Take a guided, problem-solving based approach to learning Algebra. These compilations provide unique perspectives and applications you won't find anywhere else.
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
From cracking cryptograms to calculating the top speed of a rocket, algebra gives you tools to apply mathematical reasoning to a wide range of problems. Dive in and see what you already know!
From supply and demand to position and velocity, equations model the world around us. Equip yourself with the tools necessary to solve for x.
Understanding the properties of arithmetic will allow you to simplify complex expressions and do difficult calculations in a flash. Just don't divide by 0...
Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.
Combine your skills of working with exponents and manipulating expressions for radical glory.
Arm yourself with the tools to be the king or queen of heady mathematical debates, like the age old question of whether 0.999.... = 1.
Your music collection grows by 20 songs every week. If you currently have 1,000 songs, how many weeks will it take to amass 3,500 songs?
If one Bitcoin and 100,000 Dogecoin are worth $480, and two Bitcoin and 150,000 Dogecoin are worth $948, would you rather have one Bitcoin or 100,000 Dogecoin?
Quadratic equations describe the motion of a baseball after it connects with a bat, and the acceleration of gravity at the Earth's orbit.
Learn how to eliminate the linear term and see through messy quadratics. You’ll know just how high to shoot your three-pointer to avoid tall defenders.
This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.
From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.
A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.
You might have heard of remainders in simple division, but what about remainders when dividing polynomials?
When is 2x greater than x? Compare algebraic expressions with linear inequalities.
When does revenue exceed cost? When will one race car pass another? Model these values with polynomials and use polynomial inequalities to solve questions like these and more.
Is a company's profit greater in magnitude than its loss? When is the error in the length of a candy bar acceptable to the factory?
When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.
Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.
Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.
Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.
A rational function can have a variable like "x" in the numerator AND the denominator. When this happens, there are some special rules and properties to consider.
Combine your skills of working with exponents and manipulating expressions for radical glory.
As a particle moves, its position can often be written in terms of time. Systems like these can be modeled with parametric equations, which cleverly write coordinates in terms of a parameter.
Adding 10 to itself 10 times gives 100, but how can we represent 10 multiplied to itself 10 times?
From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).
When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.
Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.
Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.
"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.
Polar coordinates are a way to describe where a point is on a plane. Instead of using x and y, you use the angle theta and radius r, to describe the angle and distance of the point from the origin.
De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.
What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.
A geometric progression is a sequence of numbers where the previous term is multiplied by a constant to get the next term. 1, 2, 4, 8,... is a geometric sequence where each term is multiplied by 2.
Induction is a fundamental proof technique that is useful for proving that statements are true for all positive integers n – for example, that 1 + 2 + 3 + ... + n = n(n+1)/2 for positive integers n.
Learn about variations on induction. For example, if you can't topple the first domino, can you prove that toppling the 4th domino topples all of the ones after it?
Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.
If a coin comes up heads you win $10, but if it comes up tails you win $0. How likely are you to win exactly $30 in five flips?
Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.
The rational root theorem describes a relationship between the roots of a polynomial and its coefficients. For a good time, use the theorem to prove that the square root of 2 is not rational.
Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.
This is completing the square on 'roids.
Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.
Suppose you have a system of 50 linear equations. It's tedious and impractical to solve them by hand. Polynomial interpolation is your better alternative.
A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.
In a symmetric polynomial, you can interchange any of the variables and get the same polynomial. Symmetric polynomials form the basis of Galois theory, which connects group theory and field theory.
Chebyshev polynomials are a sequence of orthogonal polynomials that provide recurrence relations useful for solving polynomials and approximating functions without extensive calculation.
Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.
Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.
Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."
Apply linear algebra to solve systems of linear equations, find paths in graph theory, and map rotations of points in space using matrix operations. It's a true intersection of engineering and math.
From pure math to modern cryptography to organic chemistry, abstract algebra underpins much of contemporary research. Learn the theory of groups, rings, and fields.
Explore geometric properties and spatial relations that are unaffected by continuous deformations, like stretching and bending. The next time you scan a bar code on a can of soda, thank a topologist.
Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical inequalities.
This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.
Given 5 sticks of length 1, 3, 5, 9, and 10, how many distinct triangles can be formed? Learn the techniques and develop an intuition for working with geometric inequalities.
The smell of linoleum. Apple juice and cookies at halftime. And extra hard math problems. Do it for the love, the thrill. Learn more about getting in on the action of math competitions.
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