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!

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## Comments

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TopNewestzero

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

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Telescoping series

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

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

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Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?)

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

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RMS > AM > GM > HM

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

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Terms

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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.

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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

Multiplication sign \( \prod \)

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

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Diophantine?

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

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

\(z\)-axis

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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.

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W.L.O.G

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

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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]

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Pi, \(\pi\)

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

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Power

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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.

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Exponent

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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.

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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.

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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. ;)

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Wow ! Awesome

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Summation sign \( \Sigma\)

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

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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.

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

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Denominator

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

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Numerator

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

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

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Radians

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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.

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Unknowns

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Arbitrary constant

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Algebra

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Exponential Form (of a complex number)

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Polar Form (of a complex number)

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Linear Function

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Cubic Function

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[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'].

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Quintic Function

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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.

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Quartic Function

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

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

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Matrix

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Determinant

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Vieta's Formulas

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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:

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

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

\[\displaystyle\vdots\]

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

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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.

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Rationalizing factor

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Slope of a line

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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.

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Quadratic Equation

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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 \(2\).

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Radicals

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

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weighted average

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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.

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Concatenation

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

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

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Prime Number

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This is number theory, isn't it?

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

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

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

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RRT (Rational Root Theorem)

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Binomial Theorem

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

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CRT (Chinese Remainder Theorem)

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

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Yes, that counts as Number Theory.

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PIE (Principle of Inclusion and Exclusion)

EDIT: Ignore please. Thanks Calvin.

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This would be under combinatorics

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Irrational numbers

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Already taken care of.

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Imaginary Number \( i \)

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Already taken care of.

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