Complete Number Sets, Algebra, & Equations

Part 1: The Ultimate Hierarchy of Numbers

Before manipulating numbers through algebraic operations, we must precisely define what species of numbers we are dealing with. Mathematics categorizes numbers into distinct sets, each expanding upon the limitations of the previous one. Understanding these definitions is strictly required for logical reasoning questions.

• Natural Numbers ($\mathbb{N}$): The most basic numbers used for counting items. Typically defined as $\{1, 2, 3, 4, ...\}$. Note that some mathematical conventions include zero in this set, but strictly speaking, the counting numbers start at 1.
• Integers ($\mathbb{Z}$): We introduce zero and the concept of negative quantities (debt, temperature below zero). The set is $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.
• Rational Numbers ($\mathbb{Q}$): The word rational stems from "ratio". Any number that can be expressed as a perfect ratio of two integers $\frac{p}{q}$ (where $q \neq 0$) is rational. This powerful set includes all integers (since $5 = \frac{5}{1}$), terminating decimals ($0.25 = \frac{1}{4}$), and infinitely repeating decimals.
• Irrational Numbers: Numbers that absolutely cannot be expressed as a simple fraction. Their decimal expansions are both non-terminating and non-repeating. Classic examples include $\sqrt{2}, \sqrt{3}, \pi$, and Euler's number $e$.
• Real Numbers ($\mathbb{R}$): The complete union of Rational and Irrational numbers. Every single point on an infinitely long, continuous number line corresponds to exactly one Real number.

1.1 Prime Numbers & Factorization

Within the set of Natural numbers, we distinguish between Prime and Composite numbers. A Prime Number is a natural number strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself. (e.g., 2, 3, 5, 7, 11, 13, 17...).

A Composite Number has more than two divisors. The Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime itself or can be represented uniquely as a product of prime numbers. This is called Prime Factorization.

1.2 Converting Repeating Decimals to Fractions

1.3 Scientific Notation (Standard Form)

Scientific notation is a method for expressing extremely large or extremely small numbers concisely. It is written in the form $A \times 10^n$, where $1 \le |A| < 10$ and $n$ is an integer.

• Large numbers: Move the decimal to the left, and $n$ is positive.
  Example: $4,500,000 = 4.5 \times 10^6$.
• Small numbers: Move the decimal to the right, and $n$ is negative.
  Example: $0.000032 = 3.2 \times 10^{-5}$.

Part 2: Indices, Radicals, and Rationalization

Indices (exponents) and radicals (roots) are two sides of the exact same mathematical coin. An index denotes repeated multiplication, while a radical asks for the base that was multiplied. Mastery of the laws governing these operations is non-negotiable for algebra.

2.1 Operations with Radicals

You can only add or subtract radicals if they have the exact same radicand (the number inside the root) and the same index. This is similar to combining like terms in algebra ($2x + 5x = 7x$).

2.2 Rationalizing the Denominator

In mathematics, it is standard convention to remove irrational numbers (roots) from the denominator of a fraction. This process is called rationalization. We achieve this by multiplying the fraction by a strategic form of the number $1$.

• Case 1: Single Term Denominator. If the denominator is a simple square root, multiply the numerator and the denominator by that exact square root.
  Example: $$ \frac{7}{\sqrt{3}} = \frac{7}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{7\sqrt{3}}{(\sqrt{3})^2} = \frac{7\sqrt{3}}{3} $$
• Case 2: Binomial Denominator. If the denominator is an expression with two terms (like $a + \sqrt{b}$), you must multiply the numerator and denominator by its Conjugate (change the middle sign to $a - \sqrt{b}$). This exploits the difference of squares algebraic identity $(x-y)(x+y) = x^2 - y^2$ to eliminate the square roots entirely.

2.3 Advanced Technique: Unnesting Double Radicals

An expression nested inside another root, like $\sqrt{A \pm \sqrt{B}}$, can sometimes be simplified into the much cleaner form $\sqrt{a} \pm \sqrt{b}$. The trick relies directly on reversing the binomial expansion formula: $(\sqrt{a} \pm \sqrt{b})^2 = a + b \pm 2\sqrt{ab}$.

Part 3: Comprehensive Polynomial Expansion

Expansion is the process of multiplying out brackets. Memorizing these standard expansions saves critical time during the exam. They form the foundational vocabulary of all advanced algebra.

3.1 Geometric Visualization of Expansion

Algebraic identities are not arbitrary rules; they describe fundamental geometric realities. Let us visualize why $(a+b)^2 = a^2 + 2ab + b^2$ by looking at the area of a square.

3.2 Pascal's Triangle for Higher Powers

If you need to expand $(a+b)^4$ or higher, memorizing formulas becomes impractical. Pascal's Triangle provides the exact coefficients needed for any binomial expansion $(a+b)^n$.

To build the triangle, start with 1 at the top. Every number below is the sum of the two numbers directly above it. The row index $n$ (starting at $n=0$ for the top row) corresponds to the power of the binomial.

Part 4: The Art and Strategy of Factoring

Factoring takes an expanded polynomial and breaks it down into a product of simpler parentheses. It is the exact reverse of expansion and is the most critical skill for solving equations. When faced with an expression to factor, you must follow this strict hierarchy of attack:

1. 1. Greatest Common Factor (GCF): Always check for this first before applying any complex formulas.
   
   Solve: $12x^3y^2 - 18x^2y^3$
   The GCF of the numbers is 6. For variables, take the lowest power: $x^2$ and $y^2$.
   Result: $6x^2y^2(2x - 3y)$
2. 2. Recognize Standard Formulas: Look for patterns from Part 3.
   
   Difference of Squares: $49x^2 - 25 = (7x - 5)(7x + 5)$
   Sum of Cubes: $8x^3 + 27 = (2x)^3 + 3^3 = (2x + 3)(4x^2 - 6x + 9)$
3. 3. Factoring by Grouping (4 or more terms): Group terms strategically to extract a common binomial bracket.
   
   Expression: $x^3 + 2x^2 - 4x - 8$
   Group first two and last two: $(x^3 + 2x^2) - (4x + 8)$
   Factor each group: $x^2(x + 2) - 4(x + 2)$
   Extract common bracket: $(x + 2)(x^2 - 4)$
   Factor the remaining difference of squares: $\mathbf{(x + 2)(x - 2)(x + 2) = (x + 2)^2(x - 2)}$
4. 4. The AC Method / Cross Method for Quadratics:

   When factoring a trinomial $ax^2 + bx + c$ where the leading coefficient $a \neq 1$, trial and error is frustrating. The Cross Method (AC Method) is a visual algorithm that guarantees success. You pair factors of $a$ and factors of $c$, cross-multiply them, and ensure their sum matches the middle term $b$.

   Factor: $6x^2 + 11x - 10$

   Since the sum matches the middle term $11x$, we read the factors horizontally row by row.

   Result: $(2x + 5)(3x - 2)$

4.1 The Factor Theorem

If you are dealing with a polynomial $P(x)$ of degree 3 or higher, and grouping doesn't work, you must use the Factor Theorem. The theorem states: If you plug a number $c$ into the polynomial and the result is zero (i.e., $P(c) = 0$), then $(x - c)$ is guaranteed to be a factor.

Part 5: Solving Equations & Polynomial Division

An equation $f(x) = 0$ is a question: "At what exact points does the graph of this mathematical function cross the horizontal x-axis?" Solving an equation is finding those roots.

5.1 Quadratic Equations & The Discriminant

When a quadratic equation $ax^2 + bx + c = 0$ cannot be factored easily, the Quadratic Formula is the universal tool.

The value inside the square root, $D = b^2 - 4ac$, is called the Discriminant. It fundamentally controls the nature of the roots by determining what happens when you add or subtract the square root.

5.2 Biquadratic Equations

Equations of the form $ax^4 + bx^2 + c = 0$ appear terrifying because of the 4th power, but they are secretly quadratics in disguise. You solve them using a mathematical technique called substitution.

5.3 Polynomial Long Division

When applying the Factor Theorem, you must divide a larger polynomial by a known linear factor to find the remaining polynomial. Polynomial Long Division works exactly like elementary numerical long division.

5.4 Rational and Radical Equations (The Threat of Extraneous Roots)

Part 6: Flawless Inequalities & The Wavy Curve Method

6.1 Higher-Degree Polynomial Inequalities (The Wavy Curve Method)

When dealing with inequalities involving polynomials of degree 3 or higher, or complex rational fractions, calculating test values for every single interval takes too much time. The Wavy Curve Method (Sign Chart) is a powerful visual algorithm to solve these instantly.

6.2 Fractional Inequalities: The Squaring Method

Consider the inequality $\frac{x-1}{x+3} \ge 0$. You cannot multiply both sides by the denominator $(x+3)$ because you do not know if $(x+3)$ is positive or negative. Depending on the unknown value of $x$, you might need to flip the inequality sign, which makes multiplication invalid.

6.3 Absolute Value Equations and Inequalities

The absolute value $|X|$ geometrically represents the distance of a number $X$ from zero on the number line. Because distance is always positive, absolute value equations and inequalities naturally split into two separate scenarios.

Part 7: Logarithms & Exponential Equations

If an exponent answers the question "What is the base multiplied by itself $y$ times?", a logarithm answers the reverse: "To what power $y$ must we raise the base $b$ to get the number $x$?"

Part 8: Mathematical Translation (Word Problems)

A significant portion of the IMAT tests your ability to translate real-world English sentences into abstract algebraic equations. This requires mapping verbal relationships directly to mathematical operators.

8.1 Translation Dictionary

8.2 Classic IMAT Problem Types

Part 9: Ultimate IMAT Mathematical Mastery (50 Questions)

This comprehensive diagnostic test covers every single concept taught in this module. From identifying prime numbers to expanding complex binomials, unwrapping double radicals, executing polynomial long division, and dissecting tricky word problems.

Instructions: Answer all 50 questions. Take your time. Once submitted, detailed step-by-step explanations will be revealed for every single question to help you analyze your mistakes.