This is not the only place where you will find algebra used: there is much algebra used in the higher level discussions under Number and Shape and Geometry. You will also find various number patterns under Number.
Solving equations
- 3 Linear Equations with x on both sides
- 3 Equations
- 4 Equation Solving
- 4 Hens and rabbits problem; simultaneous equations (or not)
- 4 Please help me with simultaneous equations
- 4 Biscuit and cereal prices problem; simultaneous equations
- 4 Simultaneous equations with 3/4 variables
- 4 Sum and product of digits problem
- 4 Algorithm for non-linear simultaneous equations
- 4 x=yz, y=xz, z=xy
- 4 Simultaneous equations in three variables
- 3 How many men to pump the ship dry?
- 4 Articles: Euclid's Algorithm
- 4 Quadratic Equations: where does the formula come from?
- 4 Solving x2=5x+7
- 4 Can you have an equation with these solutions?
- 4 False solution to equation involving square roots
- 4 Solve x+y=2, xy-z2=1
- 5 ax=by=(ab)xy
- 5 Completing the square
- 5 When does a quadratic have real roots?
- 5 The Anonymous Logarithm
- 5 When does a quadratic have real roots?
- 5 Solving quadratic with complex coefficients
- 5 (x+x1/2)/(1+x+x1/2)=(x2+x 1/2)/(1+x+x2)
- 5 Modular equation systems
- 5 Equation system
- 5 Solving difference equations
- 5 Spanish Problem and solving x3 + px = q
- 5 Solving cubic equations
- 5 A method for solving a cubic equation
- 5 Formula for solving cubic equations
- 5 Cubic Equations
- 6 Cubic equations and Galois theory
- 5 Cubic Equation Roots
- 5 Solving quartic equations
- 5 Quartic Equations
- 5 Are repeated roots necessarily rational?
- 5 Algorithm for finding zeros of any polynomial
- 5 Functional Equations
- 5 Solving n3n=324
- 5 x(x+1)(x+2)(x+3)=24
- 5 x1/3+x1/2+2=0
- 5 1/x - 1/xy - 1/xyz = 19/97
- 5 1/x + 1/y = 1/1995
- 5 Solving 3x - 2x = a
- 5 Solving x1/3+(2x-3)1/3=[12(x-1)]1/3
- 5 x2 +y2 +1=xyz
- 5 xn-1/(x2n-2xncosn q+1)
- 5 Reciprocal Diophantine Equation
- 6 y2+2=x3
- 6 2x2-3x=3y2
Algebraic manipulation
- 3 "Think of a number" trick - putting it into algebra
- 3 Finding a formula for Sticky Triangles
- 3 Arithmagons
- 4 BODMAS, BIDMAS, PEMDAS and PEDMSA
- 4 Difference of two squares and 39 × 41
- 4 Simplifying square root expressions
- 4 Making t the subject of an equation containing t2
- 4 Algebraic fractions and factorising quadratics
- 4 x4 + 1/x4 from x + 1/x
- 4 GCD and LCM of Polynomials
- 4 Algebraic fraction simplification
- 4 Practical uses of completing the square
- 4 Find a3+6ab+b3, given a+b=2
- 4 Unique expression of integers in a particular form
- 4 Factorising polynomials
- 5 Factorising cubic equations
- 5 Factorising polynomials
- 5 an+bn in terms of a+b and ab
- 5 Article: Telescoping Functions
- 5 Inverse of a mapping
- 5 Expanding (1 + x + 3x2)5
- 5 Quick factorisation of polynomials
- 5 Square Roots of ax+b
- 5 Binomial formula
- 5 Binomial expansion
- 5 Trinomial expansion
- 5 Binomial Expansion: (1 + px + qx2)-2
- 5 Meaning of the binominal coefficient
- 5 Coefficient of x34 in expansion of (1-3x+x2+2x4)10
- 5 Maclaurin series for tan(x), and polynomial division
- 5 Remainder Theorem and polynomial division
- 5 Polynomial division and remainders
- 5 Polynomial function problem
- 5 Polynomials, remainder theorem and Lagrange interpolation
- 5 Mobius Transformations
- 6 Mobius Transformation Problem
Graphs and coordinate geometry
- 3 Drawing graphs like y=2x-3
- 4 Graphs: y2 = x2 and y2n + x2n = r2n
- 5 Simpson's Rule
- 5 Gradient of angle bisector
- 5 Area of triangle given by coordinates
- 5 The equation of a doughnut (and volume/surface area)
- 5 Mod Graphs
- 5 Equations of circles
- 5 Equation of an ellipse
- 5 Largest perimeter of rectangle in ellipse
- 5 Focus and circumference of an ellipse
- 5 Reflecting parabolas
- 5 Parabola - focus and directrix
- 5 Coordinate geometry proof
- 5 Cartesian to polar form
- 5 Finding parametric equations
- 5 Area enclosed by x=2cost, y=sint
- 5 Equation of a Mobius Strip
- 5 Polar coordinates: area by integration
- 6 New Coordinate System
Pythagorean Triples
- 4 Pythagorean Triples (GCSE investigation)
- 4 Article: Picturing Pythagorean Triples
- 4 Articles: Pythagorean Triples
- 5 Pythagorean Triples (follows from articles)
- 4 Article: Extension of Pythagorean Triples
- 4 Pythagorean Quadruples (coursework extension)
- 6 Pythagorean triples - advanced stuff
Sequences and series
- 3 Sum of the first 10 square numbers
- 5 Sum of Squares
- 3 Sum of triangular numbers
- 3 Adding Grains of Rice
- 3 Finding a formula for Sticky Triangles
- 4 Shape/number pattern
- 4 1, 2, 3, 5, 8, 13,...
- 4 Formula for sequences like 1,4,16,64
- 4 Difference patterns and cubed numbers
- 4 Investigating expanding squares and geometric series
- 4 Fraction series sum (telescoping)
- 4 f(n)=f(n-1) + 3
- 5 Geometric Progression Puzzle
- 5 How many terms in a series?
- 5 Arithmetic sequence puzzle
- 5 2 geometric progression problems
- 5 Some geometric series questions
- 5 Geometric Progression Formula
- 5 Finding geometric means
- 5 Binomial coefficients in arithmetic progression
- 5 Coefficient of x34 in expansion of (1-3x+x2+2x4)10
- 5 Units digit of 3, 33, (33)3, ((33)3)3, etc
- 5 Summation of series questions
- 5 12 - 22 + 32 - 42 ... - (2n)2
- 5 Sequence: an = n + {sqrt(n)}
- 5 Reciprocal prime number infinite series
- 5 Sum of reciprocals of Fibonacci series
- 5 Finding the formula for a polynomial relationship
- 5 Sequences of averages
- 5 Bounded Series
- 5 New Series for Pi
- 5 Summing power series
- 5 The sum of the first n cubes
- 5 The sums of powers of integers.
- 5 Method of differences for summation of finite series
- 5 Sum of 1/n2
- 5 3/2 - 9/20 + 15/56 - 21/110 + .....
- 5 Factorials in difference triangles
- 5 Proof of Binomial theorem
- 5 Proof of a Series for Pi
- 5 Series Results proven by Induction
- 5 Factorial series
- 5 12 - 32 + 52 - 72 + ...
- 5 Sn r=0sin((2n+1)x/4)
- 6 log(1)+log(2)+log(3)+...
- 6 S1/(n2+1)
- 6 S1/n2k
- 6 Inequalities, Products and Sums of Numbers
- 6 Sn i=0 (-1)i(2i+1)3/((2i+1)4+4)
- 5 Farey Series
- 5 Series for trigonometric functions and ex
- 5 Summing nCr
- 5 Awkward proofs for series expansions
- 5 Binomial Theorem to find 1.16
- 6 S¥ n=1[1/(8n-7)]-[1/(8n-1)]
- 6 S¥ r=11/r
- 6 Utterly baffling infinite series (terms of the form nrxn)
- For series expansions, Maclaurin's series, Taylor's series, see here.
- 6 S¥ n=1nxzn ~ x!/(-ln z)x+1
- 6 S¥ n=1 {nx}/n
Sequences, series and limits
- See under Number for more on infinity and limits
- See separate index for more paradoxes
- 5 Pi's value as an infinite series
- 4 Fraction series tending to p/2
- 5 Geometric series and convergent/divergent sequences
- 5 Series Results proven by Induction
- 5 Limit of nrn
- 5 Limit of ((2x+1)/2)1/x
- 5 Limit of a summed series
- 5 X^x^x^x^x^x^...
- 5 ln2 = 0: series paradox?
- 5 Zeno's paradox
- 5 Paradoxical infinite sums
- 5 1+x+x2/(2!)+...+x2n/(2n)!=0
- 5 A Logarithmic Series
- 6 Divergent series
- 6 5=m1/m and proofs of convergence
- 6 Series for pi?
- 6 Quasiperiodic Sequences
- 6 Two Series
- 6 if limx®¥(Sx n=1ann)/x = 0 then S¥ n=1an is convergent
- 6 If S¥ n=0 akn = 0 then all am = 0?
- 6 Ramanujan Square Root
- 6 Nested radicals
Recurrence relations and iteration
- 3 Article: Try to Win
- 4 f(n)=f(n-1) + 3
- 4 Shape/number pattern
- 5 Swapping marbles
- 5 Recursive formula
- 5 Recurrence relationships
- 5 Solving difference equations
- 5 Difference equations
- 5 Recurrence relation inequality
- 5 Solving x - sinx = c
- 5 Speed of iteration
- 5 Recurrence relation problem
- 5 The logistic equation: Xn+1 = 2 Xn(1-Xn)
- 6 Recursive Sums
Inequalities
- 3 Linear Inequalities
- 4 Solving an inequality
- 4 Modulus Inequality
- 5 If a,b,c > 0 and abc=1 then a+b+c ³3
- 5 Real Number Inequality
- 5 The Triangle Inequality
- 5 A Quadratic Inequality
- 5 Solving cubic inequalities
- 5 Inequalities solving an Equation
- 5 Arithmetic mean - geometric mean inequality
- 5 Proofs of A.M. ³ G.M.
- 5 Cauchy-Schwarz inequality
- 6 Cauchy-Schwarz Inequality and Equality
- 5 Arithmetic-geometric mean inequality and Cauchy-Schwarz inequality
- 5 Jensen's Inequality
- 5 BMO inequality question
- 5 Inequality Problem
- 5 Various inequalities
- 5 Inequality proof
- 5 Prove that x + y + z £ (xyz)1/3 (x/y + y/z + z/x)
- 5 If x+y+z £ 3 then 1/x + 1/y +1/z ³ 3
- 5 n! > n3: proof by induction
- 5 Recurrence relation inequality
- 5 [1+1/3+1/5+..+1/(2n-1)]/(n+1) ³[1/2+1/4+..+1/(2n)]/n
- 5 Minimising S1381 i=1 a2i = a21 + a22 + ... + a21381
Groups and algebraic structure
- 4 Article: Small groups
- 5 Associativity
- 5 Order of element and inverse element
- 5 Lagrange's Theorem
- 5 Mathematical Systems
- 6 Group generators
- 6 Groups and isomorphism
- 6 Finding subgroups of a group
- 6 Symmetric group S(6)
- 6 Field Algebra and Simple Groups
- 6 Ring Isomorphisms
- 6 Galois theory
Matrices
- For vectors see under Shape and geometry.
- 5 Matrices - Canonical Form
- 5 Transformations and matrices
- 5 Matrices and Transformations
- 5 Rotation Matrix
- 5 Shear Matrix
- 6 What are determinants?
- 6 Definition of Determinant
- 6 Determinants, eigenvalues and eigenvectors
- 6 Matrices, Numbers
- 5 Inverse mapping
- 5 Inverse of a mapping
- 5 Singular matrices
- 6 Eigenvalues
- 6 Real eigenvalues
- 6 Symmetric Matrices
- 6 Symmetric Tensors
- 6 Matrix Polynomials
- 6 The Cayley-Hamilton theorem
Miscellaneous
- 3 Investigation: adding consecutive numbers
- 3 "Think of a number" trick - putting it into algebra
- 4 A Pile of Paper that reaches the Moon
- 4 Article: Proofs with Pictures
- 4 Missing numbers on two spun discs
- 4 3x10-y10=1991: no integral solutions?
- 4 Venn diagram problem
- 5 Golden Ratio
- 5 Partitioning {1,..,n}
- 5 Zeros of a quartic
- 5 Proof Of the Newton Raphson method for finding complex roots
- 5 Infinite products problem
- 5 Inverse mapping
- 5 Functions satisfying ff(x)=-x and gg(x)=1/x
- 5 Function directly proportional to two variables
- 5 Polynomial function problem
- 5 Floor Functions
- 5 Floor function revisited
- 5 Polynomial interpolation and polynomial extrapolation
- 5 Window cleaning problem
- 5 Three assorted problems
- 5 Dimensional equivalence
- 5 Integers satisfying 1 + x2y = x2 + 2xy + 2x + y
- 6 Algebraic Axioms
- 6 Levi Civita Symbols
- 6 Tensor Discussion
- 6 Symmetric Tensors
- 6 Fundamental Theorem of Algebra
- 6 Functional equations
- 6 Functional equation
- 6 An interesting class of polynomials
- 6 Maximising f(x1,x2,x3,...,xn) when n varies
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