Example: For $n = 3$, the average of cubes of first 3 natural numbers (1³ + 2³ + 3³) is $\frac{3 \times (3 + 1)^2}{4} = \frac{3 \times 16}{4} = 12$.

7. Correct Sum:

   $$\text{Correct Sum} = \text{Wrong Sum} + (\text{Right Value} - \text{Wrong Value})$$

Advanced Average Formulas

1. Average of squares of first $n$ consecutive even numbers:

   $$\text{Average} = \frac{2(n + 1)(2n + 1)}{3}$$

2. Average of squares of consecutive even numbers from 1 to $n$:

   $$\text{Average} = \frac{(n + 1)(n + 2)}{3}$$

3. Average of squares of consecutive odd numbers from 1 to $n$:

   $$\text{Average} = \frac{n(n + 2)}{3}$$

4. Average of combined observations:
   If the average of $n_1$ observations is $a_1$ and the average of $n_2$ observations is $a_2$, then the average of all observations is:

   $$A = \frac{n_1 a_1 + n_2 a_2 + n_3 a_3 + \dots}{n_1 + n_2 + n_3 + \dots}$$

5. Average of remaining observations:
   If the average of $m$ observations is $a$ and the average of $n$ observations taken out of $m$ is $b$, then the average of the remaining observations is:

   $$\text{Average of remaining observations} = \frac{ma - nb}{m - n}$$

Average Speed

The average speed is the total distance traveled divided by the total time taken.

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

1. For two equal distances at different speeds:
   If the distance between two points $A$ and $B$ is $d$, and the speeds from $A$ to $B$ and from $B$ to $A$ are $x$ km/h and $y$ km/h, respectively, the average speed is given by:

   $$\text{Average Speed} = \frac{2xy}{x + y}$$

   Example:
   If a person travels two equal distances at 10 km/h and 30 km/h, the average speed is:

   $$\text{Average Speed} = \frac{2 \times 10 \times 30}{10 + 30} = \frac{600}{40} = 15 \, \text{km/h}$$
   

2. For three equal distances at different speeds:
   If a person covers three equal distances at speeds $A$, $B$, and $C$ km/h, the average speed is:

   $$\text{Average Speed} = \frac{3ABC}{AB + BC + CA}$$

3. For varying distances at different speeds:
   If a person covers $P$ part of the total distance at speed $z$, $Q$ part at speed $y$, and $R$ part at speed $z$, the average speed is:

   $$\text{Average Speed} = \frac{xyz}{Pyz + Qxz + Rxy}$$

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