Basic Arithmetic: A Comprehensive Guide

Arithmetic lays the foundation for all mathematical exploration. It delves into numbers and their interactions, equipping us with essential tools for calculations, problem-solving, and understanding quantitative relationships in the world around us.

1. Numbers:

a. Natural Numbers:

• Non-negative whole numbers (1, 2, 3, ...) used for counting and basic ordering.
• Excluded in some definitions due to the absence of zero.

b. Absolute Rational Numbers:

• Can be expressed as the ratio of two integers (a/b, where b ≠ 0), representing fractions or terminating decimals.
• Include both positive and negative rational numbers.

c. Relative Rational Numbers:

• Similar to absolute rational numbers, but their ordering (greater than, less than, etc.) depends on context or a reference point.
• Used in fields like physics and economics where relative comparisons are more meaningful.

d. Prime Numbers:

• Only divisible by 1 and itself (e.g., 2, 3, 5, 7, 11, 13, ...).
• Play a crucial role in number theory and cryptography.

e. Absolute Irrational Numbers:

• Cannot be expressed as a finite or repeating decimal.
• Examples include π (ratio of a circle's circumference to its diameter), √2 (diagonal of a unit square), e (natural logarithm base).

f. Fibonacci Numbers:

• Sequence where each number is the sum of the preceding two (0, 1, 1, 2, 3, 5, 8, ...).
• Appear in nature (e.g., plant spiral patterns, seashell growth) and have applications in computer science and finance.

2. Addition:

• Combining quantities.
• a + b = c means c contains the same number of elements as a and b combined.
• Commutative (order doesn't affect the result: a + b = b + a).
• Associative ((a + b) + c = a + (b + c)).

3. Subtraction:

• Finding the difference between quantities.
• a - b = c means c is what remains when b is removed from a.
• Not commutative (a - b ≠ b - a).
• Associative ((a - b) - c = a - (b + c)).

4. Multiplication:

• Repeated addition of equal groups.
• a × b = c means c is the sum of b groups of a elements each.
• Commutative (a × b = b × a).
• Associative ((a × b) × c = a × (b × c)).
• Distributive over addition (a × (b + c) = (a × b) + (a × c)).

5. Division:

• Splitting a quantity into equal groups.
• a ÷ b = c means c is the number of equal groups of b elements each that can be formed from a.
• Not commutative (a ÷ b ≠ b ÷ a).
• Not associative ((a ÷ b) ÷ c ≠ a ÷ (b ÷ c)).
• Inverse of multiplication.

6. Indices (Exponents):

• Repeated multiplication of the same base.
• aⁿ = a ⋅ a ⋅ ... ⋅ a (n times).
• a⁰ = 1 (except for a = 0).
• a¹ = a.
• aⁿ × aᵐ = a^(n + m) (Multiplication).
• aⁿ ÷ aᵐ = a^(n - m) (Division).
• (aⁿ)ᵐ = a^(n × m) (Power of a Power).

7. Indices with Integral Exponents (Addition, Subtraction, Multiplication, Division):

• Addition:

  • aⁿ + aᵐ can be simplified when n and m are equal or have the same absolute value.
  • If n ≠ m, use aⁿ + aᵐ = a^(min(n, m)) × (aⁿ⁻ᵐ + aᵐ⁻ⁿ). This formula involves breaking down the expression into common factors using the smaller exponent and then adding or subtracting terms based on the signs of n and m.
  • Example: 3² + 3³ = 3² × (1 + 3) = 9 × 4 = 36.

• Subtraction:

  • Similar to addition, aⁿ - aᵐ can be simplified when n and m are equal or have the same absolute value.
  • If n ≠ m, use aⁿ - aᵐ = a^(min(n, m)) × (aⁿ⁻ᵐ - aᵐ⁻ⁿ).
  • Example: 5³ - 5² = 5² × (5 - 1) = 25 × 4 = 100.

• Multiplication:

  • Straightforward for positive integral exponents: aⁿ × aᵐ = a^(n + m).
  • Example: 2⁴ × 2⁵ = 2⁴⁺⁵ = 2⁹.

• Division:

  • Divide exponents with the same base: aⁿ ÷ aᵐ = a^(n - m).
  • Example: 12⁸ ÷ 12⁴ = 12⁸⁻⁴ = 12⁴.

8. Highest Common Factor (HCF) and Lowest Common Multiple (LCM):

• HCF: The largest number that is a factor of two or more given numbers.

  • Methods for finding HCF: prime factorization, Euclidean algorithm (repeated division).
  • Example: HCF(12, 18) = 6.

• LCM: The smallest number that is a multiple of two or more given numbers.

  • Methods for finding LCM: multiplication of prime factors, least common multiple method (multiplying and dividing).
  • Example: LCM(12, 18) = 36.

9. Fractions:

• Represents a part of a whole, written as a/b where a is the numerator (number of parts) and b is the denominator (total number of parts).
• Operations on fractions: addition, subtraction, multiplication, division.
  • Addition/subtraction: requires fractions to have a common denominator.
  • Multiplication: multiply numerators and denominators.
  • Division: flip the divisor (reciprocal) and multiply.

10. Surds:

• Expressions involving irrational numbers, represented by the radical symbol (√).
• Types of surds:
  • Pure surds: Composed only of a constant and a radical (e.g., √2, 5√3).
  • Mixed surds: Combination of a rational number and a pure surd (e.g., 2 + √7, 3√5 - 1).

11. Similar Surds:

• Surds with the same radical (square root, cube root, etc.).
• Can be added, subtracted, multiplied, or divided directly.
• Example: 2√5 + 3√5 = 5√5.

12. Binomial Surds:

• Surds of the form a ± √b, where a and b are rational numbers and b is not a perfect square.
• Can be rationalized using algebraic identities or the method of squaring.
• Example: Rationalize 1 + √3:
  • Multiply the numerator and denominator by the conjugate (1 - √3):
    • (1 + √3) * (1 - √3) = 1 - (√3)² = 1 - 3 = -2.
    • So, 1 + √3 = (-2 + 3√3) / -2 = √3 - 1.

13. Quadratic Surds:

• Surds of the form a ± √(b + √c), where a, b, and c are rational numbers and b + √c is not a perfect square.

• Rationalizing quadratic surds can be more intricate than binomial surds. Here's a common approach:

  1. Find a Rational Number to Multiply: Look for a conjugate with a similar structure, ideally where the radical simplifies. For example, if the surd is 8 + √61, consider the conjugate 8 - √61. Squaring both, we get 128 - 61 = 67.
  2. Multiply and Simplify: Multiply the surd by its conjugate:

  (8 + √61) * (8 - √61) = 8^2 - (√61)^2 = 64 - 61 = 3
  

  3. Isolate the Surd: Divide both sides by the rational term to isolate the surd:

  √61 = (3 - 8) / 2 = -5 / 2 = -2.5
  

  Therefore, 8 + √61 = 8 - 2.5 = 5.5.

14. Similar Surds:

• Surds with the same radical (square root, cube root, etc.) are considered similar.
• You can directly add, subtract, multiply, or divide them:
  • Addition/subtraction: Combine coefficients and radicals separately (e.g., 2√5 + 3√5 = 5√5).
  • Multiplication: Multiply coefficients and radicals (e.g., √2 × √3 = √(2 × 3) = √6).
  • Division: Divide coefficients and simplify the radical if possible (e.g., √12 ÷ √3 = √(12 / 3) = √4 = 2).

15. Binomial Surd:

• A surd of the form a ± √b, where a and b are rational numbers and b is not a perfect square.

• Rationalizing a binomial surd involves multiplying by a conjugate to eliminate the radical from the denominator.

  1. Find the Conjugate: The conjugate is the same expression with the opposite sign for the radical (e.g., if the surd is 5 - √7, the conjugate is 5 + √7).
  2. Multiply by the Conjugate: Multiply the surd and its conjugate:

  (5 - √7) * (5 + √7) = 5^2 - (√7)^2 = 25 - 7 = 18
  

  3. Simplify: Divide both sides by the rational term to isolate the irrational part:

  √7 = (18 - 5) / 2 = 13 / 2
  

  Therefore, 5 - √7 = 5 - 13/2 = -3/2.

16. Table of Multiplication Factor (multiplication factor, prefix, symbol):

Factor	Prefix	Symbol	Example
10^12	Tera-	T	1 TB = 10^12 bytes
10^9	Giga-	G	1 GB = 10^9 bytes
10^6	Mega-	M	1 MB = 10^6 bytes
10^3	Kilo-	k	1 kB = 10^3 bytes
10^2	Hecto-	h	1 hm = 10^2 meters
10	Deca-	da	1 dam = 10 meters
10^-1	Deci-	d	1 dm = 10^-1 meter
10^-2	Centi-	c	1 cm = 10^-2 meter
10^-3	Milli-	m	1 mm = 10^-3 meter
10^-6	Micro-	µ	1 µm = 10^-6 meter
10^-9	Nano-	n	1 nm = 10^-9 meter
10^-12	Pico-	p	1 pm = 10^-12 meter

17. Properties of Sets:

a. Commutative Law:

• Applies to union and intersection.
• For union: A ∪ B = B ∪ A.
• Example: {1, 2} ∪ {3, 4} = {1, 2, 3, 4} and {3, 4} ∪ {1, 2} = {1, 2, 3, 4}.
• For intersection: A ∩ B = B ∩ A.
• Example: {1, 2} ∩ {3, 4} = {} and {3, 4} ∩ {1, 2} = {}.

b. Associative Law:

• Applies to union and intersection.
• For union: (A ∪ B) ∪ C = A ∪ (B ∪ C).
• Example: ({1, 2} ∪ {3, 4}) ∪ {5, 6} = {1, 2, 3, 4} ∪ {5, 6} = {1, 2, 3, 4, 5, 6} and A ∪ ({B ∪ C}) = {1, 2} ∪ ({3, 4} ∪ {5, 6}) = {1, 2, 3, 4, 5, 6}.
• For intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C).
• Example: ({1, 2} ∩ {3, 4}) ∩ {5, 6} = {} ∩ {5, 6} = {} and A ∩ ({B ∩ C}) = {1, 2} ∩ ({3, 4} ∩ {5, 6}) = {1, 2} ∩ {} = {}.

c. Properties of Intersection of Sets:

• Identity element: The empty set ∅ acts as the identity element for intersection, meaning A ∩ ∅ = ∅ for any set A.
• Idempotent law: A ∩ A = A for any set A.
• Involution law: A ∩ (Aᶜ) = ∅, where Aᶜ is the complement of A.
• Commutative law (as mentioned above).
• Associative law (as mentioned above).
• Distributive law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

d. Distributive Law:

• Applies to both union and intersection.
• For union: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
• For intersection: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (mentioned above).

I'd be happy to continue crafting the article on basic arithmetic, incorporating the missing sections and addressing feedback:

17. Properties of Sets (continued):

d. Distributive Law:

• Applies to both union and intersection.
• Union: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
  • Example: Let A = {1, 2}, B = {2, 3}, and C = {3, 4}.
    • A ∪ (B ∩ C) = {1, 2} ∪ ({2, 3} ∩ {3, 4}) = {1, 2} ∪ {3} = {1, 2, 3}.
    • (A ∪ B) ∩ (A ∪ C) = ({1, 2} ∪ {2, 3}) ∩ ({1, 2} ∪ {3, 4}) = {1, 2, 3} ∩ {1, 2, 3, 4} = {1, 2, 3}.
• Intersection: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (mentioned above).

18. De Morgan's Laws:

• Deal with complements of sets (the elements not in the set).
• a. Disjoint Sets:
  • A ∩ Bᶜ = Aᶜ ∪ B.
  • In words, the elements in set A and not in set B are the same as the elements not in set A or in set B.
  • Example: Let A = {1, 2} and B = {3, 4}.
    • A ∩ Bᶜ = {1, 2} ∩ {3, 4}ᶜ = {1, 2} ∩ {1, 2, 4} = {1, 2}.
    • Aᶜ ∪ B = {3, 4} ∪ {3, 4} = {1, 2, 3, 4}.
• b. Difference of Two Sets:
  • Aᶜ ∪ Bᶜ = (A ∪ B)ᶜ.
  • In words, the elements not in set A or in set B are the same as the elements not in both set A and set B.
  • Example: Let A = {1, 2} and B = {3, 4}.
    • Aᶜ ∪ Bᶜ = {3, 4}ᶜ ∪ {1, 2}ᶜ = {1, 2, 3, 4}ᶜ = {}.
    • (A ∪ B)ᶜ = ({1, 2} ∪ {3, 4})ᶜ = {1, 2, 3, 4}ᶜ = {}.

19. Formulas:

a. Simple Interest:

• I = P × R × T, where:
  • I is the simple interest earned.
  • P is the principal amount (initial investment).
  • R is the annual interest rate (as a decimal).
  • T is the time period (in years).

b. Compound Interest:

• A = P(1 + R/n)^(n × T), where:
  • A is the amount accumulated after T years.
  • P is the principal amount (initial investment).
  • R is the annual interest rate (as a decimal).
  • n is the number of compounding periods per year.
  • T is the total time period (in years).

Remember that these are just some core concepts in basic arithmetic. As you delve deeper, you'll encounter more advanced topics and intricacies. Keep exploring and practicing to solidify your understanding and expand your mathematical skills!