Algebra is a broad field of mathematics that encompasses various concepts and formulas. Here are some fundamental algebraic formulas and identities:
1. Basic Arithmetic Formulas:
- Addition:
- Subtraction:
- Multiplication:
- Division:
2. Linear Equations:
- (For solving linear equations)
3. Quadratic Equations:
- The quadratic formula for :
4. Exponents:
- (a raised to the power of n)
5. Logarithms:
- (Logarithm of x to the base b)
6. Polynomials:
- (General form of a polynomial)
7. Factoring Formulas:
- (Difference of squares)
- (Difference of cubes)
- (Sum of cubes)
8. Binomial Theorem:
- (Expanding a binomial raised to a positive integer power)
9. Arithmetic Series:
- The sum of an arithmetic series:
10. Geometric Series:
- The sum of a geometric series:
11. Pythagorean Theorem:
- In a right-angled triangle, (where c is the hypotenuse)
12. Complex Numbers:
- (where is the imaginary unit)
- Complex conjugate of is
13. Inequalities:
- : a is greater than b
- : a is less than b
- : a is greater than or equal to b
- : a is less than or equal to b
14. Arithmetic Mean (Average):
- The arithmetic mean of numbers is given by:
15. Quadratic Equation (Vertex Form):
- The vertex form of a quadratic equation is: where is the vertex.
16. Distance Formula:
- The distance between two points and in a coordinate plane is given by:
17. Midpoint Formula:
- The midpoint between two points and is given by:
18. Systems of Linear Equations (Two Variables):
- For a system of equations and , the solution is given by:
19. Laws of Exponents:
- (for )
20. Matrix Multiplication:
- If is an matrix and is an matrix, then the product is an matrix. The element of is given by:
21. Discriminant of a Quadratic Equation:
- For a quadratic equation , the discriminant is given by:
- If , two distinct real solutions.
- If , one real solution (repeated).
- If , two complex conjugate solutions.
22. Permutations and Combinations:
- Permutations of distinct objects taken at a time:
- Combinations of distinct objects taken at a time:
23. Fundamental Theorem of Algebra:
- Every non-constant polynomial has at least one complex root.
24. Arithmetic Sequence:
- The -th term () of an arithmetic sequence with first term and common difference is given by:
25. Geometric Sequence:
- The -th term () of a geometric sequence with first term and common ratio is given by:
26. Binomial Coefficient Identity:
- The identity for binomial coefficients is given by:
27. Sum of the First Natural Numbers:
- The sum of the first natural numbers is given by:
28. Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality):
- For any non-negative real numbers , the inequality is:
29. Viète's Formulas:
- For a quadratic equation , the sum of roots and and the product of roots is given by:
30. De Moivre's Theorem:
- For any real number and integer ,
31. Law of Cosines:
- In a triangle with sides , , and , and angles , , and , the Law of Cosines is:
32. Law of Sines:
- In a triangle with sides , , and , and angles , , and , the Law of Sines is:
33. Euler's Formula:
- Euler's formula relates complex exponentials to trigonometric functions:
34. Wilson's Theorem:
- For a prime number ,
35. Principal Square Root:
- The principal square root of a non-negative real number is denoted by , and
36. Partial Fraction Decomposition:
- For a rational function, the process of expressing it as the sum of simpler fractions is known as partial fraction decomposition.
37. Cramer's Rule:
- Cramer's Rule is a method for solving a system of linear equations using determinants. For a system , if the determinant of the coefficient matrix is non-zero, the solution is given by: where is the matrix obtained by replacing the -th column of with vector .
38. Inverse Trigonometric Identities:
39. Pascal's Identity:
- Pascal's Identity states that
40. Distance between Point and Line:
- The distance between a point and a line is given by:
41. Sum of Cubes:
42. Bayes' Theorem:
- Bayes' Theorem relates conditional and marginal probabilities:
43. Heron's Formula:
- Heron's formula gives the area () of a triangle with sides , , and : where is the semi-perimeter of the triangle, .
44. Completing the Square:
- Completing the square is a method used to solve quadratic equations by expressing them in the form .
45. Euler's Totient Function:
- Euler's Totient Function gives the count of positive integers less than that are coprime to .
46. Laplace Transform:
- The Laplace transform of a function is given by , where is a complex number.
47. Binomial Theorem (General Term):
- The general term of the binomial expansion of is given by:
48. Mobius Inversion Formula:
- The Möbius inversion formula relates the summation of arithmetic functions: where is the Möbius function.
49. Sum of Arithmetic Series:
- The sum of an arithmetic series with terms, first term , and common difference is given by:
50. Sum of Geometric Series:
- The sum of a geometric series with terms, first term , and common ratio is given by:
No comments:
Post a Comment