Mathematical Skills and Units (Grade 10 NSC Matric Physical Sciences): Revision Notes

Mathematical Skills and Units

Mathematical skills

Mathematics forms the foundation of physical sciences. You need to be comfortable with several key mathematical concepts including scientific notation, unit conversions, and formula manipulation. These skills help you work with very large or small numbers, convert between different measurement systems, and solve problems involving rates, proportions, and ratios.

Rounding off

When working with measurements and calculations, you often need to round off numbers to a reasonable number of decimal places. This makes your answers more practical and avoids suggesting false precision.

How to round off numbers:

• Count the required number of decimal places from the decimal point
• Look at the digit immediately after your last required decimal place
• If this digit is 5 or greater, round up the last required digit
• If this digit is 4 or less, leave the last required digit unchanged
• Remove all digits after your rounded position

Scientific notation

Scientific notation allows you to write very large or very small numbers in a compact, standardised form. This is essential in physics where you encounter quantities ranging from subatomic particles to astronomical distances.

Standard form: $N \times 10^n$

Where:

• N is a decimal number between 1 and 10
• n is an integer (positive or negative)
• If n > 0, the decimal point moves right (large numbers)
• If n < 0, the decimal point moves left (small numbers)

Converting to scientific notation:

1. Move the decimal point to create a number between 1 and 10
2. Count how many places you moved the decimal
3. If you moved left, n is positive; if you moved right, n is negative

Example: Speed of light = 299 792 458 m⋅s⁻¹ = $3.00 \times 10^8$ m⋅s⁻¹

Calculations with scientific notation

Addition and subtraction: Make all exponents the same, then add or subtract the N values.

Multiplication and division: Multiply or divide the N values, then add or subtract the exponents.

Units

Imagine buying fabric without specifying whether you need 2 metres or 2 centimetres! Units give meaning to numbers and make communication in science precise and unambiguous.

What are physical quantities?

Definition: A physical quantity is anything that can be measured, such as length, mass, time, temperature, or electric current.

All physical quantities have two parts:

• A numerical value (how much)
• A unit (what type of measurement)

SI units

The SI system (Système International d'Unités) provides internationally agreed units for scientific measurements. This system ensures scientists worldwide can communicate measurements clearly and accurately.

Definition: SI units are the internationally agreed system of measurement units used in science.

The SI system is built on seven base units that cannot be expressed in terms of other units:

Base quantity	Name	Symbol
length	metre	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

Combinations of SI base units

Many physical quantities use derived units formed by combining base units. These combinations often have special names to make calculations easier.

Quantity	Formula	Unit Expressed in Base Units	Name of Combination
Force	ma	kg⋅m⋅s⁻²	N (newton)
Frequency	1/T	s⁻¹	Hz (hertz)
Work	Fs	kg⋅m²⋅s⁻²	J (joule)

Unit prefixes

Unit prefixes multiply base units by powers of 10, allowing you to express very large or small quantities conveniently.

Prefix	Symbol	Exponent	Prefix	Symbol	Exponent
yotta	Y	10²⁴	yocto	y	10⁻²⁴
zetta	Z	10²¹	zepto	z	10⁻²¹
exa	E	10¹⁸	atto	a	10⁻¹⁸
peta	P	10¹⁵	femto	f	10⁻¹⁵
tera	T	10¹²	pico	p	10⁻¹²
giga	G	10⁹	nano	n	10⁻⁹
mega	M	10⁶	micro	μ	10⁻⁶
kilo	k	10³	milli	m	10⁻³
hecto	h	10²	centi	c	10⁻²
deca	da	10¹	deci	d	10⁻¹

Examples:

• 40 000 m = 40 km (kilometre)
• 0.001 g = 1 mg (milligram)
• 250 000 A = 250 kA (kiloampere)

The importance of units

Units are essential in science because they:

• Give meaning to numerical values
• Allow clear communication between scientists
• Prevent dangerous errors in calculations
• Enable accurate comparisons between measurements

How to change units

Unit conversion is a crucial skill that allows you to change measurements from one unit to another while keeping the same physical quantity.

Length conversions

The basic pattern for metric length conversions follows powers of 1000:

• mm → m: divide by 1000
• m → km: divide by 1000
• km → m: multiply by 1000
• m → mm: multiply by 1000

Volume conversions

Volume conversions also follow the 1000-fold pattern:

• mℓ/cm³ → ℓ/dm³: divide by 1000
• ℓ/dm³ → kℓ/m³: divide by 1000

Worked examples of unit conversion

Additional useful conversions

Speed conversion:

• km⋅h⁻¹ to m⋅s⁻¹: multiply by 1000, divide by 3600 (or multiply by 1000/3600)
• m⋅s⁻¹ to km⋅h⁻¹: multiply by 3600, divide by 1000 (or multiply by 3600/1000)

Temperature conversion: $T_K = T_{°C} + 273$

Where $T_K$ is temperature in kelvin and $T_{°C}$ is temperature in Celsius.

Changing the subject of a formula

Often you need to rearrange equations to solve for different variables. This skill allows you to find any quantity when you know the others.

Method: Use algebraic operations to isolate the desired variable on one side of the equation.

Rate, proportion and ratios

Rate

Rate measures how quickly something changes over time. It always involves a change per unit time.

Examples:

• Speed: change in position per unit time ($\Delta s/\Delta t$)
• Rate of reaction: change in concentration per unit time ($\Delta C/\Delta t$)

Proportion

Proportion describes relationships between variables.

Direct proportion: When one variable increases, the other increases proportionally

• Written as: $y \propto x$ or $y = kx$ (where k is constant)
• Graph: straight line through origin

Inverse proportion: When one variable increases, the other decreases proportionally

• Written as: $y \propto \frac{1}{x}$ or $y = \frac{k}{x}$ (where k is constant)
• Graph: hyperbolic curve

Ratios and fractions

Ratios compare the relative sizes of quantities and can be written as fractions or percentages.

Example: In a chemical reaction, if reactants combine in a 2:1 ratio, this means 2 moles of one reactant for every 1 mole of another.

Constants in equations

Constants are values that remain the same in equations and have fixed numerical values.

Constant	Symbol	Value and units	SI Units
Atomic mass unit	u	$1.67 \times 10^{-24}$ g	$1.67 \times 10^{-27}$ kg
Charge on an electron	e	$-1.6 \times 10^{-19}$ C	$-1.6 \times 10^{-19}$ s⋅A
Speed of sound (in air, at 25°C)		344 m⋅s⁻¹
Speed of light	c	$3 \times 10^8$ m⋅s⁻¹
Planck's constant	h	$6.626 \times 10^{-34}$ J⋅s	$6.626 \times 10^{-34}$ kg⋅m²s⁻¹
Avogadro's number	$N_A$	$6.022 \times 10^{23}$
Gravitational acceleration	g	9.8 m⋅s⁻¹

Trigonometry

Trigonometry deals with relationships between angles and sides in right-angled triangles. These relationships are expressed as ratios.

The three basic trigonometric ratios:

Sine: $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$

Cosine: $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$

Tangent: $\tan A = \frac{\text{opposite}}{\text{adjacent}}$

Summary