Brilliant Algebra Glossary is now live

A telescoping series a series whose partial sums eventually only have a fixed number of terms after cancellation.

Example: $\displaystyle\sum^{n}_{i=1}{\dfrac{1}{i(i+1)}}=1-\dfrac{1}{n+1}$

(of a polynomial) See 'root' [in the glossary].

Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM)

I think they should be together because we usually apply one of them to obtain the other.

Let $a_1, a_2, \ldots , a_n$ be non-negative real numbers, then the

1. Arithmetic mean of these numbers is $\large \frac {a_1 + a_2 + \ldots + a_n} {n}$

2. Geometric mean of these numbers is $\large \sqrt[n] {a_1 \cdot a_2 \cdot \ldots \cdot a_n}$

3. Harmonic mean of these numbers is $\large \frac {n}{ \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n} }$

And

$\large \frac {a_1 + a_2 + \ldots + a_n} {n} \geq \sqrt[n] {a_1 \cdot a_2 \cdot \ldots \cdot a_n} \geq \frac {n}{ \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n} }$

with equality if and only if $a_1 = a_2 = \ldots = a_n$

$\Sigma$, (pronounced ˈsɪɡmə ) is a mathematical operation of adding a sequence of number of like terms, the result is their sum or total. More generally, if $m$ and $n$ are integers and $m \leq n$, then the summation from $k$ equals to $m$ to $n$ of $a_k$ is the sum of all the terms $a_m, a_{m+1}, a_{m+2}, \ldots , a_n$. We write

$\displaystyle \sum_{k=m}^n a_k = a_m + a_{m+1} + a_{m+2} + \ldots + a_n$

and call $k$ the index of the summation , $m$ the lower limit of the summation of the summation, and $n$ the upper limit of the summation.

We also can state $\displaystyle \sum_{k=m}^n$ as $\sum_{k=m}^n$

For example, $\displaystyle \sum_{k=3}^7 (k^2 + k + 10) = (3^2 + 3 + 10) + (4^2 + 4 + 10) + (5^2 + 5 + 10) + (6^2 + 6 + 10) + (7^2 + 7 + 10)$

States that the expression $xy+ax+by$ can be factorized as $(x+b)(y+a)-ab$.

Example problem: $\text{Find all integer solutions to }\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{6}$.

Solution: Adding the LHS gives $\dfrac{x+y}{xy}=\dfrac{1}{6}$.

Cross multiplying gives $xy=6x+6y$; reorder to make $xy-6x-6y=0$.

Use SFFT to get $(x-6)(y-6)=36$. Since $x,y$ are integers, that means $x-6$ and $y-6$ are also integers. Therefore, they are factors of $36$ which multiply to $36$.

The rest of the solution is left as an exercise to the reader. ;)

Exponent is a number or an unknown or a variable or an expression that denotes the measure or dimension of repeatation of multiplication of again a number or an unknown or a variable or an expression to itself.

Power of a number or an unknown or a variable or an expression is the measure or dimension of repeatation of multiplication of that number or unknown or variable or expression to itself.

$\pi$ is a mathematical constant that is defined to be the ratio of a circle's circumference to its diameter. It is approximately equals to $3.14159265358979$

W.LO.G. stands for 'without loss of generality'.

It is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others, or that all other cases are equivalent.

For example, consider the following theorem:

If three objects are each painted either red or blue, then there must be two objects of the same color.

A proof: Assume without loss of generality that the first object is red. If either of the other two objects is red, we are finished; if not, the other two objects must both be blue and we are still finished.

This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue.

[Definition and example taken from Wikipedia]

The $z$-axis is the third axis in a three-dimensional coordinate system. Typically the $x$-axis and $y$-axis are thought of as being in a horizontal plane, with the $z$-axis pointing up.

Diophantine refers to Diophantus of Alexandria. A Diophantine equation is a polynomial equation that allows two or more variables to take integer values only.

$\prod$ [upper case pi] is used to denote the product of some numbers or terms.

For example: $\displaystyle\prod_{i=2}^5 i$ simply denotes the product of $i$'s where $i$ starts out at $2$, is incremented by $1$ for each successive terms and stops at $5$

So, $\displaystyle\prod_{i=2}^5 i= 2\times 3\times 4\times 5=120$.

[See Summation sign, $\sum$ to notice the analogy between $\sum$ and $\prod$.]

Terms can be numbers, variables (alone or multiplied with numbers or constants, which is the co-efficient of the variables concerned), unknowns (alone or multiplied with numbers or constants, which is the co-efficient) that constitutes an expression alone or in addition or substraction (by $+$ or $-$ sign) to other terms.

A term is a part of a sum. For example, in the polynomial ax^2+  bx+  c, the first term is $ax^2$, the second term is $bx$, and the third term is $c$. The different terms in an expression are separated by addition (or subtraction) signs.

States that for any numbers $a,b$ and positive integer $n$, that: $(a+b)^n=\binom{n}{0}a^nb^0+\binom{n}{1}a^{n-1}b^1+\cdots + \binom{n}{n-1}a^1b^{n-1}+\binom{n}{n}a^0b^n$.

Denotes which side of the equality sign the author is talking about. $\underbrace{\text{Random Math}}_{\text{LHS}}=\underbrace{\text{More Random Math}}_{\text{RHS}}$

This is number theory, isn't it?

A prime number is a positive integer greater than $1$ that can only be divided by $1$ and itself.

In mathematics, concatenation of two or more numbers is the joining of their numerals. For example, the concatenation of $314$ and $159$ is $314159$

I honestly think this term doesn't deserve to be in the glossary. What good could come out of using that word anywhere?

A weighted average (also weighted arithmetic mean) of a group of numbers $x_1, x_2, x_3, . . . , x_n$ is: w_1x_1+  w_2x_2 + w_3x_3+ w_n+ x_n where the $w$’s are a group of positive numbers such that: w_1+  w_2 + w_3+... w_n=  1.

Each number $x_i$ has a corresponding weight $w_i$. A larger value of $w_i$ means that $x_i$ should be given greater significance in calculating the weighted average.

$\sqrt{\ \ \ }$ is the radical symbol. It is used to indicate the taking of a root of a number. $\sqrt[y]{x}$ means the $y$th root of $x$ and $(\sqrt[y]{x})^y=x$.

In the example above, $y$ is the index of the radical. If no index is specified, then the square root is meant. A radical always means to take the positive value [this is called the principal root]. For example: if you square both $5$ and $-5$ you're going to get $25$. But $\sqrt{25}=5$.

Quadratic normally refers to something with degree $2$.

For example: $ax^2+bx+c=0, a\neq 0$ is a quadratic equation because the power of$x$ is $2$.

The slope of a line is a number that measures how steep the line is. The slope of a line is defined to be \frac{\Delta y}{\Delta x}, where \Delta y is the change in the vertical coordinate and $\Delta x$ is the change in the horizontal coordinate between any two points on the line.

A horizontal line has a slope of zero. As a line approaches being a vertical line, its slope approaches infinity.

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+.......+a_1x+a_0$ be a polynomial and $x_1,x_2......,x_n$ be its roots.

Vieta's Formulas give us a relation between the coefficients of the polynomial $P(x)$ and its roots:

$(x_1+x_2......+x_n)= \frac{-a_{n-1}}{a_n}$

$(x_1x_2+x_2x_3.....+x_1x_n)=\frac{a_{n-2}}{a_n}$

$\displaystyle\vdots$

$(x_1x_2....x_n)=-1^n\frac{a_0}{a_n}$

In mathematics, the term 'quartic' describes something that pertains to the "fourth degree".

$x^4+3x^3+12x-\sqrt{2}$ is a quartic polynomial.

The word 'quintic' means 'of the fifth degree'.

For example: a quintic equation is a polynomial equation of degree $5$.

$5x^5-68x^3+89=0$ is a quintic equation.

[I suggest that you change this to just 'cubic'. That way we won't need different definitions for words like 'cubic polynomial', 'cubic equation', 'cubic field'. Dame goes for 'quintic' and 'quartic'. I'm adding my definition for 'cubic'].

Cubic refers to the third power/degree of a term. For example $P(x)=x^3+7x+13$ is a cubic polynomial for $x$ because it is a degree-$3$ polynomial [the highest power of $x$ is $3$, see 'degree'].

Radian is a unit for measuring angles.

The radian measure of an angle is found by measuring the length of the intercepted arc and dividing it by the radius of the circle. For example, the circumference of a circle is $2\pi r$, so a full circle ($360$ degrees) equals $2\pi$ radians. Also, $180$ degrees equals $\pi$ radians, and a right angle ($90$ degrees) has a measure of $\frac{\pi}{2}$ radians.

The numerator is the number above the bar in a fraction.

For example: in the fraction $\frac{7}{13}$, $7$ is the numerator

The denominator is the bottom part of a fraction. In the fraction $\frac{7}{13}$, $13$ is the denominator.

This would be under combinatorics

Yes, that counts as Number Theory.

Already taken care of.

Already taken care of.