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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
4 years ago

No vote yet
31 votes

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
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1. numbered
2. list

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Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

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paragraph 1

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[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 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} \( \boxed{123} \)

Comments

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zero

Mursalin Habib - 4 years ago

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

Mursalin Habib - 4 years ago

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Telescoping series

Pi Han Goh - 4 years ago

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

Daniel Liu - 4 years 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 - 4 years ago

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Arithmetic, Geometric, and Harmonic Mean

(should be separate definitions?)

Calvin Lin Staff - 4 years 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 \(a_1, a_2, \ldots , a_n \) 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 \( a_1 = a_2 = \ldots = a_n \)

Pi Han Goh - 4 years ago

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RMS > AM > GM > HM

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

Santanu Banerjee - 4 years ago

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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.

Mursalin Habib - 4 years 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.

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Multiplication sign \( \prod \)

Priyansh Sangule - 4 years ago

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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 \(\prod\).]

Mursalin Habib - 4 years ago

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Diophantine?

Akshat Jain - 4 years 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 - 4 years ago

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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

Mursalin Habib - 4 years ago

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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.

Mursalin Habib - 4 years ago

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W.L.O.G

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

Priyansh Sangule - 4 years 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 - 4 years ago

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Pi, \(\pi\)

Pi Han Goh - 4 years ago

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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 \(3.14159265358979 \)

Pi Han Goh - 4 years ago

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Power

Jorge Tipe - 4 years 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.

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Exponent

Jorge Tipe - 4 years 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.

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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 - 4 years ago

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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. ;)

Daniel Liu - 4 years ago

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Wow ! Awesome

Priyansh Sangule - 4 years ago

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Summation sign \( \Sigma\)

Calvin Lin Staff - 4 years ago

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

Pi Han Goh - 4 years 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 - 4 years ago

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

Soham Zemse - 3 years, 10 months ago

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Denominator

Mursalin Habib - 4 years ago

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

Mursalin Habib - 4 years ago

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Numerator

Mursalin Habib - 4 years ago

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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

Mursalin Habib - 4 years ago

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Radians

Bob Krueger - 4 years 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 \(2\pi r\), 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 - 4 years ago

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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)

Taehyung Kim - 4 years ago

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Polar Form (of a complex number)

Taehyung Kim - 4 years ago

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Linear Function

Taehyung Kim - 4 years ago

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Cubic Function

Taehyung Kim - 4 years 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 \(P(x)=x^3+7x+13\) is a cubic polynomial for \(x\) because it is a degree-\(3\) polynomial [the highest power of \(x\) is \(3\), see 'degree'].

Mursalin Habib - 4 years ago

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Quintic Function

Taehyung Kim - 4 years ago

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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.

Mursalin Habib - 4 years ago

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Quartic Function

Taehyung Kim - 4 years ago

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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.

Mursalin Habib - 4 years ago

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Matrix

Taehyung Kim - 4 years ago

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Determinant

Taehyung Kim - 4 years ago

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Vieta's Formulas

Snehdeep Arora - 4 years ago

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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:

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

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

\[\displaystyle\vdots\]

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

Snehdeep Arora - 4 years 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 - 4 years ago

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Rationalizing factor

Priyansh Sangule - 4 years ago

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Slope of a line

Priyansh Sangule - 4 years 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 \(\Delta x\) 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 - 4 years ago

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Quadratic Equation

Priyansh Sangule - 4 years ago

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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 \(2\).

Mursalin Habib - 4 years ago

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Radicals

Priyansh Sangule - 4 years ago

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

Mursalin Habib - 4 years ago

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weighted average

Mursalin Habib - 4 years ago

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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.

Mursalin Habib - 4 years ago

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Concatenation

Pi Han Goh - 4 years 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 - 4 years ago

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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 \(314159\)

Pi Han Goh - 4 years ago

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Prime Number

Pi Han Goh - 4 years ago

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This is number theory, isn't it?

Mursalin Habib - 4 years ago

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

Pi Han Goh - 4 years ago

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

Daniel Liu - 4 years ago

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

Daniel Liu - 4 years ago

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RRT (Rational Root Theorem)

Daniel Liu - 4 years ago

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Binomial Theorem

Daniel Liu - 4 years ago

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

Daniel Liu - 4 years ago

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CRT (Chinese Remainder Theorem)

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

Daniel Liu - 4 years ago

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Yes, that counts as Number Theory.

Ahaan Rungta - 4 years ago

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PIE (Principle of Inclusion and Exclusion)

EDIT: Ignore please. Thanks Calvin.

Daniel Liu - 4 years ago

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This would be under combinatorics

Calvin Lin Staff - 4 years ago

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Irrational numbers

Priyansh Sangule - 4 years ago

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Already taken care of.

Bob Krueger - 4 years ago

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Imaginary Number \( i \)

Priyansh Sangule - 4 years ago

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Already taken care of.

Bob Krueger - 4 years ago

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