Maths1_Prerequisites_Comprehensive

Maths 1: Comprehensive Prerequisites Guide

IMPORTANT

Purpose: This guide is designed to teach you the background mathematics required for Maths 1. It assumes you have forgotten most high school math and builds up from the basics.

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1. Number Systems (The Building Blocks)

Before we do math, we need to know what “numbers” we are allowed to use.

1.1 The Hierarchy of Numbers

Think of numbers like a set of nesting dolls. Each set contains the previous one.

1. Natural Numbers ($\mathbb{N}$): The counting numbers.
   • $\mathbb{N}=\{1,2,3,4,\dots \}$
   • Note: Sometimes includes 0, but usually starts at 1.
2. Integers ($\mathbb{Z}$): Natural numbers plus 0 and negatives.
   • $\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$
   • Concept: Think of debt (negative) vs. asset (positive).
3. Rational Numbers ($\mathbb{Q}$): Any number that can be written as a fraction $\frac{p}{q}$ (where $q\ne 0$).
   • Examples: $\frac{1}{2},-5(\text{which is }\frac{-5}{1}),0.75(\text{which is }\frac{3}{4})$.
   • Concept: “Ratio” - parts of a whole.
4. Real Numbers ($\mathbb{R}$): All numbers on the continuous number line.
   • Includes Rationals AND Irrationals (numbers that go on forever without repeating, like $\pi \approx 3.14159\dots$ or $\sqrt{2}\approx 1.414\dots$).

1.2 Intervals (Describing Chunks of Numbers)

Instead of listing numbers, we often want “everything between 2 and 5”.

• Closed Interval $[a,b]$: Includes endpoints. $a\le x\le b$.
  • Notation: Square brackets $[]$.
  • Example: $[2,5]$ includes 2, 3, 4, 5, 2.1, 4.999…
• Open Interval $(a,b)$: Excludes endpoints. $a<x<b$.
  • Notation: Parentheses $()$.
  • Example: $(2,5)$ includes 2.0001, 4.9999, but NOT 2 or 5.
• Mixed: $[2,5)$ means $2\le x<5$.

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2. Basic Algebra (The Language)

Algebra is just solving puzzles where $x$ is the missing piece.

2.1 Solving Linear Equations

Goal: Get $x$ by itself. Rule: Whatever you do to one side, do to the other.

Example: Solve $3x+5=11$.

1. Subtract 5: $3x=11-5⟹3x=6$.
2. Divide by 3: $x=\frac{6}{3}⟹x=2$.

2.2 Exponents (Powers)

Shorthand for repeated multiplication. $x^3=x\cdot x\cdot x$.

Key Rules:

1. Product: $x^a\cdot x^b=x^{a+b}$ (Add powers).
   • $x^2\cdot x^3=(x\cdot x)\cdot (x\cdot x\cdot x)=x^5$.
2. Power of Power: $(x^a{)}^b=x^{a\cdot b}$ (Multiply powers).
   • $(x^2{)}^3=x^2\cdot x^2\cdot x^2=x^6$.
3. Negative Power: $x^{-a}=\frac{1}{x^a}$ (Flip it).
   • $2^{-1}=\frac{1}{2}$.
4. Zero Power: $x^0=1$ (Always, except $0^0$).

2.3 Factoring Quadratics

Turning a sum ($x^2+5x+6$) into a product $(x+2)(x+3)$. Method: Find two numbers that:

1. Multiply to the last number (6).
2. Add to the middle number (5).

• Numbers are 2 and 3.
• So $(x+2)(x+3)$.

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3. Coordinate Geometry (The Map)

We draw math on a 2D plane (Cartesian Plane).

3.1 The Axes

• X-axis: Horizontal line (Left/Right).
• Y-axis: Vertical line (Up/Down).
• Origin: $(0,0)$ where they cross.

3.2 Slope (Steepness)

How fast does the line go up?

• Formula: $m=\frac{\text{Change in Y}}{\text{Change in X}}=\frac{y_2-y_1}{x_2-x_1}$.
• Positive Slope: Uphill ($\nearrow$).
• Negative Slope: Downhill ($\searrow$).
• Zero Slope: Flat (Horizontal).
• Undefined Slope: Vertical cliff.

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4. Functions (The Machines)

Think of a function $f$ as a machine.

• Input ($x$): Raw material.
• Output ($f(x)$ or $y$): Finished product.
• Rule: The machine must be consistent. One input gives exactly one output.

4.1 Domain & Range

• Domain: What you are allowed to put IN.
  • Constraint 1: No dividing by zero. (If $1/x$, then $x\ne 0$).
  • Constraint 2: No square root of negatives. (If $\sqrt{x}$, then $x\ge 0$).
• Range: What can possibly come OUT.

4.2 Composition ($f\circ g$)

Chaining machines together.

• $f(g(x))$ means: Put $x$ into $g$, get result, put result into $f$.

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5. Logarithms (The Inverse of Powers)

If $2^3=8$, how do we ask “2 to what power is 8?”

• We write: ${\log }_2(8)=3$.

Key Properties:

• $\log (xy)=\log x+\log y$ (Multiplication becomes Addition).
• $\log (x/y)=\log x-\log y$ (Division becomes Subtraction).
• $\log (x^k)=k\log x$ (Power comes down).

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6. Inequality Logic (Wavy Curve)

How to solve $(x-1)(x+2)>0$?

1. Find Roots: Where is it zero? ($1,-2$).
2. Plot on Line: $-2$ and $1$.
3. Test Regions:
   • Pick $x=100$: $(+)(+)=+$. (Right side is Positive).
   • Cross 1 (Odd power): Sign flips to Negative.
   • Cross -2 (Odd power): Sign flips to Positive.
4. Select: We want $>0$, so regions $(-\infty ,-2)\cup (1,\infty )$.