Math Foundations: quick revision, 14 bites

The Real Number System

All numbers you know fit here.

The **real number system** includes all numbers on the number line. It is divided into **rational** and **irrational** numbers. Rational numbers can be written as a fraction $\frac{p}{q}$ where $q \neq 0$. Irrational numbers cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-…

Takeaway. Every real number is either rational or irrational.

Polynomials: Definition and Degree

Expressions with many terms.

A **polynomial** is an expression of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $a_i$ are real numbers and $n$ is a non-negative integer. The **degree** is the highest power of the variable. For example, $3x^2 - 5x + 2$ has degree 2.

Takeaway. Degree tells you the highest exponent in a polynomial.

Algebraic Identities

Shortcuts for expanding expressions.

Takeaway. Memorize these identities to factor and expand quickly.

Solving Linear Equations in Two Variables

Find the point of intersection.

Takeaway. The solution is the ordered pair $(x, y)$ that satisfies both equations.

Coordinate Geometry Basics

Plotting points on a plane.

The **Cartesian plane** is divided into four quadrants by the x-axis and y-axis. A point is written as $(x, y)$. The **distance** between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Takeaway. Coordinates tell you exactly where a point is.

Heron's Formula for Triangle Area

Area from three sides.

Takeaway. Use Heron's formula when you know all three sides.

Lines and Angles

Relationships between angles.

Takeaway. Angle relationships help solve geometric problems.

Congruence of Triangles

Same shape, same size.

Two triangles are **congruent** if their corresponding sides and angles are equal. Common criteria: **SSS** (side-side-side), **SAS** (side-angle-side), **ASA** (angle-side-angle), **AAS** (angle-angle-side), and **RHS** (right angle-hypotenuse-side).

Takeaway. Congruent triangles are identical in every way.

Area of a Triangle

Half base times height.

Takeaway. This works for any triangle when you know base and height.

Circle: Circumference and Area

Round and round.

Takeaway. Remember the formulas for circumference and area.

Surface Area and Volume of Solids

3D shapes and their measures.

For a **cuboid** of length $l$, breadth $b$, height $h$: surface area = $2(lb + bh + hl)$, volume = $l \times b \times h$. For a **cylinder** of radius $r$ and height $h$: curved surface area = $2\pi rh$, total surface area = $2\pi r(r+h)$, volume = $\pi r^2 h$.

Takeaway. Know the formulas for common solids.

Statistics: Mean, Median, Mode

Summarizing data.

Takeaway. These measures describe central tendency.

Common Mistake: Cancelling Terms Incorrectly

Don't cancel across addition.

In algebra, you cannot cancel terms that are added or subtracted. For example, $\frac{x+2}{2} \neq x+1$. You can only cancel common factors in a product.

Takeaway. Cancel factors, not terms.

Exam Tip: Show Your Steps

Method marks matter.

In board exams, writing clear steps can earn you partial credit even if the final answer is wrong. Always show your work, especially in geometry and algebra.

Takeaway. Step-by-step solutions maximize your score.