NSW Syllabuses

# Glossary

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Argand plane

The Argand plane is used to represent complex numbers geometrically. The horizontal axis is the real axis $${Re}(z)$$ and the vertical axis is the imaginary axis $${Im}(z)$$. The Argand plane is also known as the Argand diagram.

argument and principal argument (of a complex number)

When a complex number $$z$$ is represented by a point $$P$$ in the complex plane then the argument of $$z$$, denoted $$\arg z$$, is the angle $$\theta$$ that $$OP$$ (where $$O$$ denotes the origin) makes with the positive real axis $$O_x$$ with the angle measured from $$O_x$$.

If the argument is restricted to the interval $$(-\pi,\pi]$$, this is called the principal argument and is denoted by $$\text{Arg}\,z$$.

arithmetic mean (average)

There are a number of different types of means used in mathematics and statistics. When dealing with a group of numbers, their arithmetic mean is defined as the sum of these values divided by the number of values. Also known as their average.

Cartesian equation

The equation connecting the points $$(x,y)$$ on a curve is called the Cartesian equation.

For example, $$y=x^2+3x-2$$, $$x=y^2$$ or $$x^2+y^2=1$$.

A Cartesian equation may be formed from two parametric equations by eliminating the parameter.

Cartesian form (of a complex number)

The Cartesian form of a complex number is $$a+ib$$, where $$a$$ and $$b$$ are real numbers and $$i$$ is the imaginary number. Also known as standard or rectangular form.

column vector notation

A vector in two or three dimensions can be represented in column vector notation. For example, the ordered vector triple $$(4,5,6)$$ can be represented in column vector notation as $$\begin{bmatrix} 4\\ 5\\ 6 \end{bmatrix}$$.

complex conjugate

For any complex number $$z=a+ib$$, its conjugate is $$\bar z = a-ib,$$ where $$a$$ and $$b$$ are real numbers. A complex number and its conjugate are called conjugate pairs.

complex number

A complex number is a number that can be expressed in the form $$a+ib$$, where $$a$$ and $$b$$ are real numbers and $$i$$ is the imaginary number which satisfies the equation $$i^2=-1$$.

component form (of a vector)

The component form of a vector expresses the vector in terms of unit vectors $$\boldsymbol i$$ (a unit vector in the $$x$$-direction), $$\,\boldsymbol j$$ (a unit vector in the $$y$$-direction) and, for three dimensional vectors, $$\boldsymbol k$$ (a unit vector in the $$z$$-direction). For example, in two dimensions the ordered vector pair $$(4,3)$$ can be represented as $$4\boldsymbol i + 3\boldsymbol j$$.

contrapositive

The contrapositive of the statement ‘If $$P$$ then $$Q$$’ is ‘If not $$Q$$ then not $$P$$’. The contrapositive of a true statement is also true.

Example:
Statement: If $$x=2$$ then $$x^2=4$$.
Contrapositive: If $$x^2 \ne4$$ then $$x\ne2$$.

converse

The converse of a statement ‘If $$P$$ then $$Q$$’ is ‘If $$Q$$ then $$P$$’.

Symbolically the converse of $$P\Rightarrow Q$$ is $$Q\Rightarrow P$$ or $$P\Leftarrow Q$$.

The converse of a true statement need not be true.

Examples:

Statement: If a quadrilateral is a rectangle then the diagonals are of equal length and they bisect each other.
Converse statement: If the diagonals of a quadrilateral are of equal length and bisect each other then the quadrilateral is a rectangle. (In this case the converse is true.)

Statement: If $$x=2$$ then $$x^2=4$$.
Converse statement: If $$x^2=4$$ then $$x=2$$. (In this case the converse is false.)

counter-example

A counter-example is an example that demonstrates that a statement is not true.

Example:

Statement: If $$x^2=4$$ then $$x=2$$.
Counter-example: $$x=-2$$ provides a counter-example.

De Moivre’s theorem

For all integers $$n$$,$$\ \left[r\left(\cos\theta+i\sin\theta\right)\right]^n=r^n\left(\cos n\theta +i\sin n\theta\right)$$ .

De Moivre’s theorem is most often used to find powers or roots of complex numbers or in trigonometric proofs.

displacement vector

A displacement vector represents the displacement from one point to another. Also known as a relative vector.

Euler’s formula

Euler’s formula for complex numbers establishes the fundamental relationship between trigonometric functions and the complex exponential function. It states that for any real number $$x$$: $$e^{ix}=\cos x+i\sin x$$.

This is generally taken as the definition of $$e^{ix}$$.

exponential form

The complex number $$z=a+ib$$ can be expressed in exponential form as $$z=re^{i\theta}$$, where $$r$$ is the modulus of the complex number and $$\theta$$ is the argument expressed in radians.

geometric mean

The geometric mean indicates the central tendency of a set of numbers by using the product of their values. The geometric mean of the positive numbers $$x_1, \ldots, x_n$$ is defined to be $$( x_1\!\!\ldots x_n )^{\frac1n}$$

imaginary number

$$i$$ is the imaginary number that satisfies the equation $$i^2=-1$$. A purely imaginary number is any number that can be expressed as $$ib$$, where $$b$$ is any real number.

implication

To say that $$P$$ implies $$Q$$ is to say that the truth of $$Q$$ may be deduced from the truth of $$P$$. In shorthand it can be written as ‘If $$P$$ then $$Q$$' and in notation form as $$P\Rightarrow Q$$.

Examples:

• If a quadrilateral is a rectangle, then the diagonals are of equal length and they bisect each other.
• If $$x=2$$ then $$x^2=4$$.
• If an animal is a kangaroo, then it is a marsupial.
• If a quadrilateral is cyclic then the opposite angles are supplementary.
Maclaurin polynomials

Maclaurin polynomials are used to give approximations to a function $$f(x)$$ near $$x=0$$, provided that all the derivatives of the function are defined.

A Maclaurin polynomial for $$f(x)$$ of degree $$n$$ is given by: $$p_n(x)=\displaystyle{\sum_{r=0}^{n}}\frac{f^{(r)}(0)}{r!}x^r$$

Maclaurin series

A Maclaurin series is the expansion series of a function $$f(x)$$ continued indefinitely and is also called a Maclaurin expansion. By cutting off the expansion after the second, third, fourth, etc… terms, you get a sequence of Maclaurin polynomials. The Maclaurin series of a function $$f(x)$$ is given by: $$f(x) = \displaystyle{\sum_{r=0}^{\infty}}\frac{f^{(r)}(0)}{r!}x^r=f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2\,+\,...\,+\,\frac{f^{(r)}(0)}{r!}x^r\,+\,...$$

A Maclaurin series may only be valid for a range of $$x$$ values.

magnitude (of a vector)

The magnitude of a vector $$a\boldsymbol{i} + b \boldsymbol{j} + c\boldsymbol{k}$$ is the length of the vector and is given by $$r = \sqrt{a^2 + b^2 + c^2}$$.

mathematical induction

Mathematical induction is a method of mathematical proof that uses deductive reasoning to show a given proposition (or statement) is true (most commonly) for all natural numbers after a starting value.

For example: A statement involving the natural number $$n$$ is true for every $$n\in \mathbb{N}$$ ($$\mathbb{N}$$ is the set of all natural numbers) provided that:

1. The statement is true in the special case $$n=1$$.

2. The truth of the statement for $$n=k$$, for some particular $$k \in \mathbb{N}$$, implies the truth of the statement for $$n=k+1$$.

Also known as proof by induction or inductive proof.

The principle of induction is an axiom and so cannot itself be proven.

modulus

The modulus of $$z=a+ib$$ is the length of the line segment joining the origin to $$z$$ in the Argand plane.

The modulus of $$z$$ is denoted as $$\left|z\right|$$, $$\left|a+ib\right|$$, or $$r$$, where $$r=\left|z\right|=\sqrt{a^2+b^2}$$.

Also referred to as the magnitude or absolute value.

negation

If $$P$$ is a statement then the statement ‘not $$P$$’ is the negation of $$P$$. The negation of $$P$$ is denoted by $$\neg P$$, $$P'$$ or $$\sim P$$.

For example, if $$P$$ is the statement ‘It is snowing’, then $$\neg P$$ is the statement ‘It is not snowing’.

ordered pairs or triples notation for vectors

A vector in two or three dimensions can be represented as an ordered pair or triple of numbers, for example, $$(1,-2)$$ or $$(4,5,6)$$.

parametric variable

A parameter is a variable in a mathematical expression. It may be thought of either as separate from the dependent and independent variable, for example, $$\theta$$ in $$y = x \cos \theta$$; or as linking the dependent and independent variable, as in $$x = a \cos \theta$$, $$y = a \sin \theta$$.

parametric equations

Parametric equations are equations connected by a variable called a parametric variable (or parameter).

For example, $$x=t^2$$ and $$y=2t$$ are connected via the parameter $$t$$. The parameter could then be eliminated by rearranging $$y=2t$$ and substituting it to form the Cartesian equation: $$x=\left(\frac{y}{2}\right)^2$$.

periodic motion

The motion of an object is referred to as periodic motion if the object repeats its motion along a certain path, around a certain point in a fixed time interval, for example, waves.

polar form (of a complex number)

The complex number $$z=a+ib$$ can be expressed in polar form as: $$z=r\cos\theta+ri\sin\theta =r\left(\cos\theta+i\sin\theta\right)= rcis\theta=re^{i\theta}$$ where $$r$$ is the modulus of the complex number and $$\theta$$ is its argument expressed in radians. This is also known as modulus-argument form.

polar form (of a vector)

The polar form of a vector $$\boldsymbol v$$, in two dimensions, represents the magnitude $$r$$ and direction $$\theta$$ (measured from the positive direction of the $$x$$-axis) of the vector as an ordered pair in the form $$\boldsymbol v=(r,\theta)$$.

position vector

The position vector of a point $$P$$ in the plane is the vector joining $$P$$ to the origin.

projectile motion

Projectile motion refers to the motion of an object thrown or projected through the air under gravity. Such motion can be analysed and modelled, with the simplest useful model assuming that:

• the projectile is a point and has no spin
• the force due to air resistance is negligible
• the only force acting on the projectile is the constant force due to gravity, $$g=9.8m/s^2$$, downwards (assuming the projectile is moving close to the Earth’s surface).

Assume the opposite (negation) of what you are trying to prove. Then proceed through a logical chain of argument until you reach a demonstrably false conclusion. Since all the reasoning is correct and a false conclusion has been reached, the only thing that could be wrong is the initial assumption. Therefore, the original statement is true. This is a proof by contradiction.

For example: the result ‘$$\sqrt 2$$ is irrational’ can be proved in this way by obtaining a contradiction from the assumption that $$\sqrt2$$ is rational.

proposition

A proposition is a statement that appears to be true but has not yet been proven.

rational functions

A rational function is a function of the form $$\frac{p(x)}{q(x)} \$$where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x)\ne0$$.

recursive formulae

Recursive formulae define a sequence in which successive terms are defined as a function of the preceding terms.

For example: $$T_{n+1}=4T_n+3$$ and $$T_1=2$$.

resisted motion

Resisted motion is motion that encounters resisting forces, for example friction and air resistance.

roots of unity

A complex number $$z$$ is an $$n$$th root of unity if $$z^n=1$$. The $$n$$th roots of unity are given by: $$\cos\frac{2k\pi}{n}+i\sin\frac{2k\pi}{n}$$ where $$k=0,1,2,\ldots,n-1$$.

The points in the complex plane representing the roots of unity lie on the unit circle.

scalar

A scalar is a quantity which has magnitude but no direction.

scalar product

The scalar product of vectors $$\boldsymbol{a}$$ and $$\boldsymbol{b}$$ is denoted by $$\boldsymbol{a} \cdot \boldsymbol{b}$$.

$$\boldsymbol{a}\cdot\boldsymbol{b} = x_1 x_2 + y_1 y_2 + z_1 z_2$$ where $$\boldsymbol{a} = x_1 \boldsymbol{i} + y_1 \boldsymbol{j} + z_1 \boldsymbol{k}$$ and $$\boldsymbol{b} = x_2 \boldsymbol{i} + y_2 \boldsymbol{j} + z_2 \boldsymbol{k}$$.

$$\boldsymbol{a} \cdot \boldsymbol{b} = \left \lvert a \right \rvert \left \lvert b \right \rvert \cos \theta$$, where $$\theta$$ is the angle between the vectors $$\boldsymbol{a}$$ and $$\boldsymbol{b}$$.

Also known as the dot product.

sigma notation

The Greek capital letter sigma $$\Sigma$$, which corresponds to the letter S in English, is used to represent the summation of a series.

For example  $\sum_{k=3}^6 k^2 =3^2+4^2+5^2+6^2 =86$

The left hand component of the equation would be read as ‘the sum of $$k^2\!$$, for $$k$$ from 3 to 6’.

The letter $$k$$ is called the index of summation and the numbers 3 and 6 are called the lower and upper limits of summation, respectively.

simple harmonic motion

Simple harmonic motion is a specific example of periodic motion.

Simple harmonic motion is modelled by the equations $$x = a \cos \left(nt + \alpha\right) + c$$ or $$x = a \sin \left(nt + \alpha\right) + c$$, where:
$$x$$ = displacement
$$a$$ = amplitude
$$\frac {2\pi} n$$ = period
$$\alpha$$ = phase shift parameter
$$c$$ = centre of motion

These equations satisfy the equation $$\ddot x = - n^2 \left(x-c\right)$$. Hence, the acceleration is proportional to the displacement and always directed towards the centre of motion.
Simple harmonic motion provides a basis for exploring more complex periodic motion involving additional forces.

statement

A statement is a sentence that is either true or false. So '3 is an odd integer' is a statement, but '$$\pi$$ is a cool number' is not a statement.

terminal velocity

The terminal velocity of a particle is the limiting value of the velocity as t becomes very large.

unit vector

A unit vector is a vector with magnitude 1.

The standard unit vectors used are $$\boldsymbol{i}$$ (a unit vector in the $$x$$-direction), $$\,\boldsymbol j$$ (a unit vector in the $$y$$-direction) and $$\boldsymbol k$$ (a unit vector in the $$z$$-direction).

Any non-zero vector $$\underset{^\sim}u$$ can be made into a unit vector $$\hat{\underset{^\sim} u}$$ by dividing this vector by its length: $$\hat{\underset{^\sim} u} = \frac {\underset{^\sim}u}{\lvert \underset{^\sim}u\rvert}$$.

vector

A vector is a quantity that has magnitude and direction.

A vector can be represented using either a bold lower case letter or using a lower case letter with a tilde underneath it. For example, $$\boldsymbol a$$ or $$\underset{^\sim}a$$.

A vector from point $$A$$ to point $$B$$ can be represented by $$\overrightarrow{AB}$$.

vector form of a straight line

The vector form of a straight line that passes through a point with position vector $$\underset{^\sim}a$$ and is parallel to a vector $$\underset{^\sim}b$$ is given by $$\underset{^\sim}r=\underset{^\sim}a + \lambda\underset{^\sim}b$$. Also known as the vector equation of a straight line.