# Community Project: Brilliant Algebra Terms Glossary

UPDATE: Great work everyone! We've locked this discussion so that we can create the (initial) glossary, which is now up and running! This looks great!

We've started a separate discussion for edits / feedback / comments / more definitions.

We think it would be helpful for Brilliant to have a glossary of terms (a collection of short definitions that explain important vocabulary). Having this available on our site will help ensure that everyone on Brilliant is using mathematical terminology in the same way, and we'd like to invite you to help us make it great. This is an opportunity for you to contribute to the community and help other students. (In the final version of the glossary, we will recognize members who have been particularly helpful.)

We will use this discussion to propose terms that should be included in the glossary and also to provide and vote on definitions.

Our goal is to gather a collection of clearly stated definitions for algebra terms. Some guidelines:

• Only ALGEBRA in this discussion, please. We'll do other topics if this work well.
• We're looking for definitions of basic terms that you'd come across while solving Brilliant problems not every term/formula/theorem you can think of. Terms are words like "absolute value" or "quadratic."
• A glossary should have definitions, not essays or proofs. Keep things short. ("Keep it to one sentence" is a good rule of thumb.)
• Well-chosen examples can be very helpful.

#### 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
6 years, 8 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
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UPDATE: Great work everyone! Things have slowed down, so we'll leave this discussion open until Friday and then close it down. We'll work on building out the first draft of our Algebra glossary next week!

Staff - 6 years, 8 months ago

Variable

Staff - 6 years, 8 months ago

A variable is a symbol, often a letter, which is used to represent a value which may change within the context of the given problem. Example: $x$ and $y$ are variables in the expression $y = x^2 + 4$.

Staff - 6 years, 8 months ago

Complex Number

- 6 years, 8 months ago

A number in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

- 6 years, 8 months ago

For example: $3+4i$.

Note that all real numbers are complex numbers as well with the imaginary part $b$ being zero.

For example: $5$ is a complex number of the form $5+0i$.

- 6 years, 8 months ago

Note that $\mathbb{R} \subseteq \mathbb{C}$.

The real numbers (symbolized by the hollow R) is a subset of the complex numbers (symbolized by the hollow C). This means that real numbers are a type of complex numbers. Stated differently, all real numbers are complex, but not all complex numbers are real.

- 6 years, 8 months ago

Good point, though should it be added? (I have no opinion)

Also, should we mention that the complex numbers are denoted by $\mathbb{C}$? Or should we verbally say that "the real numbers are a subset of the complex numbers".

Staff - 6 years, 8 months ago

I don't think the notation is necessary for someone just learning how to solve Brilliant problems.

- 6 years, 8 months ago

On the contrary, I think the notation would be useful to know. It's not a particularly unnerving thing to learn, and I have seen it used in solutions. I also think it's fairly standard across the board (world).

- 6 years, 8 months ago

I agree with Jonathan; there's a symbol for it, why not use it?

$\mathbb{C}^{*}$ would be the complex numbers without $0$.

- 6 years, 8 months ago

Absolute Value

- 6 years, 8 months ago

The absolute value of a complex number $z=a+bi$, represented by $|z|$ is the euclidean distance of that complex number from the origin in the argand plane.

In other words $|a+bi|=\sqrt{a^2+b^2}$.

- 6 years, 8 months ago

Represented by $|{x}|$, it is the distance from zero to that number $x$.

- 6 years, 8 months ago

The absolute value of a real number $x,$ denoted $|x|,$ is defined by $|x| = \begin{cases} x &\text{ if }x \ge 0 \\ -x & \text{ if } x < 0. \end{cases}$

- 6 years, 8 months ago

I don't like it for the reason that "distance" is poorly defined.

- 6 years, 8 months ago

But it's hard to define it concisely for any real, complex, or n-dimensional number. Using the intuitive idea of distance, people who aren't as familiar with terms can understand absolute value quite easily when compared to Michael's (correct) definition, or Mursalin's definition, in which the newbie would need to know about piecewise functions and complex numbers and argand planes.

- 6 years, 8 months ago

Indices and Surds

- 6 years, 8 months ago

Floor

- 6 years, 8 months ago

$\lfloor x \rfloor$ is the greatest integer function, or the floor function, which gives the greatest integer less than or equal to $x$. [example]

- 6 years, 8 months ago

For example $\lfloor 2.99 \rfloor = 2$ and $\lfloor -1.1 \rfloor = -2$

- 6 years, 8 months ago

Polynomial

Staff - 6 years, 8 months ago

A polynomial in $x$ is an algebraic expression of the form

a_nx^n+a_{n-1}x^{n -1}+\cdots a_3x^3 + a_2x^2 + a_1x + a_0 where the indices are non-negative integers.

For example, $P(x)=x^2+x+1$ is a polynomial in $x$ because the indices are non-negative integers. So, is $P(x)=5$, because the index of $x$ is $0$ [a non-negative integer]. However $P(x)=\sqrt{x}+\frac{5}{x}$ is not a polynomial in $x$ because the indices of $x$ are $\frac{1}{2}$ and$-1$ respectively, neither of which is a non-negative integer.

- 6 years, 8 months ago

Coefficient

Staff - 6 years, 8 months ago

A number or constant by which a term involving variables of an algebraic expression is being multiplied. Example: 4 and -5 are coefficients in the expression $4x+xy-5y$.

- 6 years, 8 months ago

Ceiling

- 6 years, 8 months ago

$\lceil x \rceil$ is the least integer function, or the ceiling function, which gives the smallest integer greater than or equal to $x$.

- 6 years, 8 months ago

My question: why did everyone vote up ceiling when floor is used so much more often?

- 6 years, 8 months ago

Exactly, floor doesn't need much explanation.

- 6 years, 8 months ago

Relatively Prime, Coprime

- 6 years, 8 months ago

Two integers $x,y$ are relatively prime, mutually prime, or coprime if and only if the only positive integer that they can both be divided by evenly is $1$. Equivalently, their greatest common denominator $\gcd(a,b)=1$, and no prime number divides both $a$ and $b$. Example: $6$ and $35$ are coprime, while $15$ and $35$ are not because $5$ divides them both.

- 6 years, 8 months ago

In cases where there are multiple equivalent terms, I think it would be best for the initial defining term to state the terms, but not include them in the definition. For examples,

Relatively prime, Coprime, Mutually prime. Two integers $x$ and $y$ are relatively prime if and only if ....

Vote up (on this comment) if you agree, vote down if you would rather have it as

Relatively prime, Coprime, Mutually prime. Two integers $x$ and $y$ are relatively prime, mutually prime, or coprime, if and only if ....

Staff - 6 years, 8 months ago

What would be meant by the statement: "$x$,$y$ and $z$ are coprime"?

Are the numbers $6$,$14$, $21$ coprime since $gcd(6,14,21)=1$? Or are they not coprime since $gcd(6,14) \neq 1$?

- 6 years, 8 months ago

Good point to bring up. There is ambiguity in "x, y, and z are coprime." Thus, to specify, you would want to say pairwise coprime, or gcd of all of them = 1, depending on what you mean.

- 6 years, 8 months ago

Imaginary Unit

- 6 years, 8 months ago

Represented by $i=\sqrt{-1}$

- 6 years, 8 months ago

Represented by $i^2=-1$

- 6 years, 8 months ago

This definition may be a bit confusing, because is the imaginary unit $i$ or $i^2$? You could say "The imaginary unit is represented by $i$, such that $i^2=-1$." Or you could just you my shorter definition.

- 6 years, 8 months ago

Note that your definitions are not equivalent.

My vote is for Christophers definition, as that is how $i$ was introduced to us when doing university here.

- 6 years, 8 months ago

Actually, my definition is the better one. For the statement "$i^2 = -1$" has two solutions for $i$, yet we only assign one value to $i$, that being the positive square root of negative one.

- 6 years, 8 months ago

$i^2=-1$ is THE definition of $i$.

The choice of the root doesn't matter, and it's quite arbitrary to pick one root over the other. Also, it's quite odd to talk about positive and negative when dealing with complex numbers. It's true that most people think of $i$ as the square root of $-1$, but that's not how $i$ was defined.

EDIT: re-reading my comment I feel I have to add that I'm not trying to be an ass :)

- 6 years, 8 months ago

I actually agree with Bob. The thing with Christopher's definition is that it can mislead people into thinking that the imaginary unit is represented by $i^2$ [Just read the term and the definition at once. Imaginary unit: represented by $i^2=-1$. That is confusing, at least to me.]

However, you are right. $i^2=-1$ is THE definition of $i$. So my definition [which is the same as Bob's latter definition] would be:

The imaginary unit is represented by $i$ such that $i^2=-1$.

- 6 years, 8 months ago

This would be my preference. No roots :)

- 6 years, 8 months ago

True, positive and negative values are only suitable for real numbers.

- 6 years, 8 months ago

Tuple

- 6 years, 8 months ago

A tuple is an ordered list of elements.

$(2, 3, 5, 7, 11)$ is an example of a $5$-tuple.

A $2$-tuple is generally called an ordered pair while $3$-tuples and $4$-tuples are called ordered triples and quadruples respectively.

- 6 years, 8 months ago

And a $3$-tuple an ordered triple.

- 6 years, 8 months ago

- 6 years, 8 months ago

Do the elements of a tuple have to be in order like in your example, or are $(1,2,3)$ and $(3,1,2)$ two different 3-tuples?

Editted to 3-tuples

- 6 years, 8 months ago

And this is why tuple is a hard word to define understandably. First, Mursalin did state that a tuple is ordered, so in your example you have two different tuples. And the tuples you give are actually 3-tuples, or ordered triples.

- 6 years, 8 months ago

Fixed the mistake.

It would be best to give an example where the elements of the tuple aren't in order:

$(2,7,3,11,5)$ is an example of a $5$-tuple.

- 6 years, 8 months ago

We rarely use the word "Tuple". Perhaps, it's better to call this "Ordered Pair", and then add on "Ordered triple, ordered quadruple"?

An ordered pair is also a concept that many people struggle with when they first see it. Is there a way we could make it clearer? Perhaps also explain that $(1,2)$ and $(2,1)$ are 2 different ordered pairs?

Staff - 6 years, 8 months ago

A $k$-tuple is an ordered set with $k$ elements. In math, this notation is often preferred when talking about an arbitrary $k$, or when standard naming becomes onerous, i.e. we all know "pair" or "triple" but calling an ordered set of $6$ elements a "sextuple" and so forth can be inconvenient and unnecessary.

- 6 years, 8 months ago

Constant function

- 6 years, 8 months ago

A constant function is a function whose value remains the same for all inputs.

For example: $f(x)=7$ is a constant function.

- 6 years, 8 months ago

Conjugate

- 6 years, 8 months ago

The conjugate of a binomial is the binomial obtained by multiplying the second term by $-1$. Example: the conjugates of $x+2,x+\sqrt{3},$ and $-1+4i$ are $x-2,x-\sqrt{3},$ and $-1-4i$, respectively.

I like this better because conjugates exist or can be extended in many commutative rings besides the complex numbers.

- 6 years, 8 months ago

Hmm, two of these examples contradict the definition for complex numbers.

- 6 years, 8 months ago

Wait what. At least for real or complex $x$ all of the above are complex numbers. And as i said, they don't have to be complex to have a conjugate?

- 6 years, 8 months ago

A real number $x+2$ can be written as a complex number $(x+2)+0*i$. The latter is a binomial with terms $(x+2)$ and $0*i$.

Is the conjugate $x-2$ or $(x+2)-0*i$?

- 6 years, 8 months ago

Besides the fact that such a conjugate is trivial, I do see how this might be a problem, though I've never encountered it as such.

I'd think to assume the conjugate is in the terms of the binomial as given, or the complex conjugate should be specified.

- 6 years, 8 months ago

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, $3 + 4i$ and $3 - 4i$ are complex conjugates.

The conjugate of the complex number $Z$

$Z=a+bi$

where $a$ and $b$ are real numbers, is

$\overline{Z} = a - bi$

For Example ,

$\overline{10-5i} = 10+5i$

- 6 years, 8 months ago

More concisely, the conjugate of a complex number is the complex number with the same real part but opposite imaginary part.

- 6 years, 8 months ago

Opposite Signs

- 6 years, 8 months ago

Extraneous Solution

- 6 years, 8 months ago

An extraneous solution to an equation is a solution that emerges from the process of solving the equation but isn't a valid solution to the original problem.

Take this for example.

Solve for $x$: $\sqrt{2x+7}+3=0$.

Adding $-3$ to the equation and squaring both sides give us:

$2x+7=9$.

So, $x$ should be equal to $1$. However, it is not.

Plug $x=1$ in the original equation to get $3+3=0$. Actually this equation has no solution and $1$ is an extraneous root which emerged because we squared the original equation.

- 6 years, 8 months ago

Discriminant

- 6 years, 8 months ago

Discriminant is the number $\Delta=b^2-4ac$ for the quadratic equation $ax^2+bx+c=0$.

- 6 years, 8 months ago

By the way, the term 'discriminant' isn't used only for quadratic polynomials. For example: the discriminant of the polynomial $ax^3+bx^2+cx+d$ [$a\neq 0$] is $b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd$.

- 6 years, 8 months ago

The discriminant of a polynomial, normally denoted by $\Delta$, is an expression comprised of the coefficients of the polynomial and it gives information about the nature of the roots of the polynomial.

For example if the discriminant of a polynomial is less than zero [i.e. $\Delta<0$], the polynomial has no real root.

- 6 years, 8 months ago

Distinct

- 6 years, 8 months ago

Distinct*

- 6 years, 8 months ago

Two objects are distinct if they are not exactly the same.

- 6 years, 8 months ago

For example the set {$1, 2, 3$} consists of distinct numbers, while the multiset {$1,2,1$} does not.

- 6 years, 8 months ago

Note:

Saying a collection has distinct elements means that no two of them are the same.

- 6 years, 8 months ago

Arithmetic Sequence

- 6 years, 8 months ago

A sequence of numbers in which each term is a certain amount $d$ different than its preceding term, where $d$ can be any real number.

- 6 years, 8 months ago

$d$ is called the 'difference' and can be a complex number as well.

- 6 years, 8 months ago

Geometric Sequence

- 6 years, 8 months ago

A sequence of numbers in which each term is in the same ratio $r$ with its preceding term, where $r$ can be any non zero number

- 6 years, 8 months ago

A sequence of numbers in which each subsequent number is obtained by multiplying with the same factor $r$, where $r$ can be any complex number.

- 6 years, 8 months ago

Function

- 6 years, 8 months ago

An operation or set of operations on a set called the domain, resulting in a set called the range. For every value in the domain, there is only one corresponding value in the range.

- 6 years, 8 months ago

Summation

- 6 years, 8 months ago

Monic Polynomial

- 6 years, 8 months ago

A polynomial with leading coefficient equaling 1.

- 6 years, 8 months ago

Recurrence Relation

- 6 years, 8 months ago

A process or sequence in which the next step or term is defined by one or more of the previous terms. [example]

- 6 years, 8 months ago

Example: the Fibonacci sequence is defined by $F_n=F_{n-1}+F_{n-2}, F_0=0, F_1=1$. This gives the first few terms $0,1,1,2,3,5,8,13,...$.

Could we change the term to "Recurrence Relation" then? It fits the definition you provided better. Recursion in general is a bit broad, and we're only doing algebra right now anyways.

- 6 years, 8 months ago

Done.

- 6 years, 8 months ago

Argument

- 6 years, 8 months ago

1. Of a function: an input variable. For $f(x,y)=x^2+y^2$, $x$ and $y$ are arguments.
2. Of a complex number: on the complex plane, the angle $\theta = \arg(z)$ between the vector representing the complex number $z=a+bi$ and the positive real axis, the principal value of which is taken to be the (unique) value of $\theta$ such that $-\pi < \theta \leq \pi$. Also the angle for which $z=r\cos(\theta )+ir\sin(\theta )=re^{i\theta }$, where $r=|z|=\sqrt{a^2+b^2}$.

- 6 years, 8 months ago

Rational

- 6 years, 8 months ago

Any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are both integers and $q \neq 0$

- 6 years, 8 months ago

Note also that rational numbers can be negative; also, since we can have $q = 1,$ every integer is rational.

- 6 years, 8 months ago

The set of rational numbers is denoted by $\mathbb{Q}$.

$\mathbb{Q}^{*}$ denotes the rational numbers without $0$.

- 6 years, 8 months ago

...And $q\neq 0$.

- 6 years, 8 months ago

Fixed. (and something weird just happened. I was editing the cell that Mursalin had replied to, and its contents transferred to a new comment below, and the original comment had disappeared.

- 6 years, 8 months ago

- 6 years, 8 months ago

The leading coefficient of a polynomial is the coefficient of the term with the highest degree.

For example: in the polynomial $P(x)=3x^2-5x^4+3$, the term with the highest degree is $-5x^4$. So the leading coefficient is $-5$.

- 6 years, 8 months ago

It might be better to give a polynomial where the initial term is not the term with highest degree. E.g. $P(x) = 3x^2 - 5x^4 + 3$.

Staff - 6 years, 8 months ago

Edited!

- 6 years, 8 months ago

Injective Function

- 6 years, 8 months ago

An injective function, also known as a "One-to-One function", is a function where every value in the domain has only one corresponding value in the range, and vice versa.

- 6 years, 8 months ago

A function is injective if and only if its inverse is a function.

- 6 years, 8 months ago

Not exactly true, this would be a bijective function.

An injective function is a function where all elements in the range have a unique element in the domain.

More mathematically put: For all elements in the domain we have $f(a)=f(b) \Rightarrow a=b$

The problem is that we tend to define functions and treat them on their domain and range alone, but this doesn't always have to be the case.

For example the function $f: \mathbb{R}^{*} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{1}{x}$ is an injective, but not surjective function as there is no element $a$ so that $f(a)=0$.

- 6 years, 8 months ago

Surjective Function

- 6 years, 8 months ago

A surjective function is a function where every value in the range has at least one corresponding value in the domain.

- 6 years, 8 months ago

You should probably call the codomain the range to retain the continuity with the other definitions.

- 6 years, 8 months ago

Codomain and range are not completely synonymous. I have usually taken codomain to mean the target set of a function (i.e., a constraint upon the output), whereas the range is the set of values actually mapped from the domain onto the codomain.

For example, in the function $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x) = x^2$, the codomain would be the set of all real numbers, but the range would only contain values $\geq 0$.

Staff - 6 years, 8 months ago

For example $f:\mathbb{R} \rightarrow \mathbb{R}^{+} \cup$ {$0$} defined by $f(x)=|x|$ is a surjective but not injective function.

- 6 years, 8 months ago

Bijective Function

- 6 years, 8 months ago

A bijective function is a function which is both injective and surjective.

- 6 years, 8 months ago

Alternately: A bijective function is a function where each value in its codomain is taken on exactly once.

- 6 years, 8 months ago

Also known as a one-to-one function.

- 6 years, 8 months ago

This might be replacing one definition with another. One to one is slightly clearer, but still requires some explanation for someone who has not seem it before.

Staff - 6 years, 8 months ago

True, this was meant as addition to the name, not as replacement for the definition :)

- 6 years, 8 months ago

Logarithm

- 6 years, 8 months ago

If $x=a^y$, then $y$ is the logarithm of $x$ to the base $a$ and it is denoted as $y=\log_a x$

- 6 years, 8 months ago

Or $y=$ $^{a}log(x)$

- 6 years, 8 months ago

Set

- 6 years, 8 months ago

A set is a well-defined group of objects. For example, the set of all positive integers less than $11$ consists of $\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$.

- 6 years, 8 months ago

A remark:

$0$ is a natural number.

A note:

Every element in the set is unique. The empty set exists and is denoted by {} or $\emptyset$.

- 6 years, 8 months ago

Having this available on our site will help ensure that everyone on Brilliant is using mathematical terminology in the same way

As Tim has pointed out, whether $0$ is a natural number or not is debated. So just to be on the safe side, I'm changing 'natural numbers' into 'positive integers' [unless you consider $0$ to be a positive integer. Some people do. See here.].

- 6 years, 8 months ago

These things are exactly why this glossary is needed :)

- 6 years, 8 months ago

There is a lot of debate about whether $0$ is a natural number or not.

- 6 years, 8 months ago

Agreed. I actively avoid the word natural number, and instead use positive or non-negative.

As a side note, some cultures consider 0 a positive number.....

Staff - 6 years, 8 months ago

Origin

- 6 years, 8 months ago

The origin is the point $(0, 0)$ in Cartesian coordinates. It is the point where the $x$- and the $y$-axes intersect.

- 6 years, 8 months ago

Root

- 6 years, 8 months ago

The roots (also called zeroes) of a polynomial are those values for which the polynomial is equal to zero.

For example: the roots of the polynomial $x^2-5x+6$ are $2$ and $3$.

- 6 years, 8 months ago

Function used to calculate $x$ when $x^2=a$:

$x=\sqrt{a} \Rightarrow x^2=a$.

- 6 years, 8 months ago

Interval

- 6 years, 8 months ago

An interval is a connected set of real numbers.

If $x$ is in the interval $[a, b]$, then $a\leq x\leq b$.

If $x$ is in the interval $(a, b]$, then $a< x\leq b$.

If $x$ is in the interval $[a, b)$, then $a\leq x< b$.

If $x$ is in the interval $(a, b)$, then $a< x< b$.

- 6 years, 8 months ago

The word "continuous" is wrong here - you mean "connected".

In the real line $\mathbb{R}$, an interval $I$ is characterised by the property that if $a,b \in I$, then every real number between $a$ and $b$ also belongs to $I$. This property copes with open, closed, bounded and unbounded sets.

- 6 years, 8 months ago

I have edited my definition. Could you kindly explain the difference between 'continuous' and 'connected'? I'd be grateful if you do so.

- 6 years, 8 months ago

"Continuous" in mathematics has a particular meaning, and is a property of functions. A function $f$ is continuous if $f(x_n) \to f(x)$ whenever $x_n \to x$ (if it is a map between two metric spaces. A set cannot be continuous.

"Connected" is a property of sets. Its definition is quite subtle, but in easy cases it corresponds to the idea of "all in one piece", which is the idea you were trying to convey by using "continuous". To get technical, a connected set $S$ is one that cannot be written as the union of two disjoint nonempty open (in $S$) sets. Intervals are connected; indeed, they are the only connected subsets of $\mathbb{R}$.

I was pointing out that you were using the technical word "continuous" in a non-technical manner. The aim here is to compile a glossary of technical terms, so we had better be precise.

- 6 years, 8 months ago

Thanks for clarifying!

- 6 years, 8 months ago

Perhaps this should be separated into closed and open intervals?

- 6 years, 8 months ago

- 6 years, 8 months ago

One radian ($1$ rad) is the angle given by measuring one radius along the perimeter of the circle with said radius.

Thus it is $\frac{1}{2 \pi}$ part of a full turn.

- 6 years, 8 months ago

Identities

- 6 years, 8 months ago

An identity is an equation that is true for every possible value of the unknowns. For example, the equation 4x=  x+  x+  x+  x is an identity, but 2x + 3 = 15 is not [true only for $x=6$].

- 6 years, 8 months ago

Inequality

- 6 years, 8 months ago

In mathematics, an inequality is a statement that states that two things are not equal.

The statement “$x$ is less than $y$” is written as x < y and the statement “$x$ is greater than $y$” is written as x > y.

Both of these statements are inequalities because they imply that $x$ and $y$ are not equal to each other.

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Equation

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A statement stating the equivalence of two mathematical expressions by the use of the equal sign "=".

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Expression

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A representation of a value by combining numbers and/or variables with mathematical operations.

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Abscissa and Ordinate

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The first and second entry respectively in an ordered pair.

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Sequence

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An ordered set of numbers, usually defined by some function or iterative process.

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It doesn't have to be numbers though. A sequence of functions comes to mind for algebra.

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Series

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The sum of a sequence.

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Degree

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The power of the term with the highest power. [Example may be needed]

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The angle made by taking $\frac{1}{360}$ part of a full turn.

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Domain and Range

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Domain is the set of numbers on which a function is defined, and the range is the set of values which the domain produces when evaluated with the said function.

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Cartesian Plane

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The standard x-y coordinate plane, where the two axes are perpendicular to each other.

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Arithmetic, Geometric, and Harmonic Mean (not the inequality)

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The four parts of a Cartesian plane, not including the axes; The upper right is the first quadrant, the upper left is the second quadrant, the bottom left is the third quadrant, and the bottom right is the forth quadrant.

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Integer

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Any number with no decimal or fractional part, either a whole number, a negative whole number, or zero. The set of all integers is represented by $\mathbb{Z}$.

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Domain

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The set of input values, or x-values in a relation.

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Range

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The set of output values, or y-values of a relation.

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Positive number

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A real number $x$ is called positive if $x \gt 0$.

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Negative number

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A real number $x$ is called negative if $x \lt 0$.

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Multiset

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A multiset is a collection of objects, where one object can appear more than once.

For example:

{$1,2,3,3,4$} is a multiset consisting of 5 elements.

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Natural number

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Natural numbers are a set of numbers such that

• $1$ is in the set.

• If $n$ is in the set, $(n+1)$ is also in the set.

These numbers are also called positive integers.

[By the way, I propose that the term 'natural numbers' be excluded from the glossary. Use 'positive integers' or 'non-negative integers' in appropriate places. For example, the statement '$2^{2n-1}+1$ is divisible by $3$ for all natural numbers $n$' is confusing because for $n=0$, the statement is incorrect. So to avoid all sorts of confusion it's better to use the term 'positive integers' here.]

- 6 years, 8 months ago

Not using the term 'natural numbers' because we can also describe them non-negative/positive integers is not convincing to me; we can descibe the integers with 'rationals with denominator equal to 1' ;)

Also, changing your statement to '$2^{2n+1}+1$ is divisible by $3$ for all natural numbers $n$' makes it a statement in favor of including 0 ;)

- 6 years, 8 months ago

I'm sorry. But you don't get my point. I'm saying we should avoid using the term 'natural numbers'. Not because we can. Because we can avoid confusion that way.

Do a survey and you'll find out that there's a significant number of people on both sides: people who have used $0$ as a natural number and people who have not. Yes, we can describe the integers with 'rationals with denominator equal to $1$, but what good will that do?

The example I provided was a textbook problem. People who consider $0$ to be a natural number are likely to be a little confused by it.

If you don't include $0$ in the natural numbers, it is still a monoid under the operation of multiplication ;)

Finally, I want to say that I'm neither advocating for nor against $0$ being a natural number. No matter how many reasons you have for $0$ being a natural number, in the end it's all about convention. It doesn't really matter. Using the term 'natural numbers' without being explicitly clear what you mean by it will confuse people.

Using terms like 'non-negative integers' or 'positive integers' can easily solve this problem. That's all I'm saying.

- 6 years, 8 months ago

Point taken, thanks for clarifying :)

We'll have to make sure positive and negative make it to the glossary then!

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A number from the set {$0,1,2,3,4,...$}. This set is denoted by $\mathbb{N}$.

$\mathbb{N}^{*}$ denotes the natural numbers without $0$.

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Sometimes referred to as 'counting numbers'.

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I realise there is a possible discussion about the number $0$. The glossary is meant for these kind of situation as well, so we will have to make a choice here. As the majority of mathematicians includes $0$ here, I propose to do the same.

- 6 years, 8 months ago

Interesting, how so many people have this view. I have always been taught that the natural numbers are the counting numbers, the positive numbers, all integers greater than or equal to one. The natural numbers which you propose I call the whole numbers, and for this reason: Natural numbers are natural; they are supposed to be intuitive. While humans had these counting numbers (greater than or equal to one) for eons, zero did not arise until a few thousand years ago. Zero, the concept of nothing, is a challenging one for most young kids. It is much easier to think of something, as compared to thinking of absolutely nothing, a void, because we never really experience nothing too often. It is hard (usually) to notice nothing. With zero being less intuitive, less natural then, I have never included it in the set of Natural numbers. (I am, however, very explicit when using notation involving natural numbers so that everything is clear.)

Does anyone else agree?

- 6 years, 8 months ago

As the majority of mathematicians includes $0$...

Do you have some sort of statistics? Because almost all the books I've read exclude $0$ from the set of natural numbers. The set example I mentioned above is taken from the 'Dictionary of Mathematical Terms' by Douglas Downing (3rd edition).

Also from Wikipedia:

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers $\left\{1, 2, 3, ...\right\}$, while for others the term designates the non-negative integers $\left\{0, 1, 2, 3, ...\right\}$. The former definition is the traditional one, with the latter definition having first appeared in the 19th century.

Natural numbers are called counting numbers. You can't count with zero. Can you?

In the end, it is a matter of what you prefer. I mean nothing terrible's going to happen if you don't consider $0$ to be a natural number. I'm adding my definition as well.

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I had one cow. That cow was stolen. I now have zero cows.

I made a snowman. A truck crashed into it. I now have half a snowman.

I had a pie. Friends ate a part of it. I now have a quarter of a pie.

The name 'natural numbers' is misleading as it implies that only those numbers arise in nature, which of course is not true. An even more extreme example: in quantum mechanics a lot of the formulas deal with complex numbers, suggesting that even the complex numbers are dealing with nature.

I have no statistics though, but there's a good reason to include 0: it makes the natural numbers a monoid (group without inverse).

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Real number

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Real numbers are those numbers which can be represented by points on a straight line. The set of real numbers includes all rational and irrational numbers.

Note that real numbers are complex numbers with the imaginary part equaling zero.

$2$, $0$, $5.66$, $\pi$, $-\frac{3}{5}$ are all examples of real numbers.

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According to this math program I went to:

An element of an ordered field in which the least upper bound property holds.

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Mod function

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This is more number theory.

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Golden Rule of Algebra

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What you do onto one side of an equation must be done on the other side as well.

I'm not sure if this is the proper term for this, though.

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Bijection

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More of a combinatorics thing.

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GCF

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This is Number Theory.

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The Greatest Common Factor of a set of numbers is the largest number that divides every prior number.

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GCF is also called Greatest Common Divisor (GCD) and Highest Common Factor (HCF).

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Good to point that out. Mathematical terminology is not standardized across the globe, and it is useful to state the different names for a given term.

Staff - 6 years, 8 months ago

LCM

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The Least Common Multiple of a set of numbers is the smallest number into which each of the prior numbers divide.

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This is Number Theory.

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Good Point.

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e=lim{tends to infinity}--->(1+(1/n))^n

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Sharky,

It would be better to put the term, without a definition, at this level. "e" is an appropriate algebra term, but you should let people vote on the term and definition separately.

Staff - 6 years, 8 months ago

Group

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A set $S$ with a law of composition such that the law of composition is associative, there exists an identity 1, and every $a\in S$ has an inverse $a^{-1}$

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We're looking for definitions of basic terms that you'd come across while solving Brilliant problems...

I don't think the definition of a group is something that can be considered as 'basic algebra'. I also think you'd need to know about 'association', 'identity', 'law of composition' to actually understand what your definition is trying to say.

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