Mathematical Tools

Here's a set of useful mathematical tools which you may find helpful in various situations.
A few comments regarding the tools:

Every tool has an explanation below it with more information linked. You can always return to the top by clicking on the small "return to top" button under each tool.

To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common factor.

For two numbers A and B, a common factor of A and B is a number C such that both A/C and B/C are natural numbers. The Greatest Common Factor (or Divisor) is the biggest C.

For two numbers A and B, a common multiple of A and B is a number C such that both C/A and C/B are natural numbers. The Lowest (or Least, or Smallest) Common Multiple is the smallest C.

Radians are a different way to measure angles based on the circle's perimeter - 360 degrees are equal to 2π radians. To convert degrees to radians, all you have to do is multiply by π and divide by 180°.

Degrees are a unit of measurement in which one full rotation is 360°. To get back from radians to degrees, you just need to do the reverse operation - multiply by 180° and divide by π.

To find the magnitude of the vector, think of x and y coordinates as the lengths of legs of a right triangle, where the hypotenuse is the vector length itself. Apply the Pythagorean theorem to evaluate the magnitude.

To convert Cartesian coordinates to polar coordinates, we need to separately evaluate the radius r and the angle θ. The radius is the same as the vector's length, so we just apply the vector length formula to get it. To get θ, we look for the angle whose tangent equals y over x.

Side note: You may also use this as a point converter.

In order to convert polar coordinates to Cartesian coordinates, think of x, y and r as the sides of a right triangle, where r is the hypotenuse. Use two trigonometric functions, cosine and sine, to find the values of x and y respectively.

Side note: You may also use this as a point converter.

To find the distance between two points, we first have to find the distance between the X coordinates (dx) and Y coordinates (dy) of the points. After that, we apply the Pythagorean theorem (dx and dy are the legs of a right triangle, and the distance between A and B is the hypotenuse) and evaluate the square root of the sum of dx squared and dy squared.

In order to find the coordinates of a point between two points - the midpoint - we take the mean of the X coordinates of both points and Y coordinates of both points separately.

The shortest segment which is a distance between two parallel lines is part of a line perpendicular to the two. We use this fact to find the general formula for coordinates of two points, where each of the points belongs to one of the parallel lines and the same perpendicular line. After that, we apply the distance between points formula to find and apply the formula for distance between the two lines.

To get the formula for the distance between a point and a line, we need to use the properties of a triangle where the height is the distance between the line and the point and the other two vertices are the points of intersections of the line with X and Y axes.

Complex division is more difficult than addition, subtraction or multiplication (hence I made this tool). To divide two complex numbers in rectangular form, you need to multiply the number in the denominator by its complex conjugate. To keep the same value, you also need to multiply the numerator. After a bit of simplyfing, you get the result.

Dividing complex numbers in polar form is a bit easier than in rectangular form. The magnitudes can be divided separately from the rest of the number. The arguments of the cosine and sine functions can be subtracted from one another (click on the link below to see why).

To convert a complex number from rectangular form to polar form, we need to find the magnitude and angle of the complex number. To find the magnitude, apply the Pythagorean theorem (for complex number z = a + bi and magnitude r, r² = a² + b²). To find the angle, apply the tangent function (tan θ = b/a).

To convert a complex number from polar to rectangular form, use the properties of a right triangle created from the x & y coordinates and the magnitude. Namely, apply the definition of cosine to find x, and same with sine and y.

External links

Here's a few great math tools made by different people which I highly recommend.

Desmos Graphing Calculator

Desmos is a great graphing site which offers graphing functions, equations, inequalities, plotting tables of data, evaluating equations, exploring changes with parameters and more. You can sign up to save and share your graphs. I've been using it for many years and you can see my creations in the Graphs section.

WolframAlpha

WolframAlpha is a unique engine for computing answers and providing knowledge, not only about mathematics. Among others, it can solve any equation or plot any function.

If you know any other great math tools, you can tell me about them and I'll consider linking them here.