Followers

Sunday, December 24, 2023

Some fundamental algebraic formulas and identities Part 1

 


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:

  • +=0 (For solving linear equations)

3. Quadratic Equations:

  • The quadratic formula for 2++=0: =±242

4. Exponents:

  • (a raised to the power of n)

5. Logarithms:

  • log() (Logarithm of x to the base b)

6. Polynomials:

  • +11++1+0 (General form of a polynomial)

7. Factoring Formulas:

  • 22=(+)() (Difference of squares)
  • 33=()(2++2) (Difference of cubes)
  • 3+3=(+)(2+2) (Sum of cubes)

8. Binomial Theorem:

  • (+)==0() (Expanding a binomial raised to a positive integer power)

9. Arithmetic Series:

  • The sum of an arithmetic series: =2(1+)

10. Geometric Series:

  • The sum of a geometric series: =1(1)1

11. Pythagorean Theorem:

  • In a right-angled triangle, 2+2=2 (where c is the hypotenuse)

12. Complex Numbers:

  • 2=1 (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 1,2,, is given by: Mean=1+2++

15. Quadratic Equation (Vertex Form):

  • The vertex form of a quadratic equation 2++ is: =()2+ where (,) is the vertex.

16. Distance Formula:

  • The distance between two points (1,1) and (2,2) in a coordinate plane is given by: =(21)2+(21)2

17. Midpoint Formula:

  • The midpoint between two points (1,1) and (2,2) is given by: (1+22,1+22)

18. Systems of Linear Equations (Two Variables):

  • For a system of equations += and +=, the solution is given by: =,=

19. Laws of Exponents:

  • =+
  • =
  • ()=
  • 0=1 (for 0)

20. Matrix Multiplication:

  • If is an × matrix and is an × matrix, then the product = is an × matrix. The element of is given by: ==1

21. Discriminant of a Quadratic Equation:

  • For a quadratic equation 2++=0, the discriminant is given by: Δ=24
    • If Δ>0, two distinct real solutions.
    • If Δ=0, one real solution (repeated).
    • If Δ<0, 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 1 and common difference is given by: =1+(1)

25. Geometric Sequence:

  • The -th term () of a geometric sequence with first term 1 and common ratio is given by: =1(1)

26. Binomial Coefficient Identity:

  • The identity for binomial coefficients is given by: ()+(+1)=(+1+1)

27. Sum of the First Natural Numbers:

  • The sum of the first natural numbers is given by: =(+1)2

28. Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality):

  • For any non-negative real numbers 1,2,,, the inequality is: 1+2++12

29. Viète's Formulas:

  • For a quadratic equation 2++=0, the sum of roots 1 and 2 and the product of roots is given by: 1+2=,12=

30. De Moivre's Theorem:

  • For any real number and integer , (cos+sin)=cos()+sin()

31. Law of Cosines:

  • In a triangle with sides , , and , and angles , , and , the Law of Cosines is: 2=2+22cos

32. Law of Sines:

  • In a triangle with sides , , and , and angles , , and , the Law of Sines is: sin=sin=sin

33. Euler's Formula:

  • Euler's formula relates complex exponentials to trigonometric functions: =cos+sin

34. Wilson's Theorem:

  • For a prime number , (1)!1(mod)

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:

  • sin1()+cos1()=2
  • tan1()+cot1()=2
  • sec1()+csc1()=2

39. Pascal's Identity:

  • Pascal's Identity states that (1)+()=(+1)

40. Distance between Point and Line:

  • The distance between a point (0,0) and a line ++=0 is given by: =0+0+2+2

41. Sum of Cubes:

  • 3+3=(+)(2+2)

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, =++2.

44. Completing the Square:

  • Completing the square is a method used to solve quadratic equations by expressing them in the form ()2=.

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 1, and common difference is given by: =2(21+(1))

50. Sum of Geometric Series:

  • The sum of a geometric series with terms, first term 1, and common ratio is given by: =1(1)1

No comments:

Post a Comment