(D2.2.1): Addition of natural numbers. Let \(m\) be a natural number. To add 0 to \(m\), define \(0+m=m\). Assume inductively we have...
(D2.3.1) Multiplication of natural numbers. Let \(m\) be a natural number. We define \(0\times m=0\). Now if we know \(n\times m\), we...
(D2.1.1): A natural number is any element of the set: \begin{equation*} \mathbf{N}:=\{0,1,2,3,4, \ldots\} \end{equation*} Let \(n++\)...
Suppose we want to populate \(2n\) positions with exactly \(n\) 1’s and \(n\) -1’s. But we have a constraint. For any given \(k\) where...
Every polynomial with real coefficients can be written as a product of linear or quadratic terms, each having real coefficients. Proof:...
If a polynomial \(f(x)\) has real coefficients, and has a nonzero constant term, then the number of positive real roots is the number of...
To check if a function is symmetric about the origin, replace \(y\) with \(-y\) and \(x\) with \(-x\) and see if the equation of the...
TODO: Note the equations. Parabola A parabola is the set of all points that are equidistant from a point (the focus) and a fixed line...
If two lines are perpendicular, then \(m_{1}m_{2}=-1\) where \(m_{i}\) is the slope of the line. Proof Without Trigonometry Here’s the...
Here is an example of proving something using induction in Sage. Suppose we want to prove that: \begin{equation*}...
When You Know 2 Sides and 1 Angle \begin{equation*} A=\frac{1}{2} b c \sin \alpha \end{equation*} When You Know 1 Side and 2 Angles...
Law of Sines Consider this triangle. Draw a line vertically down from \(B\). Label the length of this line as \(h\) (for height). Now...
Formula for \(\cos(\alpha-\beta)\) Consider the unit circle below: Here, we make \(\overline{AB}=\overline{CD}\), so they have the...
Some things to note: The sum of all the coefficients is \(2^{n}\). To prove this, evaluate \(\left(a+b\right)^{n}\) with \(a=1,b=1\) The...
We know that: \begin{equation*} \sum_{k=1}^{n} k=\frac{n(n+1)}{2} \end{equation*} We want to find the expression for \begin{equation*}...
\(A\ge G\ge H\) if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) { var align = "center", indent = "0em", linebreak =...
The sum of a geometric sequence is: \begin{equation*} S_{n}=\frac{a\left(r^{n}-1)}{r-1} \end{equation*} Note, this is assuming \(n\)...
The sum of an arithmetic series is: \begin{equation*} S=\frac{1}{2}\left(a+l\right) \end{equation*} where \(l\) is the last term. if...
Say we want the roots of \(z^{n}=1\). Let \(z=re^{i\theta}\). Then \(z^{n}=r^{n}e^{in\theta}=1\). We immediately see that \(r=1\). Thus...
Given \(x^{2}+bx+c=0\), let the roots be \(\alpha,\beta\). Then: \begin{equation*} \alpha+\beta=-b \end{equation*} \begin{equation*}...
Remainder Theorem If a polynomial \(p(x)\) is divided by \(x-r\), then the remainder is \(p(r)\). To prove this, let \(p(x)=Q(x)(x-r) +...
Equations of the Form \(\sqrt{ax^{2}+bx+c}\pm\sqrt{ax^{2}+bx+d}=k\) \begin{equation*} \sqrt{ax^{2}+bx+c}+\sqrt{ax^{2}+bx+d}=k...
\begin{equation*} 2*x^{2}-xy+y^{2}=4 \end{equation*} \begin{equation*} 4*x^{2}-5xy+3y^{2}=6 \end{equation*} Manipulate the two equations...
Determinant of a 2x2 matrix: \begin{equation*} \left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=ad-bc \end{equation*} The minor...
\begin{equation*} \ ||z_{1}|-|z_{2}||\le|z_{1}+z_{2}|\le|z_{1}|+|z_{2}| \end{equation*} The second inequality follows from the triangle...
\(z^{n}+\overline{z}^{n}\) is a real number for all positive \(n\). Proof 1 Treat \(z=re^{i\theta}\) and you can show that the sum...
Given: \begin{equation*} \frac{a}{b}=\frac{c}{d} \end{equation*} Componendo: \begin{equation*} \frac{a+b}{b}=\frac{c+d}{d}...
Problem How many bits are needed to represent a binary tree with \(n\) nodes? A lower bound is given by the fact that \(m\) bits can...