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Elementary Mathematics Notes

Numbers & Their Operations
🦉 Types of Numbers
Integers ($\mathbb{Z}$): $..., -3, -2, -1, 0, 1, 2, 3, 4, ...$
Prime: integers that are divisible by 1 and itself only, smallest prime number is 2
Rational numbers ($\mathbb{Q}$) $\frac{\text{integer}}{\text{integer}}$: $\frac{4}{7}, -3\frac{1}{8}, 0.3, 2.\dot{6}\dot{5}, 92, \sqrt{16}$
Irrational numbers: $\pi, \sqrt{2}, e$
Real numbers ($\mathbb{R}$): all numbers
🦉 Standard Form
$$A \times 10^n$$
where $n$ is an integer, and $1 \leq A < 10$
🦉 SI Prefix
Prefix $10^n$
pico$10^{-12}$
nano$10^{-9}$
micro$10^{-6}$
milli$10^{-3}$
kilo$10^3$
mega$10^6$
giga$10^9$
tera$10^{12}$
🦉 Indices
$$1.\,a^m \times a^n = a^{m+n}$$
$$2.\,a^m \div a^n = a^{m-n}$$
$$3.\,(a^m)^n = a^{mn}$$
$$4.\,(ab)^m = a^m b^m$$
$$5.\,\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
$$6.\,a^{-n} = \frac{1}{a^n}$$
$$7.\,a^0 = 1$$
$$8.\,a^\frac{1}{n} = \sqrt[n]{a}$$
$$9.\,a^\frac{m}{n} = (\sqrt[n]{a})^m$$
Ratio & Proportion
🦉 Map Scale
Length scale = $1:r$
Area scale = $1:r^2$
Percentage
🦉 Percentage Change
$$\text{Percentage increase/decrease} = \frac{\text{increase/decrease}}{\text{original}} \times 100\%$$
Rate & Speed
🦉 Distance-Speed-Time Triangle
D
S
T
$$\text{Average speed} = \frac{\text{total distance}}{\text{total time}}$$
Algebraic Expressions & Formulae
🦉 nth Term (Arithmetic Sequence)
$$a + (n-1)d$$
🦉 Special Algebraic Identities
$$(a+b)^2 = a^2 + 2ab + b^2$$
$$(a-b)^2 = a^2 - 2ab + b^2$$
$$(a+b)(a-b) = a^2 - b^2$$
Equations
🦉 Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Set Language & Notation
🦉 Set Symbols
$\in$: is an element of
n($A$): number of elements in set $A$
$A'$: complement of set A
$\varnothing$: empty set
$\xi$: universal set
$\cup$: union
$\cap$: intersection
$\subset$: subset
Problems In Real-World Contexts
🦉 Simple Interest
$$I = \frac{PRT}{100}$$
🦉 Compound Interest
$$A = P\left(1 + \frac{R}{100}\right)^n$$
Angles, Triangles & Polygons
🦉 Types of Polygons
No. of sides Polygons
3triangle
4quadrilateral
5pentagon
6hexagon
7heptagon
8octagon
9nonagon
10decagon
🦉 Sum of Interior & Exterior Angles
Sum of interior angles = $(n-2) \times 180°$
Sum of exterior angles = $360°$
Congruence & Similarity
🦉 Congruent & Similar Triangles
Congruent triangles Similar triangles
SSS, SAS, AAS, RHSSSS, SAS, AAA
🦉 Ratio of Area & Volume
$$\frac{A_1}{A_2} = \left(\frac{l_1}{l_2}\right)^2$$
$$\frac{V_1}{V_2} = \left(\frac{l_1}{l_2}\right)^3$$
Pythagoras' Theorem & Trigonometry
🦉 Pythagoras' Theorem
$$a^2 + b^2 = c^2$$
🦉 Trigonometric Ratios
$$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$
$$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
TOA CAH SOH is applicable for only right-angled triangles
🦉 Obtuse Angles
$$\sin(180° - \theta) = \sin\theta$$
$$\cos(180° - \theta) = -\cos\theta$$
🦉 Sine Rule
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
🦉 Cosine Rule
$$c^2 = a^2 + b^2 - 2ab\cos C$$
🦉 Area of Triangle
$$\text{Area of triangle} = \frac{1}{2}ab\sin C$$
🦉 Bearings
000°
180°
090°
270°
A bearing is a 3-digit positive number with units of degree to show direction clockwise from the north direction.
Mensuration
🦉 Conversion
$1\text{ m} = 100\text{ cm}$
$1\text{ m}^2 = 10,000\text{ cm}^2$
$1\text{ m}^3 = 1,000,000\text{ cm}^3$
🦉 Radian & Degree
$$180° = \pi \text{ rad}$$
🦉 Arc Length & Sector Area (Degree)
$$s = \frac{\theta}{360°} \times 2\pi r$$
where $\theta$ is in degrees
$$A = \frac{\theta}{360°} \times \pi r^2$$
where $\theta$ is in degrees
🦉 Arc Length & Sector Area (Radian)
$$s = r\theta$$
where $\theta$ is in radians
$$A = \frac{1}{2}r^2\theta$$
where $\theta$ is in radians
Coordinate Geometry
🦉 Cartesian Coordinate
$$(x, y)$$
🦉 Gradient
$$m = \frac{y_1 - y_2}{x_1 - x_2}$$
🦉 Equation of Line
$$y - y_1 = m(x - x_1)$$
$$y = mx + c$$
*Vertical line: $x = a$
*Horizontal line: $y = b$
🦉 Distance Between Points
$$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Vectors In 2 Dimensions
🦉 Vector Representation
Vectors can be represented by $\begin{pmatrix}x\\y\end{pmatrix}$, $\overrightarrow{AB}$, a or $\underline{a}$.
🦉 Vector Magnitude
$$|\overrightarrow{AB}| \text{ or } |\mathbf{a}| = \sqrt{x^2 + y^2}$$
Data Analysis
🦉 Mode
Mode is the most frequently occurring number. A set of data can have more than one mode.
🦉 Mean
$$\text{mean} = \frac{\text{sum of all numbers}}{\text{number of numbers}}$$
🦉 Median
Median is the centre number when the numbers are arranged from smallest to largest.
🦉 Range
Range = maximum - minimum
🦉 Quartiles & Percentiles
0th
25th
50th
75th
100th
min
lower quartile
median
upper quartile
max
🦉 Interquartile Range
Interquartile range = upper quartile - lower quartile
🦉 Box-and-Whisker Plot
minimum
lower quartile
median
upper quartile
maximum
🦉 Mean & Standard Deviation (Ungrouped)
$$\text{Mean}, \bar{x} = \frac{\sum x}{N}$$
$$\text{Standard deviation}, \sigma = \sqrt{\frac{\sum(x-\bar{x})^2}{N}}$$
🦉 Mean & Standard Deviation (Grouped)
$$\text{Mean}, \bar{x} = \frac{\sum fx}{\sum f}$$
$$\text{Standard deviation}, \sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$$