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!

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

- 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.

## Comments

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TopNewestzero – Mursalin Habib · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 months ago

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

Telescoping series – Pi Han Goh · 3 years, 9 months ago

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– Daniel Liu · 3 years, 9 months ago

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

– Pi Han Goh · 3 years, 9 months ago

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

Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?) – Calvin Lin Staff · 3 years, 9 months ago

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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 \( a_1 = a_2 = \ldots = a_n \) – Pi Han Goh · 3 years, 9 months ago

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where RMS is the Root of Mean of Square of the given numbers – Santanu Banerjee · 3 years, 9 months ago

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Terms – Sheikh Asif Imran Shouborno · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 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.Log in to reply

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. – Sheikh Asif Imran Shouborno · 3 years, 9 months agoLog in to reply

Multiplication sign \( \prod \) – Priyansh Sangule · 3 years, 9 months ago

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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 \(\prod\).] – Mursalin Habib · 3 years, 9 months ago

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Diophantine? – Akshat Jain · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 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.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 – Mursalin Habib · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 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.Log in to reply

W.L.O.G

(Posting for rookies like me who din't get abbreviations) – Priyansh Sangule · 3 years, 9 months ago

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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] – Mursalin Habib · 3 years, 9 months ago

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Pi, \(\pi\) – Pi Han Goh · 3 years, 9 months ago

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– Pi Han Goh · 3 years, 9 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 \(3.14159265358979 \)Log in to reply

Power – Jorge Tipe · 3 years, 9 months ago

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– Sheikh Asif Imran Shouborno · 3 years, 9 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.Log in to reply

Exponent – Jorge Tipe · 3 years, 9 months ago

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– Sheikh Asif Imran Shouborno · 3 years, 9 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.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. – Daniel Liu · 3 years, 9 months ago

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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. ;) – Daniel Liu · 3 years, 9 months ago

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– Priyansh Sangule · 3 years, 9 months ago

Wow ! AwesomeLog in to reply

Summation sign \( \Sigma\) – Calvin Lin Staff · 3 years, 9 months ago

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

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, \( \displaystyle \sum_{k=3}^7 (k^2 + k + 10) = (3^2 + 3 + 10) + (4^2 + 4 + 10) + (5^2 + 5 + 10) + (6^2 + 6 + 10) + (7^2 + 7 + 10) \) – Pi Han Goh · 3 years, 9 months ago

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– Jonathan Wong · 3 years, 9 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.Log in to reply

there is a typo in "argument" section. in pi/2 = 45. it must be pi/4. – Soham Zemse · 3 years, 7 months ago

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Denominator – Mursalin Habib · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 months ago

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

Numerator – Mursalin Habib · 3 years, 9 months ago

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For example: in the fraction \(\frac{7}{13}\), \(7\) is the numerator – Mursalin Habib · 3 years, 9 months ago

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Radians – Bob Krueger · 3 years, 9 months ago

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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. – Mursalin Habib · 3 years, 9 months ago

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Unknowns – Sheikh Asif Imran Shouborno · 3 years, 9 months ago

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Arbitrary constant – Sheikh Asif Imran Shouborno · 3 years, 9 months ago

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Algebra – Sheikh Asif Imran Shouborno · 3 years, 9 months ago

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Exponential Form (of a complex number) – Taehyung Kim · 3 years, 9 months ago

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Polar Form (of a complex number) – Taehyung Kim · 3 years, 9 months ago

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Linear Function – Taehyung Kim · 3 years, 9 months ago

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Cubic Function – Taehyung Kim · 3 years, 9 months ago

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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']. – Mursalin Habib · 3 years, 9 months ago

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Quintic Function – Taehyung Kim · 3 years, 9 months ago

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For example: a quintic equation is a polynomial equation of degree \(5\).

\(5x^5-68x^3+89=0\) is a quintic equation. – Mursalin Habib · 3 years, 9 months ago

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Quartic Function – Taehyung Kim · 3 years, 9 months ago

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\(x^4+3x^3+12x-\sqrt{2}\) is a quartic polynomial. – Mursalin Habib · 3 years, 9 months ago

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Matrix – Taehyung Kim · 3 years, 9 months ago

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Determinant – Taehyung Kim · 3 years, 9 months ago

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Vieta's Formulas – Snehdeep Arora · 3 years, 9 months ago

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

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

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

\[\displaystyle\vdots\]

\[(x_1x_2....x_n)=-1^n\frac{a_0}{a_n}\] – Snehdeep Arora · 3 years, 9 months ago

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– Bob Krueger · 3 years, 9 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.Log in to reply

Rationalizing factor – Priyansh Sangule · 3 years, 9 months ago

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Slope of a line – Priyansh Sangule · 3 years, 9 months ago

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A horizontal line has a slope of zero. As a line approaches being a vertical line, its slope approaches infinity. – Mursalin Habib · 3 years, 9 months ago

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Quadratic Equation – Priyansh Sangule · 3 years, 9 months ago

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For example: \(ax^2+bx+c=0, a\neq 0\) is a quadratic equation because the power of\(x\) is \(2\). – Mursalin Habib · 3 years, 9 months ago

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Radicals – Priyansh Sangule · 3 years, 9 months ago

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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 \(\sqrt{25}=5\). – Mursalin Habib · 3 years, 9 months ago

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weighted average – Mursalin Habib · 3 years, 9 months ago

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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. – Mursalin Habib · 3 years, 9 months ago

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Concatenation – Pi Han Goh · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 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?Log in to reply

– Pi Han Goh · 3 years, 9 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 \(314159\)Log in to reply

Prime Number – Pi Han Goh · 3 years, 9 months ago

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– Mursalin Habib · 3 years, 9 months ago

This is number theory, isn't it?Log in to reply

– Pi Han Goh · 3 years, 9 months ago

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

LHS and RHS (Left Hand Side and Right Hand Side) – Daniel Liu · 3 years, 9 months ago

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– Daniel Liu · 3 years, 9 months ago

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

RRT (Rational Root Theorem) – Daniel Liu · 3 years, 9 months ago

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Binomial Theorem – Daniel Liu · 3 years, 9 months ago

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– Daniel Liu · 3 years, 9 months ago

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

CRT (Chinese Remainder Theorem)

EDIT: ignore, this should be in Number Theory, correct? – Daniel Liu · 3 years, 9 months ago

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– Ahaan Rungta · 3 years, 9 months ago

Yes, that counts as Number Theory.Log in to reply

PIE (Principle of Inclusion and Exclusion)

EDIT: Ignore please. Thanks Calvin. – Daniel Liu · 3 years, 9 months ago

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– Calvin Lin Staff · 3 years, 9 months ago

This would be under combinatoricsLog in to reply

Irrational numbers – Priyansh Sangule · 3 years, 9 months ago

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– Bob Krueger · 3 years, 9 months ago

Already taken care of.Log in to reply

Imaginary Number \( i \) – Priyansh Sangule · 3 years, 9 months ago

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– Bob Krueger · 3 years, 9 months ago

Already taken care of.Log in to reply