Main post link -> https://brilliant.org/assessment/techniques-trainer/algebra-glossary/

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

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!

- 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. - Reply to the term you want to define with a defintion you'd like to propose.
**One definition per reply.** - Vote up terms and definitions you like. If you see a definition you disagree with, vote it down and write a better one.
- IMPORTANT: Only one term or definition per post, please.

No vote yet

31 votes

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestTelescoping series

Log in to reply

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

Log in to reply

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

Log in to reply

zero

Log in to reply

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

Log in to reply

Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?)

Log in to reply

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

Arithmetic mean of these numbers is \( \large \frac {a_1 + a_2 + \ldots + a_n} {n} \)

Geometric mean of these numbers is \( \large \sqrt[n] {a_1 \cdot a_2 \cdot \ldots \cdot a_n} \)

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 \( a_1 = a_2 = \ldots = a_n \)

Log in to reply

RMS > AM > GM > HM

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

Log in to reply

Summation sign \( \Sigma\)

Log in to reply

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

Log in to reply

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.

Log in to reply

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.

Log in to reply

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. ;)

Log in to reply

Wow ! Awesome

Log in to reply

Exponent

Log in to reply

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.

Log in to reply

Power

Log in to reply

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.

Log in to reply

Pi, \(\pi\)

Log in to reply

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

Log in to reply

W.L.O.G

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

Log in to reply

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:

A proof: Assume

without loss of generalitythat 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]

Log in to reply

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

\(z\)-axis

Log in to reply

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.

Log in to reply

Diophantine?

Log in to reply

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

Log in to reply

Multiplication sign \( \prod \)

Log in to reply

\(\prod\) [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 \(\prod\).]

Log in to reply

Terms

Log in to reply

Terms can be numbers, variables (alone or multiplied with numbers or constants, which is the

co-efficientof thevariablesconcerned), unknowns (alone or multiplied with numbers or constants, which is the co-efficient) that constitutes anexpressionalone or in addition or substraction (by \(+\) or \(-\) sign) to other terms.Log in to reply

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.

Log in to reply

Binomial Theorem

Log in to reply

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

Log in to reply

RRT (Rational Root Theorem)

Log in to reply

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

Log in to reply

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

Log in to reply

Prime Number

Log in to reply

This is number theory, isn't it?

Log in to reply

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

Log in to reply

Concatenation

Log in to reply

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

Log in to reply

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

Log in to reply

weighted average

Log in to reply

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.

Log in to reply

Radicals

Log in to reply

\(\sqrt{\ \ \ }\) 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 \(\sqrt{25}=5\).

Log in to reply

Quadratic Equation

Log in to reply

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 \(2\).

Log in to reply

Slope of a line

Log in to reply

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.

Log in to reply

Rationalizing factor

Log in to reply

Vieta's Formulas

Log in to reply

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:

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

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

\[\displaystyle\vdots\]

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

Log in to reply

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.

Log in to reply

Determinant

Log in to reply

Matrix

Log in to reply

Quartic Function

Log in to reply

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

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

Log in to reply

Quintic Function

Log in to reply

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.

Log in to reply

Cubic Function

Log in to reply

[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'].

Log in to reply

Linear Function

Log in to reply

Polar Form (of a complex number)

Log in to reply

Exponential Form (of a complex number)

Log in to reply

Algebra

Log in to reply

Arbitrary constant

Log in to reply

Unknowns

Log in to reply

Radians

Log in to reply

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.

Log in to reply

Numerator

Log in to reply

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

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

Log in to reply

Denominator

Log in to reply

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

Log in to reply

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

Log in to reply

PIE (Principle of Inclusion and Exclusion)

EDIT: Ignore please. Thanks Calvin.

Log in to reply

This would be under combinatorics

Log in to reply

CRT (Chinese Remainder Theorem)

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

Log in to reply

Yes, that counts as Number Theory.

Log in to reply

Imaginary Number \( i \)

Log in to reply

Already taken care of.

Log in to reply

Irrational numbers

Log in to reply

Already taken care of.

Log in to reply