Maths AA terms in Number and algebra

If a task can be done in one of several separate cases that cannot happen together, with $m$ ways in one case and $n$ ways in the other, then the task can be done in $m+n$ ways.

A plane with a horizontal real axis (Re) and a vertical imaginary axis (Im), on which the complex number $a+b\mathrm{i}$ is shown as the point $(a,b)$ or as the vector from the origin to that point.

The angle $\arg z$, in radians, from the positive real axis anticlockwise to the line joining the origin to $z$. The principal argument satisfies $-\pi <\arg z\le \pi$; $\arg 0$ is undefined.

A sequence in which each term after the first is found by adding the same number, the common difference $d$, to the previous term.

The sum of the terms of an arithmetic sequence. The sum of the first $n$ terms is $S_n=\frac{n}{2}(u_1+u_n)$.

Solving a system in triangular form by finding the unknown in the last equation first, then substituting it into the equations above it in turn.

The number that is raised to an exponent. In $a^n$ the base is $a$; in ${\log }_{a}b$ the base $a$ is the number being raised to a power, with $a>0$ and $a\mathrm{\ne }1$.

The first step of a proof by induction: a direct check that the statement $P(n)$ is true for the smallest value of $n$ it claims, usually $n=1$.

An algebraic expression with exactly two terms, such as $a+b$, $2x-3$ or $x^2+\frac{1}{x}$.

The number $\left(\genfrac{}{}{0px}{}{n}{r}\right)={}^n{C}_r=\frac{n!}{r!(n-r)!}$: the coefficient of $a^{n-r}{b}^r$ in the expansion of $(a+b{)}^n$, and the number of ways to choose $r$ objects from $n$.

The result $(a+b{)}^n={\sum }_{r=0}^n\left(\genfrac{}{}{0px}{}{n}{r}\right)a^{n-r}{b}^r$ for $n\in \mathbb{N}$, which expands any power of a binomial without multiplying out the brackets.

The way of writing a complex number as $z=a+b\mathrm{i}$ with $a,b\in \mathbb{R}$, giving its real and imaginary parts. It corresponds to the point $(a,b)$ on an Argand diagram.

The rule ${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$, which rewrites a logarithm in any base $a$ in terms of logarithms in another base $b$, usually $10$ or $e$.

A selection of objects in which order does not matter. The number of combinations of $r$ objects chosen from $n$ different objects is ${}^n{C}_r=\frac{n!}{r!(n-r)!}$.

The fixed amount $d=u_{n+1}-u_n$ added to each term of an arithmetic sequence to get the next term. It can be positive, negative or zero.

A logarithm with base $10$. In IB notation $\log x$ with no base written means ${\log }_{10}x$, so $\log 1000=3$ because ${10}^3=1000$.

The fixed number $r=\frac{u_{n+1}}{u_n}$ that each term of a geometric sequence is multiplied by to give the next term.

For $z=a+b\mathrm{i}$, the number $z^{\ast }=a-b\mathrm{i}$, with the sign of the imaginary part changed. On an Argand diagram it is the reflection of $z$ in the real axis, and $z{z}^{\ast }=a^2+b^2$ is real.

A number of the form $z=a+b\mathrm{i}$, where $a,b\in \mathbb{R}$ and ${\mathrm{i}}^2=-1$. The set of all complex numbers is $\mathbb{C}$; it contains every real number (the case $b=0$).

Interest calculated on the original amount plus all the interest already added, so the balance is multiplied by the same factor each period. The balances form a geometric sequence.

The length of time between two consecutive additions of interest, for example one month for monthly compounding. The number of periods per year is written $k$.

A statement that appears to be true, usually from a pattern in examples, but has not yet been proved. A conjecture becomes a theorem once it is proved and is discarded if a counterexample is found.

If a polynomial equation with real coefficients has a root $a+b\mathrm{i}$ with $b\mathrm{\ne }0$, then its conjugate $a-b\mathrm{i}$ is also a root. Non-real roots therefore come in conjugate pairs.

Integers that follow each other without gaps, each $1$ more than the one before, such as $7,8,9$. In algebra they are written $n,n+1,n+2,\dots$ for an integer $n$.

A system of equations that has at least one solution (a unique solution or infinitely many). A system with no solution is called inconsistent.

The term of an expression or expansion that does not contain the variable, that is, the term in $x^0$. In $(2x-\frac{1}{x}{)}^6$ it is $-160$.

An infinite series whose partial sums approach a finite limit as the number of terms increases. A geometric series converges exactly when $\mathrm{\mid }r\mathrm{\mid }<1$.

A single specific case that satisfies the conditions of a general statement but for which its conclusion is false. One counterexample is enough to disprove the statement.

A quick way to find a partial fraction numerator: substitute the value of $x$ that makes one linear factor zero into the rest of the fraction, with that factor covered up.

For any integer $n$, $[r(\cos \theta +\mathrm{i}\sin \theta ){]}^n=r^n(\cos n\theta +\mathrm{i}\sin n\theta )$: to raise a complex number to a power, raise the modulus to that power and multiply the argument by it.

The number of digits written after the decimal point. Rounding to 2 decimal places keeps two digits after the point, as in $3.14159\approx 3.14$.

The loss in value of an asset, such as a car or a machine, over time. With a fixed annual percentage rate the values form a geometric sequence with ratio less than 1.

An infinite series whose partial sums do not approach a finite limit: they grow without bound or keep oscillating. A geometric series with $\mathrm{\mid }r\mathrm{\mid }\ge 1$ and $u_1\mathrm{\ne }0$ diverges.

Setting the coefficients of each power of $x$ on the two sides of an identity equal to each other, to find unknown constants.

A statement that two expressions are equal, which may be true for only some values of the variable. Solving an equation means finding those values; for example $2x+1=7$ is true only when $x=3$.

The way of writing a complex number with modulus $r$ and argument $\theta$ as $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, using Euler's formula ${\mathrm{e}}^{\mathrm{i}\theta }=\cos \theta +\mathrm{i}\sin \theta$.

The identity ${\mathrm{e}}^{\mathrm{i}\theta }=\cos \theta +\mathrm{i}\sin \theta$ for real $\theta$, linking the exponential function with the circular functions. With $\theta =\pi$ it gives Euler's identity, ${\mathrm{e}}^{\mathrm{i}\pi }+1=0$.

The irrational constant $e\approx 2.71828$, the limit of ${(1+\frac{1}{n})}^n$ as $n$ grows without bound. It is the base of the natural logarithm and of the exponential function $e^x$.

An integer that is divisible by $2$, so it can be written as $2n$ for some integer $n$. Examples are $-4$, $0$ and $18$.

The number written as a superscript in a power such as $a^n$, showing how many times the base is used as a factor when $n$ is a positive integer. For example, in $2^5$ the exponent is $5$.

An equation in which the unknown appears in an exponent, such as $2^{x+1}=5$. It is solved by writing both sides with the same base or by taking logarithms of both sides.

The infinite series $(1+x{)}^n=1+nx+\frac{n(n-1)}{2!}{x}^2+\frac{n(n-1)(n-2)}{3!}{x}^3+\cdots$ for a negative or fractional index $n\in \mathbb{Q}$, valid for $\mathrm{\mid }x\mathrm{\mid }<1$.

A value produced while solving an equation, for example after squaring both sides or splitting into cases, that does not satisfy the original equation and must be rejected.

The product of all positive integers from 1 up to $n$: $n!=n(n-1)(n-2)\cdots 2\times 1$ for $n\in {\mathbb{Z}}^{+}$, with $0!=1$ by definition.

A shape built by repeating the same construction at smaller and smaller scales, so that parts of it look like the whole. The Koch snowflake is an example.

The value that an amount of money will have at a later date after interest has been added, written $FV$.

The expression $\left(\genfrac{}{}{0px}{}{n}{r}\right)a^{n-r}{b}^r$, which gives the $(r+1)$th term of the expansion of $(a+b{)}^n$ for $r=0,1,\dots ,n$.

A sequence in which each term after the first is found by multiplying the previous term by the same non-zero number, the common ratio $r$.

The sum of the terms of a geometric sequence. For $r\mathrm{\ne }1$, the sum of the first $n$ terms is $S_n=\frac{u_1(r^n-1)}{r-1}$.

An equality that is true for every value of the variable for which both sides are defined, such as $(x+1{)}^2\equiv x^2+2x+1$. The symbol $\equiv$ means "is identically equal to".

For $z=a+b\mathrm{i}$ with $a,b\in \mathbb{R}$, the real number $b$, written $\mathrm{I}\mathrm{m}(z)$. It is $b$, not $b\mathrm{i}$.

The number $\mathrm{i}$ defined by ${\mathrm{i}}^2=-1$, so that $\mathrm{i}=\sqrt{-1}$. Its powers repeat in the cycle $\mathrm{i},-1,-\mathrm{i},1$.

An algebraic fraction whose numerator has a degree greater than or equal to that of its denominator, such as $\frac{x^2+1}{x^2-4}$. Dividing first writes it as a polynomial plus a proper fraction.

The assumption, made inside a proof by induction, that the statement $P(n)$ is true for one particular value $n=k$.

The part of a proof by induction that uses the assumption that $P(k)$ is true to show that $P(k+1)$ is true.

A general rise in prices over time, usually given as an annual percentage rate. Inflation reduces what a fixed sum of money can buy.

A real number that cannot be written as $\frac{p}{q}$ with $p,q\in \mathbb{Z}$ and $q\mathrm{\ne }0$, such as $\sqrt{2}$ or $\pi$. Its decimal expansion neither ends nor repeats.

An equation in which each unknown appears only to the first power and is multiplied only by a constant, such as $2x-y+3z=7$.

The exponent to which a base $a$ must be raised to give a number $b$: ${\log }_{a}b=x$ exactly when $a^x=b$, for $a>0$, $a\mathrm{\ne }1$ and $b>0$.

An equation in which the unknown appears inside a logarithm, such as ${\log }_2(x+3)=5$. Every solution must make the argument of each logarithm positive.

The distance $\mathrm{\mid }z\mathrm{\mid }=\sqrt{a^2+b^2}$ from the origin to the point representing $z=a+b\mathrm{i}$ on an Argand diagram. It is always a non-negative real number.

A logarithm with base $e$, written $\ln x$. So $\ln x=y$ exactly when $e^y=x$, for $x>0$.

A number which, raised to the power $n$, gives $a$; written $\sqrt[n]{a}=a^{1\mathrm{/}n}$. For example $\sqrt[3]{8}=2$ because $2^3=8$. For even $n$ the symbol means the non-negative root of $a\ge 0$.

The $n$ distinct solutions of $z^n=w$ for a non-zero complex number $w$. They have equal modulus $\mathrm{\mid }w{\mathrm{\mid }}^{1\mathrm{/}n}$ and arguments spaced $\frac{2\pi }{n}$ apart, forming a regular polygon centred on the origin.

An integer that leaves remainder $1$ when divided by $2$, so it can be written as $2n+1$ for some integer $n$. Examples are $-3$, $1$ and $17$.

The power of 10 that a quantity is closest to, given by $k$ when it is written as $a\times {10}^k$. Two quantities differ by one order of magnitude when one is about 10 times the other.

Simpler fractions whose sum equals a given algebraic fraction, for example $\frac{5x-1}{(x+1)(x-2)}=\frac{2}{x+1}+\frac{3}{x-2}$.

The sum $S_n$ of the first $n$ terms of a series. The sum of an infinite series is the value the partial sums approach as $n$ increases, if there is one.

A triangular array of numbers with 1 at the ends of each row, in which every other entry is the sum of the two entries directly above it. Row $n$ (counting from row 0) holds the binomial coefficients $\left(\genfrac{}{}{0px}{}{n}{0}\right),\dots ,\left(\genfrac{}{}{0px}{}{n}{n}\right)$.

An ordered arrangement of objects. The number of permutations of $r$ objects chosen from $n$ different objects is ${}^n{P}_r=\frac{n!}{(n-r)!}$.

The way of writing a non-zero complex number by its modulus $r$ and argument $\theta$: $z=r(\cos \theta +\mathrm{i}\sin \theta )=r\mathrm{cis}\theta$.

The value of an amount of money now: the amount invested or borrowed at the start, written $PV$.

A positive integer greater than 1 whose only positive factors are 1 and itself, such as 2, 3, 5, 7 and 11.

If one choice can be made in $m$ ways and, after it, a second choice can be made in $n$ ways, then the two choices together can be made in $m\times n$ ways.

A chain of logical steps that starts from definitions, known facts or assumptions and shows that a statement must be true in every case it covers. Checking examples is not a proof.

A method of proof that assumes the statement is false and shows, by valid steps, that this assumption leads to something impossible, so the statement must be true.

A method of proving a statement $P(n)$ for every integer $n$ from a starting value: show that $P(n)$ is true for the first value, and that whenever $P(k)$ is true, $P(k+1)$ is also true.

An algebraic fraction whose numerator has a lower degree than its denominator, such as $\frac{3x+1}{x^2-4}$.

A complex number whose real part is zero, $z=b\mathrm{i}$ with $b\in \mathbb{R}$, $b\mathrm{\ne }0$. It lies on the imaginary axis of an Argand diagram.

An exponent that is a fraction $\frac{m}{n}$. It means a root followed by a power: $a^{m\mathrm{/}n}={\left(\sqrt[n]{a}\right)}^m=\sqrt[n]{a^m}$, so $8^{2\mathrm{/}3}=2^2=4$.

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

The value of an amount of money measured by what it can buy, after allowing for inflation, rather than by the number written on it.

A decimal in which a digit or block of digits repeats forever, such as $0.\dot{3}=0.333\dots$ or $0.\dot{1}\dot{2}=0.1212\dots$. Every recurring decimal is a rational number.

A rule that gives each term of a sequence from the term before it, such as $u_{n+1}=u_n+3$, together with a starting value such as $u_1=2$.

The $n$ solutions of $z^n=1$, given by $z={\mathrm{e}}^{2k\pi \mathrm{i}\mathrm{/}n}$ for $k=0,1,\dots ,n-1$. They lie on the unit circle at the vertices of a regular $n$-sided polygon, and their sum is $0$ for $n\ge 2$.

Solving a system of linear equations by swapping equations, multiplying an equation by a non-zero constant and adding a multiple of one equation to another, until the unknowns can be found one at a time.

A way of writing a number as $a\times {10}^k$, where $1\le a<10$ and $k$ is an integer. It makes very large and very small numbers easy to read, compare and calculate with.

An ordered list of numbers, such as $2,5,8,11,\dots$, in which each number has a position: first, second, third and so on.

The sum of the terms of a sequence, such as $2+5+8+11$. The sum of the first $n$ terms is written $S_n$.

A short way of writing a sum with the Greek capital letter $\mathrm{\Sigma }$. For example, ${\sum }_{k=1}^n{u}_k$ means $u_1+u_2+\cdots +u_n$.

The digits of a number that carry meaning about its size, counted from the first non-zero digit. For example, $0.004070$ has four significant figures: 4, 0, 7 and the final 0.

Interest calculated only on the original amount, so the same interest is added every year. The balances form an arithmetic sequence.

The limit $S_{\mathrm{\infty }}$ that the partial sums $S_n$ of a series approach as $n$ increases without bound. For a geometric series with $\mathrm{\mid }r\mathrm{\mid }<1$, $S_{\mathrm{\infty }}=\frac{u_1}{1-r}$.

A set of two or more linear equations in the same unknowns, to be satisfied at the same time. A solution is a set of values of the unknowns that satisfies every equation.

A series whose terms can be written as differences, so that most of them cancel when added and only a few terms at the start and end remain.

One number in a sequence. The term in position $n$ is written $u_n$, so $u_1$ is the first term; a formula for $u_n$ in terms of $n$ is called the general term.

A mathematical statement that has been proved to be true, such as Pythagoras' theorem or the binomial theorem.

The form of a system after row reduction in which each equation has fewer unknowns than the one before, for example the last equation contains only $z$, so the system can be solved by back substitution.