Community Project: Brilliant Algebra Terms Glossary

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 $y = x^2 + 4$.

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

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.

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

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

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

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.

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 $4x+xy-5y$.

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

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

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.

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

Represented by $i^2=-1$

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.

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?

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.

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

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

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.

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 ,

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

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.

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

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.

Distinct*

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

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.

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

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.

A polynomial with leading coefficient equaling 1.

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

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 $r=|z|=\sqrt{a^2+b^2}$.

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

...And $q\neq 0$.

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 $-5$.

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.

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

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

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

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

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

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 $3$.

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

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

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 $a< x< b$.

Perhaps this should be separated into closed and open intervals?

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.

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 $x=6$].

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.

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

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

The first and second entry respectively in an ordered pair.

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

The sum of a sequence.

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

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

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.

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

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.

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 $\mathbb{Z}$.

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

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

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

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

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.

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.]

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

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

Sometimes referred to as 'counting numbers'.

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.

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.

According to this math program I went to:

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

This is more number theory.

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.

More of a combinatorics thing.

This is Number Theory.

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

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

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

This is Number Theory.

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.

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 $a^{-1}$