📘 Arithmetic & Geometry: A Visual Journey

🔢 Arithmetic

Natural Numbers: $1, 2, 3, \ldots$
Integers: $\ldots, -2, -1, 0, 1, 2, \ldots$
Fractions: $\frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4}$
Decimals: $0.5 + 0.25 = 0.75$
Percentages: $25\% = \frac{1}{4} = 0.25$
Order of Operations: $3 + 6 \div 2 = 3 + 3 = 6$
Exponents: $2^3 = 8,\ \sqrt{16} = 4$
Ratios: $2:3 = \frac{2}{3}$
GCF of 12 & 18: $6$ (using Euclidean algorithm: $18 = 12 \cdot 1 + 6$, $12 = 6 \cdot 2 + 0$)
LCM of 4 & 6: $12$ (using GCF: $\text{LCM} = \frac{4 \cdot 6}{\text{GCF}(4,6)} = \frac{24}{2} = 12$)
Prime Numbers: $2, 3, 5, 7, 11$ (numbers divisible only by 1 and themselves)
Modular Arithmetic: $15 \mod 4 = 3$ (since $15 = 4 \cdot 3 + 3$)
Basic Algebra: Solve $2x + 3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2$

Types of Angles: $45^\circ$ (Acute), $90^\circ$ (Right), $135^\circ$ (Obtuse)
Triangle Sum Theorem: $50^\circ + 60^\circ + 70^\circ = 180^\circ$
Rectangle Area: $A = l \times w = 5 \times 3 = 15$
Circle Area: $A = \pi r^2,\ r = 3 \Rightarrow A = 9\pi$
Volume of Cube: $V = a^3 = 4^3 = 64$
Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, e.g., points $(1,2)$ and $(4,6)$: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5$
Symmetry: A square has 4 lines of symmetry
Pythagorean Theorem: $a^2 + b^2 = c^2 \Rightarrow 3^2 + 4^2 = 9 + 16 = 25 \Rightarrow c = 5$
Circle Circumference: $C = 2\pi r,\ r = 5 \Rightarrow C = 10\pi$
Surface Area of Sphere: $A = 4\pi r^2,\ r = 2 \Rightarrow A = 16\pi$
Parallel Lines: If $l \parallel m$ and a transversal intersects, corresponding angles are equal, e.g., $\angle 1 = \angle 2$