IB Maths AA Topic 1 Definitions

This page contains our IB Maths AA definitions for topic 1. By learning each one of these definitions, you will fully cover the content for IB Maths AA 'Number & Algebra'.

All Topic 1 Sub-topics 1.1: Scientific Notation 1.2: Arithmetic Sequences 1.3: Geometric Sequences 1.4: Interest 1.5: Exponents & Logs 1.6: Deductive Proofs 1.7: Binomial Theorem (HL)1.8: Counting principles (HL)1.9: Partial fractions (HL)1.10: Complex numbers (HL)1.11: Additional proofs (HL)1.12: Systems of equations

AND

Indicating that separate stages both occur, so the total number of outcomes is found by multiplying the numbers of ways for each stage, giving

m\times n

argument

The angle a complex number makes with the positive real axis on an Argand diagram; for

z=r\operatorname{cis}\theta

, it is

\arg\left(z\right)=\theta

arithmetic series

The sum of the terms of an arithmetic sequence, used to total repeated equal increases over time in simple-interest contexts.

augmented

Describes a matrix that includes both the coefficient matrix and the constant terms in one array, separated by a vertical bar.

base

The initial value of

n

for which a claim is first verified in an induction argument, commonly

n=1

binomial coefficient

A number written as

{}^nC_r

\binom{n}{r}

, given by

{}^nC_r=\frac{n!}{r!\left(n-r\right)!}

, which counts combinations: the number of ways to choose

r

objects from

n

when order does not matter.

binomial theorem

A rule for expanding

(a+b)^n

for

n\in\mathbb{N}

by generating every term and coefficient systematically, rather than multiplying out repeatedly.

Cartesian form

A way to write a complex number as

z=a+bi

with

a,b\in\mathbb{R}

, separating its real and imaginary components.

change

A method for rewriting a logarithm in a different base using

\log_ax=\frac{\log_bx}{\log_ba}

, often with

b=10

b=e

cis

Shorthand for

\cos\theta+i\sin\theta

, so a complex number in polar form can be written as

z=r\operatorname{cis}\theta

coefficient

The numerical factor multiplying a term in an expansion, for example the number multiplying

x^2

in a polynomial term.

combinations

The number of ways to pick

r

objects from

n

distinct objects when order does not matter, given by

{}^nC_r=\frac{n!}{r!\left(n-r\right)!}

common

Describes a logarithm with base

10

, written as

\log x

common ratio

The constant multiplier between consecutive terms of a geometric sequence, found using $r=\frac{u_{n+1}}{u_n}$ .

complex

A number that can be written as

a+bi

with

a,b\in\mathbb{R}

, extending the real numbers by including multiples of

i

compounded

Describes interest that is repeatedly added to the balance at regular intervals, so future interest is calculated on both the original amount and previously added interest.

conjugate

The complex number obtained by changing the sign of the imaginary part; if

z=a+bi

then

\overline{z}=a-bi

, which reflects the point in the real axis on an Argand diagram.

consecutive

Next to each other in order, such as

u_n

and

u_{n+1}

constants

Fixed numbers (not depending on '

x

') introduced in a partial fraction form, such as '

A

' and '

B

', which are determined to make two expressions identical.

contradiction

An impossibility or inconsistency reached in an argument, such as deducing two statements that cannot both be true.

convergent

Approaching a finite limit; an infinite geometric series is convergent only when $|r|\lt1$ .

counterexample

A single example that makes a universal claim fail, showing the statement is not always true.

A theorem stating that for a positive integer

n

\left[r\left(\cos\theta+i\sin\theta\right)\right]^n=r^n\left(\cos n\theta+i\sin n\theta\right)

, enabling powers and roots to be found using modulus and argument.

decay

A multiplicative decrease in a quantity over time; for a decrease of

r\%

per period the multiplier is

1-\frac{r}{100}

, giving a geometric pattern.

decimal

The point in a base-10 numeral that separates the whole-number part from the fractional part; moving it left or right changes the power of '

10

' in '

a \times10^{k}

decomposed

Written in an equivalent form as a sum of simpler fractional terms, typically by introducing constants such as '

A

' and '

B

' and determining their values.

deductive

Based on reasoning that starts from known facts or definitions and uses logically valid steps to reach a conclusion that must follow.

degree

The highest power of

x

with a non-zero coefficient in a polynomial.

denominator

The polynomial or expression written below the fraction bar in a rational expression; its factorisation determines the form of the decomposition.

depreciation

A decrease in value over time; if the same percentage is lost each year, the values follow a geometric model with multiplier

1-\frac{r}{100}

equality

A relationship showing that two expressions have the same value; written using the symbol '

=

Euler

A numerical method that generates successive approximations to a solution by stepping forward with a fixed step size

h

using update rules such as

x_{n+1}=x_n+h y_n

expand

To rewrite a product such as

A(x+2)

as a sum by distributing multiplication, for example

A(x+2)=Ax+2A

exponent

A number or expression written as a superscript that tells how many times the base is multiplied by itself, for example

3^4=3\times3\times3\times3

factorial

A product written as

n!

meaning

n\times\left(n-1\right)\times\cdots\times2\times1

, with

0!=1

, used to define binomial coefficients.

factorials

Products of consecutive positive integers used in counting formulas, with

n!=n\times\left(n-1\right)\times\dots\times2\times1

and

0!=1

factorise

To rewrite an expression as a product of simpler factors, for example '

x^2+x-2=\left(x-1\right)\left(x+2\right)

finite sums

Totals found by adding only a fixed number of terms from a series; for a geometric series this is written $S_n$ and can be calculated using $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$ when $r\neq1$ .

fractional

Describing an index

n

that is not an integer, so the expansion of

\left(1+x\right)^n

uses the general series

1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+\dots

and does not terminate.

frequency

A table that shows how often each value (discrete) or each class interval (continuous) occurs.

fundamental

Stating that if one task can be done in

m

ways and a second task can be done in

n

ways, then doing both tasks can be done in

m\times n

ways.

general term

A single term in the binomial expansion written as

T_{r+1}={}^nC_r a^{n-r}b^r

, used to identify a specific term without expanding fully.

geometric

Changing by a constant multiplier each step, so terms follow the form

a,ar,ar^2,\dots

and match compound-interest and constant-percentage decrease models.

geometric sequence

A sequence in which each term is obtained by multiplying the previous term by a constant value called the common ratio.

geometric series

A series formed by adding the terms of a geometric sequence, for example $u_1+u_1r+u_1r^2+\dots$ .

growth

The multiplicative change in a quantity over time; for an increase of

r\%

per period the multiplier is

1+\frac{r}{100}

, giving a geometric pattern.

Identity

A square matrix with ones on the main diagonal and zeros elsewhere, with the property that

AI=IA=A

for any compatible matrix

A

imaginary

A number involving

i

; in

a+bi

this corresponds to the coefficient

b

i

, and if

a=0

the number is purely imaginary.

index

The variable that counts through the terms in sigma notation, such as

k

\sum_{k=1}^{n}u_k

Induction hypothesis

The assumption that

P(k)

is true for some fixed

k\in\mathbb{N}

, used as a step in proving

P(k+1)

Inductive step

The part of an induction proof where, using the induction hypothesis,

P(k+1)

is proved to follow from

P(k)

inflation

A sustained rise in general price levels that reduces purchasing power, so real value compares an investment’s growth factor with the inflation growth factor.

integer

A whole number that can be positive, negative, or zero.

integration by substitution

Replacing one expression with another to simplify a differential equation, such as using '

y=vx

' so that '

\frac{dy}{dx}=v+x\frac{dv}{dx}

interest

Money paid or earned for borrowing or investing; in sequence models it represents the change in an amount over time according to a fixed rule.

intersect

A calculator feature or graphical method that finds the approximate

x

-value where

y=f(x)

and

y=g(x)

meet, corresponding to a solution of

f(x)=g(x)

inverse

Describes an operation that undoes another operation; for example, if

a^x=b

then applying

\log_a

b

returns

x

laws

Rules that allow expressions to be rewritten in an equivalent form, for example to simplify or solve equations while keeping the same value.

LHS

Abbreviation for the left-hand side of an equation or identity.

logarithm

The inverse of an exponential: if

a^x=b

, then

\log_ab=x

, meaning it gives the power needed on base

a

to produce

b

matrix

A rectangular array of numbers used to represent the coefficients and constants of a system so it can be manipulated systematically.

natural

Describes a logarithm with base

e

, written as

\ln x=\log_ex

nCr

Combinatorial notation for the binomial coefficient, written as

{}^nC_r

, used as the coefficient of terms in binomial expansions and to count combinations.

notation

A system of symbols used to represent mathematical ideas concisely, such as sigma notation for adding terms.

nth term

A formula for the general term of a sequence; for a geometric sequence it is given by $u_n=u_1r^{n-1}$ .

numerator

The expression written above the fraction bar; in this topic its degree is lower than the degree of the expression below the fraction bar.

operations

Calculations such as multiplication, division, addition, and subtraction carried out on numbers written in the form '

a \times10^{k}

opposite

The side that lies across from the chosen acute angle in a right-angled triangle.

Partial fractions

A method for rewriting a rational expression as a sum of simpler fractions, which are usually easier to manipulate and integrate.

Pascal's triangle

A triangular array of numbers where each interior entry is the sum of the two entries directly above it, and each row gives the coefficients for expanding

(a+b)^n

for a particular value of

n

permutation

An arrangement of objects where order matters; the number of ways to arrange

r

objects chosen from

n

distinct objects is

{}^nP_r=\frac{n!}{\left(n-r\right)!}

plane

A two-dimensional coordinate system used to represent complex numbers, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.

precision

How exact a value is, indicated by how many significant figures are used.

principal

The original amount borrowed (or the remaining loan balance) before interest is added; part of each payment reduces this amount.

probability

Measures how likely an event is to happen, taking a value between $0$ and $1$ , where $0$ means impossible and $1$ means certain.

proof

A sequence of logically valid steps showing that a statement is true.

Proof by contradiction

A method where the opposite of the desired statement is assumed and then logical consequences are derived until an impossibility or inconsistency is reached, forcing the original statement to be true.

Proof by mathematical induction

A method for proving a statement

P(n)

for all relevant positive integers by verifying an initial case and then showing that

P(k)

implies

P(k+1)

for some

k\in\mathbb{N}

quotient

A result of division; in a logarithmic identity this refers to combining a fraction inside a single log, such as

\log_a\left(\frac{x}{y}\right)=\log_ax-\log_ay

for suitable positive

x

and

y

quotients

Results of dividing one complex number by another; in Cartesian form, division can be carried out by multiplying numerator and denominator by the conjugate of the denominator to remove

i

from the denominator.

rational

Expressible as a ratio of integers, written as $n=\frac{p}{q}$ with integers $p,q$ and $q\neq0$ , allowing the series form of $(1+x)^n$ to be used.

real

A number on the real number line, with no imaginary component; in

a+bi

this corresponds to

b=0

reciprocal

The multiplicative inverse of a non-zero number, so multiplying a value by its reciprocal gives

1

(for example, the reciprocal of

a

\frac{1}{a}

RHS

Abbreviation for the right-hand side of an equation or identity.

roots

Values of

w

that satisfy an equation of the form

w^n=z

; for

z=r\operatorname{cis}\theta

, the

n

th roots are

w_k=\sqrt[n]{r}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)

for

k=0,1,\dots,n-1

S n

The notation for the total of the first $n$ terms of a series; for a geometric series with $r\neq1$ , $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$ , and if $r=1$ then $S_n=nu_1$ .

scientific notation

A way of writing very large or very small numbers in the form '

a \times10^{k}

', where '

1 \leq a \lt 10

' and '

k \in \mathbb{Z}

sequence

An ordered set of steps written so that each line follows logically from the previous one.

series

A sum of terms written in a continuing pattern, such as $1+nx+\frac{n(n-1)}{2!}x^2+\dots$ , which may be infinite when $n$ is not a positive integer.

sigma

A summation symbol used to write a series compactly, for example

(a+b)^n=\sum_{r=0}^{n}{}^nC_r a^{n-r}b^r

means add the terms for all integer values of

r

from

0

n

sigma notation

A compact way to write a sum using the symbol $\sum$ , for example $\sum_{k=0}^{n-1}u_1r^k$ for the first $n$ terms of a geometric series.

significant figures

Digits that indicate the precision of a value; counted from the first non-zero digit, with every digit after that counted as significant.

sum of the first n terms

The total of the first $n$ terms of a series, written $S_n$ ; for a geometric series with $r\neq1$ , $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$ .

sum to infinity

The finite value approached by an infinite geometric series when it converges; if $|r|\lt1$ then $S_\infty=\frac{u_1}{1-r}$ .

system

A set of equations involving the same variables, considered together so that all equations must be satisfied at the same time.

technology

Digital tools such as a graphic display calculator (GDC) or a spreadsheet used to carry out calculations, solve equations, and model financial situations efficiently.

term

A single value in a sequence, such as

u_1

u_n

theorem

A mathematical result that can be proved logically from definitions and earlier results, then used to solve problems.

universal

Describing a claim that applies to every element in a set, typically signalled by words such as “all”, “every”, “always”, or “no”.

unknowns

Variables whose values are to be found so that all equations in the system are true simultaneously.

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