IB Maths AI Topic 1 Definitions

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

All Topic 1 Sub-topics 1.1: Scientific Notation 1.2: Arithmetic Sequences 1.3: Geometric Sequences 1.4: Interest 1.5: Introduction To Exponents & Logs 1.6: Handling Numbers 1.7: Use Of Technology (HL)1.8: Advanced exponents & logs (HL)1.9: Introduction to complex numbers (HL)1.10: Advanced complex numbers (HL)1.11: Introduction to matrices (HL)1.12: Advanced matrices

absolute error

The size of the difference between a measured value and the true value, calculated as

|\text{measured value}-\text{true value}|

amortization

The process of repaying a loan through regular payments over a fixed period of time, reducing the outstanding balance to

0

by the end.

amplitude

The distance from the principal axis to a maximum or minimum of a sinusoidal graph, given by '

\left|a\right|

annuity

A sequence of equal payments made at regular intervals over a fixed period of time, with payments made at the end of each period in this topic.

approximation

Using a truncated Maclaurin series (a finite number of terms) to estimate a function value near

x=0

Argand

A diagram where a complex number

a+bi

is plotted as the point

(a,b)

, with the real part on the horizontal axis and the imaginary part on the vertical axis.

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.

A matrix product used to represent a system of linear equations in the form

Ax=b

, where

A

is the coefficient matrix and

x

is the column vector of variables.

base

The initial value of

n

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

n=1

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

characteristic equation

The equation

\det\left(A-\lambda I\right)=0

that must be satisfied for

A\mathbf{v}=\lambda\mathbf{v}

to have a non-zero solution.

characteristic polynomial

The polynomial expression

\det\left(A-\lambda I\right)

whose roots are the eigenvalues of

A

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.

column

A vertical line of elements in a matrix.

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}

convergent

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

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}

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}

determinant

For a

2\times2

matrix

A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}

, the scalar

ad-bc

, which equals the product of the eigenvalues and is used in

\det\left(A-\lambda I\right)

diagonal matrix

A square matrix with all off-diagonal entries equal to

0

, so powers are found by raising each diagonal entry to the power, e.g.

D^n=\begin{pmatrix}\lambda_1^n&&0\\0&&\lambda_2^n\end{pmatrix}

diagonalizable

Able to be expressed as

A=PDP^{-1}

for some invertible matrix

P

and diagonal matrix

D

, which occurs for a

2\times2

matrix with distinct real eigenvalues.

diagonalization

Writing a matrix in the form

A=PDP^{-1}

, where

P

has eigenvectors as columns and

D

is diagonal with the corresponding eigenvalues on the diagonal; for

2\times2

matrices this is restricted to distinct real eigenvalues.

eigenvalue

A scalar

\lambda

in the equation

A\mathbf{v}=\lambda\mathbf{v}

that gives the scale factor by which an eigenvector is stretched, shrunk, reversed, or mapped to the zero vector.

eigenvector

A non-zero vector

\mathbf{v}

that satisfies

A\mathbf{v}=\lambda\mathbf{v}

for some scalar

\lambda

, meaning the matrix transformation changes only its size (and possibly reverses it) but not its direction.

element

A single entry in a matrix; the entry in row

i

and column

j

is written as

a_{ij}

estimation

A quick approximate calculation found by rounding numbers to easy values, used to check whether an answer is reasonable.

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

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

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 $r\neq1$ .

frequency

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

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

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.

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.

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.

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

lower bound

The smallest possible true value consistent with a rounded value, found by subtracting half of the rounding unit and using an inclusive endpoint.

multiplication

An operation on complex numbers that, in polar form, multiplies moduli and adds arguments: if

z_1=r_1\operatorname{cis}\theta_1

and

z_2=r_2\operatorname{cis}\theta_2

, then

z_1z_2=r_1r_2\operatorname{cis}\left(\theta_1+\theta_2\right)

natural

Describes a logarithm with base

e

, written as

\ln x=\log_ex

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 $u_n=u_1r^{n-1}$ .

operations

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

a \times10^{k}

order

The size of a matrix written as

m \times n

, where

m

is the number of rows and

n

is the number of columns.

percentage error

The size of an error compared with the true value, expressed as a percentage:

\frac{|\text{measured value}-\text{true value}|}{\text{true value}}\times100\%

phase

The angle that determines the horizontal shift of a sinusoidal function; in

a\cos\left(\omega t+\alpha\right)

it is

\alpha

phasors

Complex-number representations of sinusoidal functions with a fixed frequency, where the modulus gives the amplitude and the argument gives the phase angle.

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.

polynomial

An algebraic expression made from constants and variables using addition, subtraction, and multiplication, with variables raised to non-negative integer powers.

powers

Repeated multiplication of a matrix by itself, written

A^n

, which can be computed efficiently using diagonalization via

A^n=PD^nP^{-1}

when

A=PDP^{-1}

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.

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}

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

row

A horizontal line of elements in a matrix.

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 $S_n=nu_1$ .

scalar

A single number used to multiply a matrix by multiplying every element by that number.

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 $S_\infty=\frac{u_1}{1-r}$ .

systems

Sets of equations considered together, where a solution must satisfy every equation in the set 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.

trace

For a

2\times2

matrix

A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}

, the sum of the diagonal entries

a+d

, which equals the sum of the eigenvalues.

upper bound

The greatest possible true value consistent with a rounded value, found by adding half of the rounding unit and using an exclusive endpoint.

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