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
5 years, 10 months ago

34 votes

  Easy Math Editor

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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!

Calvin Lin Staff - 5 years, 10 months ago

Variable

Arron Kau Staff - 5 years, 10 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 x and y y are variables in the expression y=x2+4 y = x^2 + 4 .

Arron Kau Staff - 5 years, 10 months ago

Complex Number

Bob Krueger - 5 years, 10 months ago

A number in the form a+bia+bi, where aa and bb are real numbers and ii is the imaginary unit.

Bob Krueger - 5 years, 10 months ago

For example: 3+4i3+4i.

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

For example: 55 is a complex number of the form 5+0i5+0i.

Mursalin Habib - 5 years, 10 months ago

Note that RC\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.

Michael Tong - 5 years, 10 months ago

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

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

Calvin Lin Staff - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

@Bob Krueger 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).

Jonathan Wong - 5 years, 10 months ago

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

C\mathbb{C}^{*} would be the complex numbers without 00.

Ton de Moree - 5 years, 10 months ago

Absolute Value

Bob Krueger - 5 years, 10 months ago

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

In other words a+bi=a2+b2|a+bi|=\sqrt{a^2+b^2}.

Mursalin Habib - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

The absolute value of a real number x,x, denoted x,|x|, is defined by x={x if x0x if x<0.|x| = \begin{cases} x &\text{ if }x \ge 0 \\ -x & \text{ if } x < 0. \end{cases}

Michael Tang - 5 years, 10 months ago

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

Jonathan Wong - 5 years, 10 months ago

@Jonathan Wong 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.

Bob Krueger - 5 years, 10 months ago

Indices and Surds

Bob Krueger - 5 years, 10 months ago

Floor

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Polynomial

Arron Kau Staff - 5 years, 10 months ago

A polynomial in xx 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)=x2+x+1P(x)=x^2+x+1 is a polynomial in xx because the indices are non-negative integers. So, is P(x)=5P(x)=5, because the index of xx is 00 [a non-negative integer]. However P(x)=x+5xP(x)=\sqrt{x}+\frac{5}{x} is not a polynomial in xx because the indices of xx are 12\frac{1}{2} and1-1 respectively, neither of which is a non-negative integer.

Mursalin Habib - 5 years, 10 months ago

Coefficient

Arron Kau Staff - 5 years, 10 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+xy5y4x+xy-5y.

Jonathan Wong - 5 years, 10 months ago

Ceiling

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

Exactly, floor doesn't need much explanation.

Edward Lemur - 5 years, 10 months ago

Relatively Prime, Coprime

Bob Krueger - 5 years, 10 months ago

Two integers x,yx,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 11. Equivalently, their greatest common denominator gcd(a,b)=1\gcd(a,b)=1, and no prime number divides both aa and bb. Example: 66 and 3535 are coprime, while 1515 and 3535 are not because 55 divides them both.

Jonathan Wong - 5 years, 10 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 xx and yy 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 xx and yy are relatively prime, mutually prime, or coprime, if and only if ....

Calvin Lin Staff - 5 years, 10 months ago

What would be meant by the statement: "xx,yy and zz are coprime"?

Are the numbers 66,1414, 2121 coprime since gcd(6,14,21)=1gcd(6,14,21)=1? Or are they not coprime since gcd(6,14)1gcd(6,14) \neq 1?

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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.

Bob Krueger - 5 years, 10 months ago

Imaginary Unit

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

Represented by i2=1i^2=-1

Christopher Boo - 5 years, 10 months ago

This definition may be a bit confusing, because is the imaginary unit ii or i2i^2? You could say "The imaginary unit is represented by ii, such that i2=1i^2=-1." Or you could just you my shorter definition.

Bob Krueger - 5 years, 10 months ago

@Bob Krueger Note that your definitions are not equivalent.

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

Ton de Moree - 5 years, 10 months ago

@Ton de Moree Actually, my definition is the better one. For the statement "i2=1i^2 = -1" has two solutions for ii, yet we only assign one value to ii, that being the positive square root of negative one.

Bob Krueger - 5 years, 10 months ago

@Bob Krueger i2=1i^2=-1 is THE definition of ii.

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 ii as the square root of 1-1, but that's not how ii was defined.

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

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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 i2i^2 [Just read the term and the definition at once. Imaginary unit: represented by i2=1i^2=-1. That is confusing, at least to me.]

However, you are right. i2=1i^2=-1 is THE definition of ii. So my definition [which is the same as Bob's latter definition] would be:

The imaginary unit is represented by ii such that i2=1i^2=-1.

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib This would be my preference. No roots :)

Ton de Moree - 5 years, 10 months ago

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

Christopher Boo - 5 years, 10 months ago

Tuple

Bob Krueger - 5 years, 10 months ago

A tuple is an ordered list of elements.

(2,3,5,7,11)(2, 3, 5, 7, 11) is an example of a 55-tuple.

A 22-tuple is generally called an ordered pair while 33-tuples and 44-tuples are called ordered triples and quadruples respectively.

Mursalin Habib - 5 years, 10 months ago

And a 33-tuple an ordered triple.

Michael Tang - 5 years, 10 months ago

@Michael Tang Added!

Mursalin Habib - 5 years, 10 months ago

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

Editted to 3-tuples

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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.

Bob Krueger - 5 years, 10 months ago

@Bob Krueger 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)(2,7,3,11,5) is an example of a 55-tuple.

Ton de Moree - 5 years, 10 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) (1,2) and (2,1) (2,1) are 2 different ordered pairs?

Calvin Lin Staff - 5 years, 10 months ago

A kk-tuple is an ordered set with kk elements. In math, this notation is often preferred when talking about an arbitrary kk, or when standard naming becomes onerous, i.e. we all know "pair" or "triple" but calling an ordered set of 66 elements a "sextuple" and so forth can be inconvenient and unnecessary.

Michael Tong - 5 years, 10 months ago

Constant function

Mursalin Habib - 5 years, 10 months ago

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

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

Mursalin Habib - 5 years, 10 months ago

Conjugate

Justin Wong - 5 years, 10 months ago

The conjugate of a binomial is the binomial obtained by multiplying the second term by 1-1. Example: the conjugates of x+2,x+3,x+2,x+\sqrt{3}, and 1+4i-1+4i are x2,x3,x-2,x-\sqrt{3}, and 14i-1-4i, respectively.


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

Jonathan Wong - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

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

Jonathan Wong - 5 years, 10 months ago

@Jonathan Wong A real number x+2x+2 can be written as a complex number (x+2)+0i(x+2)+0*i. The latter is a binomial with terms (x+2)(x+2) and 0i0*i.

Is the conjugate x2x-2 or (x+2)0i(x+2)-0*i?

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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.

Jonathan Wong - 5 years, 10 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+4i3 + 4i and 34i3 - 4i are complex conjugates.

The conjugate of the complex number ZZ

Z=a+biZ=a+bi

where aa and bb are real numbers, is

Z=abi\overline{Z} = a - bi

For Example ,

105i=10+5i\overline{10-5i} = 10+5i

Gabriel Merces - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

@Bob Krueger Opposite Signs

Gabriel Merces - 5 years, 10 months ago

Extraneous Solution

Justin Wong - 5 years, 10 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 xx: 2x+7+3=0\sqrt{2x+7}+3=0.

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

2x+7=92x+7=9.

So, xx should be equal to 11. However, it is not.

Plug x=1x=1 in the original equation to get 3+3=03+3=0. Actually this equation has no solution and 11 is an extraneous root which emerged because we squared the original equation.

Mursalin Habib - 5 years, 10 months ago

Discriminant

Brilliant Member - 5 years, 10 months ago

Discriminant is the number Δ=b24ac\Delta=b^2-4ac for the quadratic equation ax2+bx+c=0ax^2+bx+c=0.

Christopher Boo - 5 years, 10 months ago

By the way, the term 'discriminant' isn't used only for quadratic polynomials. For example: the discriminant of the polynomial ax3+bx2+cx+dax^3+bx^2+cx+d [a0a\neq 0] is b2c24ac34b3d27a2d2+18abcdb^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.

Mursalin Habib - 5 years, 10 months ago

The discriminant of a polynomial, normally denoted by Δ\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. Δ<0\Delta<0], the polynomial has no real root.

Mursalin Habib - 5 years, 10 months ago

Distinct

Ton de Moree - 5 years, 10 months ago

Distinct*

Brilliant Member - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

For example the set {1,2,31, 2, 3} consists of distinct numbers, while the multiset {1,2,11,2,1} does not.

Ton de Moree - 5 years, 10 months ago

@Ton de Moree Note:

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

Ton de Moree - 5 years, 10 months ago

Arithmetic Sequence

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Geometric Sequence

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Function

Bob Krueger - 5 years, 10 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.

Bob Krueger - 5 years, 10 months ago

Summation

Bob Krueger - 5 years, 10 months ago

Monic Polynomial

Bob Krueger - 5 years, 10 months ago

A polynomial with leading coefficient equaling 1.

Bob Krueger - 5 years, 10 months ago

Recurrence Relation

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

Example: the Fibonacci sequence is defined by Fn=Fn1+Fn2,F0=0,F1=1F_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,...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.

Jonathan Wong - 5 years, 10 months ago

@Jonathan Wong Done.

Bob Krueger - 5 years, 10 months ago

Argument

Bob Krueger - 5 years, 10 months ago

  1. Of a function: an input variable. For f(x,y)=x2+y2f(x,y)=x^2+y^2, xx and yy are arguments.
  2. Of a complex number: on the complex plane, the angle θ=arg(z) \theta = \arg(z) between the vector representing the complex number z=a+biz=a+bi and the positive real axis, the principal value of which is taken to be the (unique) value of θ\theta such that π<θπ-\pi < \theta \leq \pi. Also the angle for which z=rcos(θ)+irsin(θ)=reiθz=r\cos(\theta )+ir\sin(\theta )=re^{i\theta }, where r=z=a2+b2r=|z|=\sqrt{a^2+b^2}.

Jonathan Wong - 5 years, 10 months ago

Rational

Bob Krueger - 5 years, 10 months ago

Any number that can be written as a fraction pq\frac{p}{q}, where pp and qq are both integers and q0q \neq 0

Bob Krueger - 5 years, 10 months ago

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

Michael Tang - 5 years, 10 months ago

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

Q\mathbb{Q}^{*} denotes the rational numbers without 00.

Ton de Moree - 5 years, 10 months ago

...And q0q\neq 0.

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib 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.

Bob Krueger - 5 years, 10 months ago

Leading Coefficient

Justin Wong - 5 years, 10 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)=3x25x4+3P(x)=3x^2-5x^4+3, the term with the highest degree is 5x4-5x^4. So the leading coefficient is 5-5.

Mursalin Habib - 5 years, 10 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)=3x25x4+3 P(x) = 3x^2 - 5x^4 + 3 .

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin Edited!

Mursalin Habib - 5 years, 10 months ago

Injective Function

Shendy Marcello Yuniar - 5 years, 10 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.

Shendy Marcello Yuniar - 5 years, 10 months ago

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

Michael Tang - 5 years, 10 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)a=bf(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:RRf: \mathbb{R}^{*} \rightarrow \mathbb{R} defined by f(x)=1xf(x)=\frac{1}{x} is an injective, but not surjective function as there is no element aa so that f(a)=0f(a)=0.

Ton de Moree - 5 years, 10 months ago

Surjective Function

Shendy Marcello Yuniar - 5 years, 10 months ago

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

Shendy Marcello Yuniar - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

@Bob Krueger 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:RR f: \mathbb{R} \rightarrow \mathbb{R} where f(x)=x2 f(x) = x^2 , the codomain would be the set of all real numbers, but the range would only contain values 0 \geq 0 .

Arron Kau Staff - 5 years, 10 months ago

For example f:RR+f:\mathbb{R} \rightarrow \mathbb{R}^{+} \cup {00} defined by f(x)=xf(x)=|x| is a surjective but not injective function.

Ton de Moree - 5 years, 10 months ago

Bijective Function

Shendy Marcello Yuniar - 5 years, 10 months ago

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

Shendy Marcello Yuniar - 5 years, 10 months ago

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

Michael Tang - 5 years, 10 months ago

Also known as a one-to-one function.

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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.

Calvin Lin Staff - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Logarithm

Mursalin Habib - 5 years, 10 months ago

If x=ayx=a^y, then yy is the logarithm of xx to the base aa and it is denoted as y=logaxy=\log_a x

Mursalin Habib - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Set

Mursalin Habib - 5 years, 10 months ago

A set is a well-defined group of objects. For example, the set of all positive integers less than 1111 consists of {1,2,3,4,5,6,7,8,9,10}\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}.

Mursalin Habib - 5 years, 10 months ago

A remark:

00 is a natural number.

A note:

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

Ton de Moree - 5 years, 10 months ago

@Ton de Moree

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 00 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 00 to be a positive integer. Some people do. See here.].

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib These things are exactly why this glossary is needed :)

Ton de Moree - 5 years, 10 months ago

@Ton de Moree There is a lot of debate about whether 00 is a natural number or not.

Tim Vermeulen - 5 years, 10 months ago

@Tim Vermeulen 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.....

Calvin Lin Staff - 5 years, 10 months ago

Origin

Mursalin Habib - 5 years, 10 months ago

The origin is the point (0,0)(0, 0) in Cartesian coordinates. It is the point where the xx- and the yy-axes intersect.

Mursalin Habib - 5 years, 10 months ago

Root

Mursalin Habib - 5 years, 10 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 x25x+6x^2-5x+6 are 22 and 33.

Mursalin Habib - 5 years, 10 months ago

Function used to calculate xx when x2=ax^2=a:

x=ax2=ax=\sqrt{a} \Rightarrow x^2=a.

Ton de Moree - 5 years, 10 months ago

Interval

Mursalin Habib - 5 years, 10 months ago

An interval is a connected set of real numbers.

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

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

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

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

Mursalin Habib - 5 years, 10 months ago

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

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

Mark Hennings - 5 years, 10 months ago

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

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib "Continuous" in mathematics has a particular meaning, and is a property of functions. A function ff is continuous if f(xn)f(x)f(x_n) \to f(x) whenever xnxx_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 SS is one that cannot be written as the union of two disjoint nonempty open (in SS) sets. Intervals are connected; indeed, they are the only connected subsets of R\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.

Mark Hennings - 5 years, 10 months ago

@Mark Hennings Thanks for clarifying!

Mursalin Habib - 5 years, 10 months ago

Perhaps this should be separated into closed and open intervals?

Bob Krueger - 5 years, 10 months ago

Radian measure of an angle

Snehal Shekatkar - 5 years, 10 months ago

One radian (11 rad) is the angle given by measuring one radius along the perimeter of the circle with said radius.

Thus it is 12π\frac{1}{2 \pi} part of a full turn.

Ton de Moree - 5 years, 10 months ago

Identities

Brilliant Member - 5 years, 10 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=6x=6].

Mursalin Habib - 5 years, 10 months ago

Inequality

Ahaan Rungta - 5 years, 10 months ago

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

The statement “xx is less than yy” is written as x < y and the statement “xx is greater than yy” is written as x > y.

Both of these statements are inequalities because they imply that xx and yy are not equal to each other.

Mursalin Habib - 5 years, 10 months ago

Equation

Mindren Lu - 5 years, 10 months ago

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

Mindren Lu - 5 years, 10 months ago

Expression

Mindren Lu - 5 years, 10 months ago

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

Mindren Lu - 5 years, 10 months ago

Abscissa and Ordinate

Bob Krueger - 5 years, 10 months ago

The first and second entry respectively in an ordered pair.

Bob Krueger - 5 years, 10 months ago

Sequence

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Series

Bob Krueger - 5 years, 10 months ago

The sum of a sequence.

Bob Krueger - 5 years, 10 months ago

Degree

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

The angle made by taking 1360\frac{1}{360} part of a full turn.

Ton de Moree - 5 years, 10 months ago

Domain and Range

Bob Krueger - 5 years, 10 months ago

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.

Bob Krueger - 5 years, 10 months ago

Cartesian Plane

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

Quadrants

Bob Krueger - 5 years, 10 months ago

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.

Bob Krueger - 5 years, 10 months ago

Integer

Bob Krueger - 5 years, 10 months ago

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 Z\mathbb{Z}.

Bob Krueger - 5 years, 10 months ago

Domain

Justin Wong - 5 years, 10 months ago

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

Justin Wong - 5 years, 10 months ago

Range

Justin Wong - 5 years, 10 months ago

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

Justin Wong - 5 years, 10 months ago

Positive number

Ton de Moree - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Negative number

Ton de Moree - 5 years, 10 months ago

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

Ton de Moree - 5 years, 10 months ago

Multiset

Ton de Moree - 5 years, 10 months ago

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

For example:

{1,2,3,3,41,2,3,3,4} is a multiset consisting of 5 elements.

Ton de Moree - 5 years, 10 months ago

Natural number

Ton de Moree - 5 years, 10 months ago

Natural numbers are a set of numbers such that

  • 11 is in the set.

  • If nn is in the set, (n+1)(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 '22n1+12^{2n-1}+1 is divisible by 33 for all natural numbers nn' is confusing because for n=0n=0, the statement is incorrect. So to avoid all sorts of confusion it's better to use the term 'positive integers' here.]

Mursalin Habib - 5 years, 10 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 '22n+1+12^{2n+1}+1 is divisible by 33 for all natural numbers nn' makes it a statement in favor of including 0 ;)

Ton de Moree - 5 years, 10 months ago

@Ton de Moree 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 00 as a natural number and people who have not. Yes, we can describe the integers with 'rationals with denominator equal to 11, but what good will that do?

The example I provided was a textbook problem. People who consider 00 to be a natural number are likely to be a little confused by it.

If you don't include 00 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 00 being a natural number. No matter how many reasons you have for 00 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.

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib Point taken, thanks for clarifying :)

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

Ton de Moree - 5 years, 10 months ago

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

N\mathbb{N}^{*} denotes the natural numbers without 00.

Ton de Moree - 5 years, 10 months ago

Sometimes referred to as 'counting numbers'.

Ton de Moree - 5 years, 10 months ago

I realise there is a possible discussion about the number 00. 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 00 here, I propose to do the same.

Ton de Moree - 5 years, 10 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?

Bob Krueger - 5 years, 10 months ago

As the majority of mathematicians includes 00...

Do you have some sort of statistics? Because almost all the books I've read exclude 00 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 {1,2,3,...}\left\{1, 2, 3, ...\right\}, while for others the term designates the non-negative integers {0,1,2,3,...}\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 00 to be a natural number. I'm adding my definition as well.

Mursalin Habib - 5 years, 10 months ago

@Mursalin Habib 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).

Ton de Moree - 5 years, 10 months ago

Real number

Ton de Moree - 5 years, 10 months ago

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.

22, 00, 5.665.66, π\pi, 35-\frac{3}{5} are all examples of real numbers.

Mursalin Habib - 5 years, 10 months ago

According to this math program I went to:

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

Mindren Lu - 5 years, 10 months ago

Mod function

Shubham Srivastava - 5 years, 10 months ago

This is more number theory.

Bob Krueger - 5 years, 10 months ago

Golden Rule of Algebra

Mindren Lu - 5 years, 10 months ago

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.

Mindren Lu - 5 years, 10 months ago

Bijection

Fan Zhang - 5 years, 10 months ago

More of a combinatorics thing.

Bob Krueger - 5 years, 10 months ago

GCF

Bob Krueger - 5 years, 10 months ago

This is Number Theory.

Zi Song Yeoh - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

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

Brilliant Member - 5 years, 10 months ago

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.

Calvin Lin Staff - 5 years, 10 months ago

LCM

Bob Krueger - 5 years, 10 months ago

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

Bob Krueger - 5 years, 10 months ago

This is Number Theory.

Zi Song Yeoh - 5 years, 10 months ago

Good Point.

Bob Krueger - 5 years, 10 months ago

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

Sharky Kesa - 5 years, 10 months ago

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.

Arron Kau Staff - 5 years, 10 months ago

Group

Taehyung Kim - 5 years, 10 months ago

A set SS with a law of composition such that the law of composition is associative, there exists an identity 1, and every aSa\in S has an inverse a1a^{-1}

Taehyung Kim - 5 years, 10 months ago

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.

Mursalin Habib - 5 years, 10 months ago
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