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# Brilliant Algebra Glossary is now live

Thanks to the members who contributed to our glossary discussion, we now have an Algebra glossary available to everyone on Brilliant:

BRILLIANT ALGEBRA GLOSSARY

Please take a look and let us know what you think. This is still very much a community project, so please feel free to reply to this discussion with feedback, as well as terms that you think still need to be added or definitions that should be revised or improved.

Thanks again to all of our original contributors for their help!

#### Rules to keep things organized and civil:

1. Top level replies (replies in the box directly below this message) should only contain a single term that you think belongs in the glossary. One term per post. Make sure your term isn't already listed (you might want to use your browser's search function), so we can avoid duplicates.
2. Reply to the term you want to define with a defintion you'd like to propose. One definition per reply.
3. Vote up terms and definitions you like. If you see a definition you disagree with, vote it down and write a better one.
4. IMPORTANT: Only one term or definition per post, please.

Note by Arron Kau
4 years, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
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1. numbered2. list
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# up as a code block.

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# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

zero

- 4 years, 3 months ago

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

- 4 years, 3 months ago

Telescoping series

- 4 years, 3 months ago

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

- 4 years, 3 months ago

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

- 4 years, 3 months ago

Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?)

Staff - 4 years, 3 months ago

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 $$a_1 = a_2 = \ldots = a_n$$

- 4 years, 3 months ago

RMS > AM > GM > HM

where RMS is the Root of Mean of Square of the given numbers

- 4 years, 3 months ago

Terms

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

Multiplication sign $$\prod$$

- 4 years, 3 months ago

$$\prod$$ [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 $$\prod$$.]

- 4 years, 3 months ago

Diophantine?

- 4 years, 3 months ago

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

- 4 years, 3 months ago

[I see that some alphabets aren't getting much love! So I'm going to add a few more.]

$$z$$-axis

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

W.L.O.G

(Posting for rookies like me who din't get abbreviations)

- 4 years, 3 months ago

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]

- 4 years, 3 months ago

Pi, $$\pi$$

- 4 years, 3 months ago

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

- 4 years, 3 months ago

Power

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

Exponent

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

SFFT (Simon's Favorite Factoring Trick/Simon's Favorite Factoring Theorem)

Just a note: I'm posting these theorems because they are commonly abbreviated, and rookie problem solvers might get confused as to what they mean.

Of course, Binomial Theorem is too important for me to exclude.

- 4 years, 3 months ago

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. ;)

- 4 years, 3 months ago

Wow ! Awesome

- 4 years, 3 months ago

Summation sign $$\Sigma$$

Staff - 4 years, 3 months ago

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

- 4 years, 3 months ago

I'm used to calling the limits as lower and upper bounds, so I guess it should be included for the odd ones like me.

- 4 years, 3 months ago

there is a typo in "argument" section. in pi/2 = 45. it must be pi/4.

- 4 years, 1 month ago

Denominator

- 4 years, 3 months ago

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

- 4 years, 3 months ago

Numerator

- 4 years, 3 months ago

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

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

- 4 years, 3 months ago

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

Unknowns

- 4 years, 3 months ago

Arbitrary constant

- 4 years, 3 months ago

Algebra

- 4 years, 3 months ago

Exponential Form (of a complex number)

- 4 years, 3 months ago

Polar Form (of a complex number)

- 4 years, 3 months ago

Linear Function

- 4 years, 3 months ago

Cubic Function

- 4 years, 3 months ago

[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'].

- 4 years, 3 months ago

Quintic Function

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

Quartic Function

- 4 years, 3 months ago

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

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

- 4 years, 3 months ago

Matrix

- 4 years, 3 months ago

Determinant

- 4 years, 3 months ago

Vieta's Formulas

- 4 years, 3 months ago

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:

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

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

$\displaystyle\vdots$

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

- 4 years, 3 months ago

The problem with Vieta's formulas at this level is that it is not clear what the information represented by the elipses is . This doesn't seem particularly helpful when solving problems, just by looking at it.

- 4 years, 3 months ago

Rationalizing factor

- 4 years, 3 months ago

Slope of a line

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

- 4 years, 3 months ago

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 $$2$$.

- 4 years, 3 months ago

- 4 years, 3 months ago

$$\sqrt{\ \ \ }$$ 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 $$\sqrt{25}=5$$.

- 4 years, 3 months ago

weighted average

- 4 years, 3 months ago

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.

- 4 years, 3 months ago

Concatenation

- 4 years, 3 months ago

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

- 4 years, 3 months ago

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

- 4 years, 3 months ago

Prime Number

- 4 years, 3 months ago

This is number theory, isn't it?

- 4 years, 3 months ago

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

- 4 years, 3 months ago

LHS and RHS (Left Hand Side and Right Hand Side)

- 4 years, 3 months ago

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

- 4 years, 3 months ago

RRT (Rational Root Theorem)

- 4 years, 3 months ago

Binomial Theorem

- 4 years, 3 months ago

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

- 4 years, 3 months ago

CRT (Chinese Remainder Theorem)

EDIT: ignore, this should be in Number Theory, correct?

- 4 years, 3 months ago

Yes, that counts as Number Theory.

- 4 years, 3 months ago

PIE (Principle of Inclusion and Exclusion)

- 4 years, 3 months ago

This would be under combinatorics

Staff - 4 years, 3 months ago

Irrational numbers

- 4 years, 3 months ago

- 4 years, 3 months ago

Imaginary Number $$i$$

- 4 years, 3 months ago