Brilliant Algebra Glossary is now live

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:

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
5 years, 11 months ago

No vote yet
31 votes

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Telescoping series

Pi Han Goh - 5 years, 11 months ago

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

Pi Han Goh - 5 years, 11 months ago

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Example: i=1n1i(i+1)=11n+1\displaystyle\sum^{n}_{i=1}{\dfrac{1}{i(i+1)}}=1-\dfrac{1}{n+1}

Daniel Liu - 5 years, 11 months ago

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zero

Mursalin Habib - 5 years, 11 months ago

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

Mursalin Habib - 5 years, 11 months ago

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Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?)

Calvin Lin Staff - 5 years, 11 months ago

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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 a1,a2,,ana_1, a_2, \ldots , a_n be non-negative real numbers, then the

  1. Arithmetic mean of these numbers is a1+a2++ann \large \frac {a_1 + a_2 + \ldots + a_n} {n}

  2. Geometric mean of these numbers is a1a2ann \large \sqrt[n] {a_1 \cdot a_2 \cdot \ldots \cdot a_n}

  3. Harmonic mean of these numbers is n1a1+1a2++1an \large \frac {n}{ \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n} }

And

a1+a2++anna1a2annn1a1+1a2++1an \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 a1=a2==an a_1 = a_2 = \ldots = a_n

Pi Han Goh - 5 years, 11 months ago

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RMS > AM > GM > HM

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

Santanu Banerjee - 5 years, 11 months ago

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Summation sign Σ \Sigma

Calvin Lin Staff - 5 years, 11 months ago

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Σ \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 mm and nn are integers and mnm \leq n , then the summation from kk equals to mm to nn of aka_k is the sum of all the terms am,am+1,am+2,,ana_m, a_{m+1}, a_{m+2}, \ldots , a_n . We write

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

and call kk the index of the summation , mm the lower limit of the summation of the summation, and nn the upper limit of the summation.

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

For example, k=37(k2+k+10)=(32+3+10)+(42+4+10)+(52+5+10)+(62+6+10)+(72+7+10) \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 - 5 years, 11 months ago

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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.

Jonathan Wong - 5 years, 11 months ago

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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.

Daniel Liu - 5 years, 11 months ago

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

Example problem: Find all integer solutions to 1x+1y=16\text{Find all integer solutions to }\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{6}.

Solution: Adding the LHS gives x+yxy=16\dfrac{x+y}{xy}=\dfrac{1}{6}.

Cross multiplying gives xy=6x+6yxy=6x+6y; reorder to make xy6x6y=0xy-6x-6y=0.

Use SFFT to get (x6)(y6)=36(x-6)(y-6)=36. Since x,yx,y are integers, that means x6x-6 and y6y-6 are also integers. Therefore, they are factors of 3636 which multiply to 3636.

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

Daniel Liu - 5 years, 11 months ago

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Wow ! Awesome

Priyansh Sangule - 5 years, 11 months ago

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Exponent

Jorge Tipe - 5 years, 11 months ago

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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.

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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Power

Jorge Tipe - 5 years, 11 months ago

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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.

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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Pi, π\pi

Pi Han Goh - 5 years, 11 months ago

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π\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 3.141592653589793.14159265358979

Pi Han Goh - 5 years, 11 months ago

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W.L.O.G

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

Priyansh Sangule - 5 years, 11 months ago

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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:

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]

Mursalin Habib - 5 years, 11 months ago

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

zz-axis

Mursalin Habib - 5 years, 11 months ago

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The zz-axis is the third axis in a three-dimensional coordinate system. Typically the xx-axis and yy-axis are thought of as being in a horizontal plane, with the zz-axis pointing up.

Mursalin Habib - 5 years, 11 months ago

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Diophantine?

Akshat Jain - 5 years, 11 months ago

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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.

Mursalin Habib - 5 years, 11 months ago

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Multiplication sign \prod

Priyansh Sangule - 5 years, 11 months ago

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

For example: i=25i\displaystyle\prod_{i=2}^5 i simply denotes the product of ii's where ii starts out at 22, is incremented by 11 for each successive terms and stops at 55

So, i=25i=2×3×4×5=120\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 - 5 years, 11 months ago

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Terms

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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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.

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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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 ax2ax^2, the second term is bxbx, and the third term is cc. The different terms in an expression are separated by addition (or subtraction) signs.

Mursalin Habib - 5 years, 11 months ago

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Binomial Theorem

Daniel Liu - 5 years, 11 months ago

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States that for any numbers a,ba,b and positive integer nn, that: (a+b)n=(n0)anb0+(n1)an1b1++(nn1)a1bn1+(nn)a0bn(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.

Daniel Liu - 5 years, 11 months ago

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RRT (Rational Root Theorem)

Daniel Liu - 5 years, 11 months ago

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

Daniel Liu - 5 years, 11 months ago

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

Daniel Liu - 5 years, 11 months ago

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Prime Number

Pi Han Goh - 5 years, 11 months ago

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This is number theory, isn't it?

Mursalin Habib - 5 years, 11 months ago

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A prime number is a positive integer greater than 11 that can only be divided by 11 and itself.

Pi Han Goh - 5 years, 11 months ago

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Concatenation

Pi Han Goh - 5 years, 11 months ago

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In mathematics, concatenation of two or more numbers is the joining of their numerals. For example, the concatenation of 314314 and 159159 is 314159314159

Pi Han Goh - 5 years, 11 months ago

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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?

Mursalin Habib - 5 years, 11 months ago

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weighted average

Mursalin Habib - 5 years, 11 months ago

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A weighted average (also weighted arithmetic mean) of a group of numbers x1,x2,x3,...,xnx_1, x_2, x_3, . . . , x_n is: w_1x_1+  w_2x_2 + w_3x_3+ w_n+ x_n where the ww’s are a group of positive numbers such that: w_1+  w_2 + w_3+... w_n=  1.

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

Mursalin Habib - 5 years, 11 months ago

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Radicals

Priyansh Sangule - 5 years, 11 months ago

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

In the example above, yy 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 55 and 5-5 you're going to get 2525. But 25=5\sqrt{25}=5.

Mursalin Habib - 5 years, 11 months ago

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Quadratic Equation

Priyansh Sangule - 5 years, 11 months ago

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Quadratic normally refers to something with degree 22.

For example: ax2+bx+c=0,a0ax^2+bx+c=0, a\neq 0 is a quadratic equation because the power ofxx is 22.

Mursalin Habib - 5 years, 11 months ago

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Slope of a line

Priyansh Sangule - 5 years, 11 months ago

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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 Δx\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.

Mursalin Habib - 5 years, 11 months ago

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Rationalizing factor

Priyansh Sangule - 5 years, 11 months ago

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Vieta's Formulas

Snehdeep Arora - 5 years, 11 months ago

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Let P(x)=anxn+an1xn1+.......+a1x+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+.......+a_1x+a_0 be a polynomial and x1,x2......,xnx_1,x_2......,x_n be its roots.

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

(x1+x2......+xn)=an1an(x_1+x_2......+x_n)= \frac{-a_{n-1}}{a_n}

(x1x2+x2x3.....+x1xn)=an2an(x_1x_2+x_2x_3.....+x_1x_n)=\frac{a_{n-2}}{a_n}

\displaystyle\vdots

(x1x2....xn)=1na0an(x_1x_2....x_n)=-1^n\frac{a_0}{a_n}

Snehdeep Arora - 5 years, 11 months ago

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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.

Bob Krueger - 5 years, 11 months ago

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Determinant

Taehyung Kim - 5 years, 11 months ago

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Matrix

Taehyung Kim - 5 years, 11 months ago

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Quartic Function

Taehyung Kim - 5 years, 11 months ago

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

x4+3x3+12x2x^4+3x^3+12x-\sqrt{2} is a quartic polynomial.

Mursalin Habib - 5 years, 11 months ago

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Quintic Function

Taehyung Kim - 5 years, 11 months ago

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

For example: a quintic equation is a polynomial equation of degree 55.

5x568x3+89=05x^5-68x^3+89=0 is a quintic equation.

Mursalin Habib - 5 years, 11 months ago

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Cubic Function

Taehyung Kim - 5 years, 11 months ago

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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)=x3+7x+13P(x)=x^3+7x+13 is a cubic polynomial for xx because it is a degree-33 polynomial [the highest power of xx is 33, see 'degree'].

Mursalin Habib - 5 years, 11 months ago

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Linear Function

Taehyung Kim - 5 years, 11 months ago

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Polar Form (of a complex number)

Taehyung Kim - 5 years, 11 months ago

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Exponential Form (of a complex number)

Taehyung Kim - 5 years, 11 months ago

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Algebra

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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Arbitrary constant

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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Unknowns

Sheikh Asif Imran Shouborno - 5 years, 11 months ago

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Radians

Bob Krueger - 5 years, 11 months ago

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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 2πr2\pi r, so a full circle (360360 degrees) equals 2π2\pi radians. Also, 180180 degrees equals π\pi radians, and a right angle (9090 degrees) has a measure of π2\frac{\pi}{2} radians.

Mursalin Habib - 5 years, 11 months ago

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Numerator

Mursalin Habib - 5 years, 11 months ago

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

For example: in the fraction 713\frac{7}{13}, 77 is the numerator

Mursalin Habib - 5 years, 11 months ago

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Denominator

Mursalin Habib - 5 years, 11 months ago

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The denominator is the bottom part of a fraction. In the fraction 713\frac{7}{13}, 1313 is the denominator.

Mursalin Habib - 5 years, 11 months ago

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

Soham Zemse - 5 years, 9 months ago

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PIE (Principle of Inclusion and Exclusion)

EDIT: Ignore please. Thanks Calvin.

Daniel Liu - 5 years, 11 months ago

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This would be under combinatorics

Calvin Lin Staff - 5 years, 11 months ago

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CRT (Chinese Remainder Theorem)

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

Daniel Liu - 5 years, 11 months ago

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Yes, that counts as Number Theory.

Ahaan Rungta - 5 years, 11 months ago

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Imaginary Number i i

Priyansh Sangule - 5 years, 11 months ago

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Already taken care of.

Bob Krueger - 5 years, 11 months ago

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Irrational numbers

Priyansh Sangule - 5 years, 11 months ago

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Already taken care of.

Bob Krueger - 5 years, 11 months ago

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