Maths AI terms in Number and algebra

The size of the difference between an approximate value and the exact value, $\mathrm{\mid }{v}_A-v_E\mathrm{\mid }$, measured in the same units as the quantity.

The distance of a real number from zero on the number line, written $\mathrm{\mid }x\mathrm{\mid }$; it equals $x$ when $x\ge 0$ and $-x$ when $x<0$.

An electric current, or the voltage that drives it, that reverses direction periodically and is modelled by a sinusoidal function of time such as $V=A\cos (\omega t+\varphi )$.

The repayment of a loan by equal payments at regular intervals, each of which pays the interest due for that period and repays part of the principal, so that the debt is cleared at the end of the loan term.

A table showing, for each payment of an amortized loan, the payment, the interest charged, the part of the principal repaid and the outstanding balance after the payment.

The constant $\omega$ in a sinusoidal function $A\cos (\omega t+\varphi )$, measured in radians per second, equal to $2\pi f$ where $f$ is the number of cycles per second; the period is $\frac{2\pi }{\omega }$.

The yearly cost of a loan expressed as a single percentage that includes the interest and certain compulsory fees, published by lenders so that different loans can be compared.

A series of equal payments made at equal time intervals, either paid into a fund to build up savings or paid out of a fund to provide an income.

A value used in place of the exact value of a quantity, such as a rounded number, an estimate or a measurement, written $v_A$.

A diagram of the complex plane in which the complex number $a+b\mathrm{i}$ is represented by the point $(a,b)$, with real parts on the horizontal real axis and imaginary parts on the vertical imaginary axis.

The angle $\theta$, measured anticlockwise in radians from the positive real axis, between that axis and the line joining the origin to the point representing $z$ on the Argand diagram; it is written $\arg z$.

The number or expression whose logarithm is taken, $x$ in ${\log }_{a}x$; it must be positive for the logarithm to be defined.

A sequence in which each term after the first is found by adding the same fixed number, the common difference, to the term before it.

The sum of the terms of an arithmetic sequence.

The property that $(AB)C=A(BC)$ for any matrices $A$, $B$ and $C$ whose products are defined, so a product of several matrices can be written $ABC$ without brackets.

A unit of length used for distances in the solar system, defined as exactly 149,597,870,700 m, about the mean distance from the Earth to the Sun.

The number of particles in one mole of a substance, fixed in the SI as exactly $6.02214076\times {10}^{23}$ per mole.

The number that is multiplied by itself in a power, written $a$ in $a^n$; in a logarithm ${\log }_{a}b$, the base $a$ is the number that is raised to a power.

The way a calculator or computer displays a number in standard form, with the letter E (or a small ×10) standing for "times ten to the power", as in 2.88E4 for $2.88\times {10}^4$.

The form $z=a+b\mathrm{i}$ of a complex number, with $a,b\in \mathbb{R}$, in which the number is written as a real part plus a real multiple of $\mathrm{i}$.

The equation $\det (M-\lambda I)=0$, whose solutions are the eigenvalues of the square matrix $M$.

For a square matrix $M$, the polynomial in $\lambda$ given by $\det (M-\lambda I)$; for a 2 × 2 matrix it is a quadratic.

The abbreviation $\mathrm{c}\mathrm{i}\mathrm{s}\theta =\cos \theta +\mathrm{i}\sin \theta$, so that a complex number in polar form is written $r\mathrm{c}\mathrm{i}\mathrm{s}\theta$.

The constant that multiplies a variable, or a power of a variable, in a term of an expression or equation.

The fixed amount $d=u_{n+1}-u_n$ by which each term of an arithmetic sequence differs from the term before it; it can be positive, negative or zero.

The logarithm with base 10, written ${\log }_{10}x$ or simply $\log x$; it is the power to which 10 must be raised to give $x$.

The constant $r$ by which each term of a geometric sequence is multiplied to give the next term, so that $r=\frac{u_{n+1}}{u_n}$ for every $n$.

For a complex number $z=a+b\mathrm{i}$ with $a,b\in \mathbb{R}$, the number $z^{\ast }=a-b\mathrm{i}$, which has the same real part and the opposite imaginary part.

A number of the form $a+b\mathrm{i}$, where $a$ and $b$ are real numbers and ${\mathrm{i}}^2=-1$; the set of all complex numbers is written $\mathbb{C}$.

Interest that is added to the balance at the end of each compounding period, so that in later periods interest is earned on the original amount and on the interest already added.

The length of time between the moments when interest is added to a balance; the number of compounding periods in a year is written $k$.

Two complex numbers of the form $p+q\mathrm{i}$ and $p-q\mathrm{i}$, with $p,q$ real and $q\mathrm{\ne }0$, each the conjugate of the other.

The term of a polynomial that does not contain the variable, equal to the value of the polynomial when the variable is 0.

The limiting case of compound interest in which interest is added at every instant, so that an amount $PV$ at a nominal annual rate $r\mathrm{\%}$ grows to $PV\times {\mathrm{e}}^{rn\mathrm{/}100}$ after $n$ years.

An infinite series whose partial sums $S_n$ approach a finite limit as $n$ increases without bound; that limit is the sum of the series.

A polynomial equation of degree 3, which can be written as $a{x}^3+b{x}^2+cx+d=0$ with $a\mathrm{\ne }0$.

The result that for any integer $n$, $(r(\cos \theta +\mathrm{i}\sin \theta ){)}^n=r^n(\cos n\theta +\mathrm{i}\sin n\theta )$, so a power of a complex number has the modulus raised to that power and the argument multiplied by it.

The number between 0 and 1 by which a quantity is multiplied in each time period when it decreases by a fixed percentage; for a decrease of $p\mathrm{\%}$ it equals $1-\frac{p}{100}$.

The unit of a logarithmic scale of the ratio of two powers or intensities; a sound of intensity $I$ has level $L=10{\log }_{10}\left(\frac{I}{I_0}\right)$ decibels, where $I_0={10}^{-12}$ W/m² is the reference intensity, about the threshold of human hearing.

The number of digits written after the decimal point; rounding to $n$ decimal places keeps $n$ digits after the point.

The highest power of the variable that appears with a non-zero coefficient in the polynomial.

The precision to which a value is given, stated as a number of decimal places, a number of significant figures or a unit of rounding such as the nearest 10 or the nearest cent.

The part of the price of an item paid at the time of purchase from the buyer's own money, so that only the remainder is borrowed.

The fall in the value of an asset over time; it is often modelled as a fixed percentage of the current value lost each year.

A number calculated from the elements of a square matrix $A$, written $\det A$ or $\mathrm{\mid }A\mathrm{\mid }$; for $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ it is $ad-bc$.

A square matrix in which every entry off the leading diagonal is zero.

Writing a square matrix $M$ as $M=PD{P}^{-1}$, where $D$ is a diagonal matrix whose leading-diagonal entries are the eigenvalues of $M$ and $P$ is an invertible matrix whose columns are corresponding eigenvectors.

For the quadratic $a{x}^2+bx+c$ with $a\mathrm{\ne }0$, the number $\mathrm{\Delta }=b^2-4ac$, whose sign determines whether the equation $a{x}^2+bx+c=0$ has two distinct real roots, one repeated real root or two non-real complex roots.

The property that $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$ for matrices whose sums and products are defined.

An infinite series whose partial sums do not approach a finite limit, either because they grow without bound or because they never settle on one value.

The time taken for a quantity that increases by a constant factor to double in size.

The percentage by which a balance actually grows in one year once compounding within the year is included; for a nominal rate $r\mathrm{\%}$ compounded $k$ times a year it equals $({(1+\frac{r}{100k})}^k-1)\times 100\mathrm{\%}$.

A number $\lambda$ for which a square matrix $M$ has a non-zero vector $\mathit{v}$ with $M\mathit{v}=\lambda \mathit{v}$; it is the factor by which $M$ scales its eigenvectors.

A non-zero vector $\mathit{v}$ for which multiplying by a square matrix $M$ gives a scalar multiple of $\mathit{v}$, so that $M\mathit{v}=\lambda \mathit{v}$ for some number $\lambda$.

One of the numbers in a matrix; the element in row $i$ and column $j$ of matrix $A$ is written $a_{ij}$.

A non-singular square matrix used to convert a message, written as columns of numbers, into code by multiplication; its inverse is the decoding matrix that recovers the message.

Two matrices that have the same order and whose corresponding elements are all equal.

Finding an approximate value of a calculation or quantity, usually by rounding the numbers involved to 1 significant figure, in order to judge whether an exact result is reasonable.

The identity ${\mathrm{e}}^{\mathrm{i}\theta }=\cos \theta +\mathrm{i}\sin \theta$, valid for every real $\theta$ measured in radians.

The equation ${\mathrm{e}}^{\mathrm{i}\pi }+1=0$, the case $\theta =\pi$ of Euler's formula.

The irrational constant $\mathrm{e}=2.718281828\dots$, the value that ${(1+\frac{1}{n})}^n$ approaches as $n$ becomes very large; it is the base of the natural logarithm.

The true or accepted value of a quantity, written $v_E$, against which an approximation is compared.

The amount of one currency that is equal in value to one unit of another currency.

The number written as a raised superscript on a base, which, when it is a positive integer $n$, counts how many copies of the base are multiplied together: $a^n=a\times a\times \cdots \times a$ with $n$ factors.

An equation in which the unknown appears in an exponent.

The form $z=r{\mathrm{e}}^{\mathrm{i}\theta }$ of a non-zero complex number, where $r=\mathrm{\mid }z\mathrm{\mid }>0$ is its modulus and $\theta$ (in radians) is an argument; it is also called the Euler form.

A sequence with a last term, so that it has a fixed, countable number of terms.

The value of an investment or a loan at the end of a stated time, after interest has been added.

A formula for $u_n$ in terms of the position $n$, from which any term of a sequence can be calculated directly.

A sequence in which each term after the first is found by multiplying the term before it by the same non-zero constant, called the common ratio.

The sum of the terms of a geometric sequence, $u_1+u_{1}r+u_1{r}^2+\cdots$; the sum of the first $n$ terms is written $S_n$.

The number ${10}^{100}$, a 1 followed by one hundred zeros.

The number ${10}^{\text{googol}}={10}^{{10}^{100}}$, a 1 followed by a googol of zeros.

The number greater than 1 by which a quantity is multiplied in each time period when it increases by a fixed percentage; for an increase of $p\mathrm{\%}$ it equals $1+\frac{p}{100}$.

The time taken for a quantity that decreases by a constant factor to fall to half of its value.

A square matrix with every element on the leading diagonal equal to 1 and every other element equal to 0, written $I$; for any square matrix $A$ of the same order, $AI=IA=A$.

For a complex number $z=a+b\mathrm{i}$ with $a,b\in \mathbb{R}$, the real number $b$ (the coefficient of $\mathrm{i}$), written $\mathrm{Im}(z)$.

The number $\mathrm{i}$ defined by ${\mathrm{i}}^2=-1$, so that $\mathrm{i}=\sqrt{-1}$.

A sequence that continues without end, so that every term is followed by another term.

The sum of the terms of an infinite sequence, $u_1+u_2+u_3+\cdots$, written ${\sum }_{k=1}^{\mathrm{\infty }}{u}_k$.

The percentage increase in the general level of prices over a year, usually measured by the change in the price of a fixed basket of goods and services.

A whole number that may be positive, negative or zero; the set of integers is written $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$.

The money paid by a borrower to a lender, or earned by a saver, for the use of money over a period of time.

For a square matrix $A$, the matrix $A^{-1}$ of the same order such that $A{A}^{-1}=A^{-1}A=I$.

A real number that cannot be written as a fraction of two integers; its decimal expansion neither terminates nor repeats.

The rules for calculating with powers: $a^m\times a^n=a^{m+n}$, $a^m÷a^n=a^{m-n}$, $(a^m{)}^n=a^{mn}$, $(ab{)}^n=a^n{b}^n$ and ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$, together with $a^0=1$ and $a^{-n}=\frac{1}{a^n}$ for $a\mathrm{\ne }0$.

The three rules ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$, ${\log }_a\frac{x}{y}={\log }_{a}x-{\log }_{a}y$ and ${\log }_a{x}^m=m{\log }_{a}x$, valid for $a,x,y>0$ and $a\mathrm{\ne }1$, which follow from the laws of exponents.

The elements of a square matrix whose row number equals their column number, running from the top left to the bottom right.

An equation in which every term is either a constant or a constant multiplied by a single variable raised to the power 1.

An amount of money borrowed from a lender, such as a bank, that the borrower agrees to repay with interest under agreed conditions.

The length of time over which a loan is to be fully repaid, usually given in years.

For a base $a>0$, $a\mathrm{\ne }1$, and a positive number $b$, the logarithm ${\log }_{a}b$ is the exponent to which $a$ must be raised to give $b$; so $x={\log }_{a}b$ means exactly the same as $a^x=b$.

An equation in which the unknown appears in the argument of a logarithm.

The results ${\log }_{a}1=0$, ${\log }_{a}a=1$, ${\log }_a(a^x)=x$ and $a^{{\log }_{a}x}=x$ (for $x>0$), which hold for every base $a>0$, $a\mathrm{\ne }1$ and follow directly from the definition of a logarithm.

A scale on which equal distances represent multiplication by the same factor, so that a value is placed according to its logarithm (usually its common logarithm) rather than its size.

The smallest value that a rounded number could have had before it was rounded; for a value rounded to the nearest unit $u$, it is the rounded value minus $\frac{u}{2}$.

The number $a$ in a number written in standard form $a\times {10}^k$, with $1\le a<10$.

A rectangular array of numbers arranged in rows and columns, enclosed in brackets and treated as a single object.

A system of linear equations written as $A\mathit{x}=\mathit{b}$, where $A$ is the matrix of coefficients, $\mathit{x}$ is the column of unknowns and $\mathit{b}$ is the column of constants.

The product $AB$ of an $m\times n$ matrix $A$ and an $n\times p$ matrix $B$ is the $m\times p$ matrix whose element in row $i$, column $j$ is the sum of the products of the elements of row $i$ of $A$ with the corresponding elements of column $j$ of $B$.

The difference between a measured value and the true value of a quantity, caused by the limited precision of the measuring instrument, rounding of the reading, or the way the measurement is made.

The distance of the point representing $z=a+b\mathrm{i}$ from the origin of the Argand diagram, written $\mathrm{\mid }z\mathrm{\mid }$ and equal to $\sqrt{a^2+b^2}$.

A long-term loan used to buy property, in which the property is security for the loan and can be repossessed by the lender if the payments are not made.

The logarithm with base e, written $\ln x$, so that $\ln x={\log }_{\mathrm{e}}x$ is the power to which e must be raised to give $x$.

One of the counting numbers 0, 1, 2, 3, …; the set of natural numbers is written $\mathbb{N}$.

An exponent less than zero, defined by $a^{-n}=\frac{1}{a^n}$ for $a\mathrm{\ne }0$, so that a negative power is the reciprocal of the corresponding positive power.

The yearly interest rate $r\mathrm{\%}$ as quoted, before it is divided by the number of times interest is added in a year.

The property that the order of the factors in a matrix product matters, so that in general $AB\mathrm{\ne }BA$.

A square matrix whose determinant is not 0, so that it has an inverse.

For a positive integer $n$, the $n$th root of a number $a$, written $\sqrt[n]{a}$, is the number whose $n$th power equals $a$; when $n$ is even it is taken as the non-negative root and exists only for $a\ge 0$.

The number of rows and the number of columns of a matrix, written $m\times n$ for a matrix with $m$ rows and $n$ columns.

The power of ten that describes the size of a quantity, usually the exponent $k$ when the quantity is written in standard form $a\times {10}^k$; two quantities differ by $n$ orders of magnitude when one is about ${10}^n$ times the other.

An annuity in which each payment is made at the end of a payment period.

The amount of a loan still owed at a given time, after the payments made so far.

The sum $S_n=u_1+u_2+\cdots +u_n$ of the first $n$ terms of a sequence.

The number of regular payments made in one year, entered as $P\mathrm{/}Y$ in the TVM solver; 12 for monthly payments, 4 for quarterly, 1 for annual.

The change in a quantity expressed as a percentage of its original value, $\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100\mathrm{\%}$, positive for an increase and negative for a decrease.

The absolute error of an approximation expressed as a percentage of the exact value, $\epsilon =\mid \frac{v_A-v_E}{v_E}\mid \times 100\mathrm{\%}$.

A logarithmic measure of the acidity of a solution, given by $\text{pH}=-{\log }_{10}[{\mathrm{H}}^{+}]$, where $[{\mathrm{H}}^{+}]$ is the hydrogen-ion concentration in moles per litre.

The complex number $A{\mathrm{e}}^{\mathrm{i}\varphi }$ that represents the sinusoid $A\cos (\omega t+\varphi )$ of a known angular frequency $\omega$, with modulus equal to the amplitude and argument equal to the phase angle $\varphi$.

The form $z=r(\cos \theta +\mathrm{i}\sin \theta )$ of a non-zero complex number, where $r=\mathrm{\mid }z\mathrm{\mid }>0$ is its modulus and $\theta$ is an argument of $z$.

An expression in one variable made of one or more terms, each a constant multiplied by a non-negative integer power of the variable, such as $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$.

An equation that can be rearranged into the form $p(x)=0$, where $p(x)$ is a polynomial; its degree is the degree of $p(x)$.

For $a,x>0$ with $a\mathrm{\ne }1$ and any real number $m$, ${\log }_a(x^m)=m{\log }_{a}x$; the logarithm of a power equals the exponent multiplied by the logarithm of the base.

Rounding a value in the middle of a calculation and then using the rounded value in later steps, which can make the final answer inaccurate.

The value of money at the start of a financial calculation, such as the amount invested or borrowed now.

The amount of money originally invested or borrowed, before any interest is added.

The value of the argument of a non-zero complex number that lies in the interval $-\pi <\theta \le \pi$.

For $a,x,y>0$ with $a\mathrm{\ne }1$, ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$; the logarithm of a product equals the sum of the logarithms of its factors.

A complex number of the form $b\mathrm{i}$, with $b$ real and $b\mathrm{\ne }0$, whose real part is zero.

A polynomial equation of degree 2, which can be written as $a{x}^2+bx+c=0$ with $a\mathrm{\ne }0$.

The formula $x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$ for the solutions of the quadratic equation $a{x}^2+bx+c=0$, where $a\mathrm{\ne }0$.

For $a,x,y>0$ with $a\mathrm{\ne }1$, ${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y$; the logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.

An exponent that is a fraction $\frac{m}{n}$, with $m$ and $n$ integers and $n>0$; for $a>0$, $a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m=\sqrt[n]{a^m}$.

A number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q\mathrm{\ne }0$; the set of rational numbers is written $\mathbb{Q}$.

Rewriting a fraction with a surd in its denominator as an equal fraction whose denominator is a rational number, by multiplying the numerator and denominator by a suitable expression.

The rate at which the buying power of an investment grows once inflation is taken into account; it is approximately the nominal interest rate minus the inflation rate.

Any number that can be represented as a point on the number line, rational or irrational; the set of real numbers is written $\mathbb{R}$.

For a complex number $z=a+b\mathrm{i}$ with $a,b\in \mathbb{R}$, the real number $a$, written $\mathrm{Re}(z)$.

The value of a future amount of money measured in the prices of an earlier time (usually the start of the investment), so that it shows what the money can actually buy.

The reciprocal of a non-zero number $x$ is $\frac{1}{x}$, the number that gives 1 when multiplied by $x$.

The fixed amount paid at equal time intervals to repay a loan or to build up or draw from an annuity, written $PMT$ in the TVM solver.

A root of a polynomial equation whose factor $(x-r)$ appears more than once in the factorised polynomial; a root that appears twice is called a double root.

A logarithmic scale of earthquake magnitude, $M={\log }_{10}\left(\frac{A}{A_0}\right)$, where $A$ is the maximum amplitude recorded by a seismograph and $A_0$ is a fixed reference amplitude, so that each increase of 1 in magnitude is a tenfold increase in measured amplitude.

A value of the unknown that makes an equation true; the roots of $p(x)=0$ are its solutions.

Replacing a number by the closest value that has a stated accuracy, such as a given number of decimal places or significant figures.

Multiplying a matrix by a number $k$, which multiplies every element of the matrix by $k$; the result $kA$ has the same order as $A$.

An ordered list of numbers, in which each number has a definite position (first, second, third and so on).

The sum of the terms of a sequence, written with plus signs between the terms.

A unit of the International System of Units, the internationally agreed system of measurement whose base units include the metre (length), the kilogram (mass) and the second (time).

A way of writing a sum as ${\sum }_{k=a}^b{u}_k$, meaning the sum of the terms $u_k$ for every whole number $k$ from $a$ to $b$ inclusive.

The rule in a finance solver that money paid out by the person whose point of view is taken is entered as a negative number, and money received by them as a positive number.

The digits of a number that carry information about its size and precision, counted from the first non-zero digit.

Interest calculated only on the original principal, so that the same amount is added in every period; on a principal $C$ at $r\mathrm{\%}$ a year, the total interest over $n$ years is $I=\frac{Crn}{100}$.

A square matrix whose determinant is 0, so that it has no inverse.

A matrix with the same number of rows as columns, so its order is $n\times n$.

A way of writing a number as $a\times {10}^k$, where $1\le a<10$ and $k$ is an integer.

The limit of the partial sums $S_n$ of an infinite series as $n$ increases without bound, when that limit exists; for a geometric series with $\mathrm{\mid }r\mathrm{\mid }<1$ it is $S_{\mathrm{\infty }}=\frac{u_1}{1-r}$.

An irrational root of a positive rational number, written in root form so that its value stays exact.

A set of two or more linear equations in the same unknowns, all of which must be satisfied by the same values at the same time.

One of the numbers in a sequence, written $u_n$ where the whole number $n$ is its position in the sequence.

A calculator or software tool that links the number of periods, the interest rate, the present value, the regular payment and the future value of an investment or loan, and finds any one of them from the others.

The difference between the total amount repaid on a loan and the amount borrowed; equivalently, the total cost of an item bought with a loan minus its price.

The sum of the entries on the leading diagonal of a square matrix; for $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ the trace is $a+d$.

Shortening a number by removing all digits beyond a chosen position, without rounding; a positive number is never increased by truncation.

A single set of values of the unknowns that satisfies every equation in a system, with no other set of values doing so.

The boundary above which a number would no longer round to the recorded value; for a value rounded to the nearest unit $u$, it is the rounded value plus $\frac{u}{2}$, and the true value is strictly less than it.

An input of a function at which the output is 0; the zeros of $f$ are the roots of $f(x)=0$ and the $x$-coordinates of the points where the graph of $y=f(x)$ meets the $x$-axis.

The rule that any non-zero number raised to the power 0 equals 1, so $a^0=1$ for $a\mathrm{\ne }0$.

A matrix in which every element is 0, written $0$ (or $O$); it can have any order.