Math basic formulas and notes.

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How are Whole Numbers classified based on their factors?

• Neither prime nor composite: The numbers 0 and 1. • Prime numbers: Whole numbers with exactly 2 different factors (1 and itself). • Composite numbers: Whole numbers with more than 2 different factors.

What are the definitions and examples for Perfect Squares, Perfect Cubes, HCF, and LCM?

• Perfect Square: A whole number whose square root is also a whole number (e.g., sqrt(25) = 5). • Perfect Cube: A whole number whose cube root is also a whole number (e.g., cbrt(125) = 5).

• HCF: The largest factor common to all numbers (e.g., HCF of 18 and 30 is 6). • LCM: The smallest multiple common to all numbers (e.g., LCM of 12, 18, 56 is 504). • Prime Factorisation: Expressing a composite number as a product of primes (e.g., 18 = 2 x 3^2).

What is the complete Real Numbers classification tree?

• Irrational numbers: Non-terminating/non-recurring decimals like pi and sqrt(7).

• Rational numbers: Terminating or recurring decimals.

• Non-integer rational numbers: Fractions and decimals like -3/8 or -0.3. • Integers: Split into Negative integers (-4, -7) and Whole numbers.

• Whole numbers: Split into Zero and Positive integers (2, 8).

What are the rules and examples for Adding and Subtracting Negative Numbers?

• Addition of two negatives: Add values, keep negative, e.g., (-2)+(-1) = -3.

• Addition of pos/neg: Subtract smaller from larger, keep sign of larger, e.g., 5+(-2) = 3.

• Subtraction between two positives: Can yield a negative if subtracting a larger number, e.g., 2-5 = -3.

• Subtraction of positive from negative: Make it more negative, e.g., -5-2 = -7.

• Subtraction of a negative: Turns into a direct addition, e.g., 5-(-2) = 5+2 = 7.

What are the unique rules for Square Roots and Cube Roots of positive vs. negative numbers?

• Positive numbers: Have two square roots (+/-sqrt(64) = +/-8) but only one cube root (cbrt(64) = 4).

• Negative numbers: Have no real square root, but have exactly one negative cube root (cbrt(-64) = -4).

What are the Five Rules to identify digits which are significant (sig. fig.)?

• Rule 1: All non-zero digits are significant.

• Rule 2: All zeroes between non-zero digits are significant.

• Rule 3: In a decimal, all zeroes before a non-zero digit are NOT significant.

• Rule 4: In a decimal, all zeroes after a non-zero digit are significant.

• Rule 5: Zeroes at the end of a whole number may or may not be significant depending on rounding.

What are the Follow-through Error rules for intermediate working vs. final answers?

• For presentation: Write 4 or 5 significant figures for intermediate working steps

. • Final answer accuracy: Ensure the final recorded answer is rounded to 3 significant figures.

• For calculation: Always use the exact unrounded values stored directly in your calculator memory.

• Estimation definition: The process of guessing the value of an unknown quantity in a real-world scenario.

What is a Linear Expression in one variable vs. two variables? Give examples.

• In one variable x: An algebraic expression containing only one term in x, with or without a constant, e.g., 2x + 7 or -1/3 x.

• In two variables x and y: Contains terms with both x and y variables, e.g., 4x + y - 8 or 1/2 x + 3/4 y.

What are the core laws of Expansion, Factorisation, and Negatives in Chapter 4?

• Distributive Law: a(b + c) = ab + ac.

• Factorisation: Expressing an algebraic expression as the product of two or more factors (reverse of expansion).

• Negative of a bracket: -(x + y) = -x - y and -(x - y) = -x + y.

• Extracting negative common factor: -x - y = -(x + y).

• Simplification constraint: Unlike terms cannot be further added or subtracted together.

What does it mean to solve an equation, and how do you handle fractional equations?

• Solving an equation in x: Finding the value of x that makes the equation true by isolating x using balanced operations.

• Fractional equations: Clear fractions by multiplying both sides by the LCM of the denominators.

• Formula definition: An equation that algebraically expresses a relationship between distinct quantities or variables (e.g., A = l x b).

What is a Function, and what are the 4 ways to represent it?

• Definition: A relationship between two variables x and y such that every single input x produces exactly one output y (e.g., y = 2x - 3).

• 4 Representations: Words, An equation, A table of values, A graph.

What are the rules for Straight Line Equations, Gradients, and Distance-Time Graphs?

• Equation of a straight line: y = mx + c.

• m meaning: The gradient, calculated as rise / run = (vertical change) / (horizontal change).

• c meaning: The y-intercept, where the line hits the vertical axis.

• Gradient direction: Sloping upwards left-to-right is Positive; sloping downwards is Negative.

• Distance-time graph: The gradient of the line measures the exact speed of the moving object.

What is a Number Sequence, a General Term, and the three methods to find a pattern?

• Number Sequence: A set of terms governed and ordered by a specific rule.

• General Term (Tn): An algebraic expression written in terms of its position index n. • Method A: Examining the common difference between consecutive terms.

• Method B: Transforming the original sequence to another more familiar sequence.

• Method C: By direct observation based on the structural relationship between a sequence and its graph.

What are the definitions, formulas for Percentage Change, and New Value in Chapter 8?

• Percentage: x% = x/100 (representing a part per hundred).

• Expressing percentage: (B / A) x 100% (both quantities must use identical units). • Percentage Change: (Increase or Decrease / Original Value) x 100%.

• Change value: Increase or Decrease = Percentage Change x Original Value.

• New Value: Final Percentage x Original Value.

What are the formulas for Profit, Loss, Discount, GST, and Commission?

• Profit: Selling Price - Cost Price.

• Loss: Cost Price - Selling Price.

• Profit/Loss %: (Profit or Loss / Cost Price) x 100%.

• Discount: Marked Price - Sale Price.

• Percentage Discount: (Discount / Marked Price) x 100%.

• GST: Tax paid in addition to the base price of goods or services, including service charges.

• Commission: Payment an agent receives for conducting a transaction on behalf of another party.

What are the specific definitions and rules for Ratios and Rates?

• Ratio: Comparing two or more quantities of the exact same kind; the ratio a : b has no units and can be written as a/b.

• Rate: Comparing how one quantity changes relative to another quantity of same or different kinds.

• Rate expression: Expressed as units of one quantity per unit of another quantity.

• Conversion rule: A rate that compares two quantities of the same kind can be converted into a unitless ratio.

What are the definitions and formulas for Constant Speed and Average Speed?

• Constant Speed: An object travels at constant speed when its instantaneous speed does not change throughout the entire journey.

• Average Speed Formula: Total Distance Travelled / Total Time Taken.

What are the definitions of the 4 primary types of angles, plus complementary and supplementary angles?

• Acute angle: Less than 90 degrees.

• Right angle: Exactly equal to 90 degrees.

• Obtuse angle: More than 90 degrees but less than 180 degrees.

• Reflex angle: More than 180 degrees but less than 360 degrees.

• Complementary angles: Two angles that add up to exactly 90 degrees.

• Supplementary angles: Two angles that add up to exactly 180 degrees.

What are the notation symbols and properties of lines, intersecting angles, and points?

• Adjacent angles on a straight line: add up to 180 degrees, a + b = 180 (adj. ∠s on a str. line).

• Vertically opposite angles: are equal to each other, a = b and c = d (vert. opp. ∠s).

• Angles at a point: add up to 360 degrees, a + b + c + d = 360 (∠s at a point).

What are the parallel line transversal theorems and basic triangle properties from Chapter 10?

• Parallel Lines (AB || CD):

Corresponding angles are equal (corr. ∠s);

Alternate angles are equal (alt. ∠s);

Interior angles add up to 180 (int. ∠s).

• Triangle Interior: Sum of interior angles of a triangle is always 180 degrees (∠ sum of Δ).

• Triangle Exterior: An exterior angle equals the sum of its two opposite interior angles (ext. ∠ of Δ).

• Isosceles Triangle: The base angles opposite the equal sides are always equal (base ∠s of isos. Δ).

What are the properties of Triangles, Quadrilaterals, General Polygons, and the 3 conditions for constructing a triangle?

• Angle Sums: Triangle interior = 180 deg, Quadrilateral interior = 360 deg.

• General Polygon: Interior angle + exterior angle = 180 deg.

• Polygon Interior Sum: Sum of interior angles of an n-sided polygon = (n - 2) x 180 deg.

• Polygon Exterior Sum: Sum of exterior angles of any convex n-sided polygon = 360 deg.

• Construction Cases: (a) 3 sides given, (b) 1 side and 2 angles given, (c) 2 sides and 1 angle given.

What are the diagonal, symmetry, and rotational properties of the 5 Special Quadrilaterals?

• Parallelogram: Diagonals bisect; symmetry lines = 0; rotational order = 2.

• Rectangle: Diagonals bisect and are equal length; symmetry lines = 2; rotational order = 2.

• Rhombus: Diagonals bisect at 90 deg and bisect interior angles; symmetry lines = 2; rotational order = 2.

• Square: Diagonals bisect at 90 deg, equal length, bisect angles; symmetry lines = 4; rotational order = 4.

• Kite: Diagonals cross at 90 deg, one diagonal bisects angles; symmetry lines = 1; rotational order = 1.

What are the perimeter and area formulas for Rectangles, Squares, Triangles, and Circles?

• Rectangle: Perimeter = 2(l + b), Area = l x b.

• Square: Perimeter = 4l, Area = l^2.

• Triangle: Perimeter = a + b + c, Area = 1/2 x b x h (where h is the perpendicular height).

• Circle: Perimeter (Circumference) = 2 x pi x r OR pi x d, Area = pi x r^2.