Real Number System

Hello there, Grade 10 students! Welcome to the world of the "Real Number System"

If mathematics were like building a house, the "Real Number System" would be the bricks and mortar—the most essential foundation. Whether you go on to study functions, calculus, or statistics, everything rests on the foundation of real numbers.

In this chapter, we will organize the numbers you've known since childhood into clear categories and learn how to solve equations and inequalities like a pro. Don't worry if math has felt difficult before; we’ll take it one step at a time, together!

1. The Structure of the Real Number System

Imagine the real numbers as a "big family" living together. We can group the members as follows:

1.1 Real Numbers (\( \mathbb{R} \)): Any number that can be plotted on a number line.

1.2 Rational Numbers (\( \mathbb{Q} \)): Numbers that can be written as fractions, such as \( \frac{1}{2}, 5, -3 \), or repeating decimals like \( 0.333... \)

1.3 Irrational Numbers (\( \mathbb{Q}' \)): Numbers that cannot be expressed as fractions. They are non-terminating and non-repeating decimals, such as \( \pi, \sqrt{2}, \sqrt{3} \)

1.4 Integers (\( \mathbb{I} \) or \( \mathbb{Z} \)): Divided into:

• Positive integers: \( 1, 2, 3, ... \)
• Zero: \( 0 \)
• Negative integers: \( -1, -2, -3, ... \)

Key Note:

Did you know? All repeating decimals are rational numbers! For example, \( 0.121212... \) can be written as the fraction \( \frac{12}{99} \)

Quick Summary: Real Numbers = Rational + Irrational

2. Properties of Real Numbers

These properties help us calculate numbers more easily and accurately:

• Commutative Property: \( a + b = b + a \) and \( a \cdot b = b \cdot a \) (You can swap terms in addition and multiplication, but not in subtraction or division!)
• Associative Property: \( (a + b) + c = a + (b + c) \)
• Distributive Property: \( a(b + c) = ab + ac \) (Use this often when solving equations!)
• Additive Identity: \( 0 \), because adding it to any number leaves the value unchanged.
• Multiplicative Identity: \( 1 \), because multiplying any number by it leaves the value unchanged.
• Additive Inverse: The number that results in \( 0 \) when added to the original. For example, the additive inverse of \( 5 \) is \( -5 \).
• Multiplicative Inverse: The number that results in \( 1 \) when multiplied by the original. For example, the multiplicative inverse of \( 2 \) is \( \frac{1}{2} \) (Note: \( 0 \) has no multiplicative inverse).

3. Solving Single-Variable Polynomial Equations

If it feels tricky at first, don't worry! The secret to solving equations is "Factoring."

Must-Know Techniques:

1. Difference of Squares: \( x^2 - a^2 = (x - a)(x + a) \)

2. Perfect Square Trinomials: \( (x + a)^2 = x^2 + 2ax + a^2 \)

3. Factoring Trinomials (The 2-Bracket Method): Find two numbers that "multiply to the constant term and add up to the middle term."

Example: Solve the equation \( x^2 - 5x + 6 = 0 \)
We need numbers that multiply to \( 6 \) and add to \( -5 \), which are \( -2 \) and \( -3 \).
This gives us \( (x - 2)(x - 3) = 0 \)
Therefore, \( x = 2 \) or \( 3 \).

4. Intervals and Number Line Representation

In this chapter, we won't just find single answers; we will often find answers as "intervals":

• Closed Interval [a, b]: Includes endpoints (represented by a solid dot \( \bullet \)). This means \( a \leq x \leq b \)
• Open Interval (a, b): Does not include endpoints (represented by an open circle \( \circ \)). This means \( a < x < b \)
• Half-Open Interval: For example, \( [a, b) \) means it includes \( a \) but excludes \( b \).

5. Inequalities

Inequalities are expressions involving signs like \( <, >, \leq, \geq \)

The Golden Rule (Common Pitfall):

"Whenever you multiply or divide both sides by a negative number, you must reverse the inequality sign!"
For instance, if \( -2x < 10 \), dividing by \( -2 \) results in \( x > -5 \).

Steps to solve polynomials with degree > 1:

1. Set one side to \( 0 \).
2. Factor the polynomial into brackets.
3. Find the "critical points" (the values of \( x \) that make each bracket \( 0 \)).
4. Plot the critical points on a number line.
5. Mark the intervals as + , - , + starting from the right moving left.
6. If the sign is \( > \), pick the positive (+) intervals. If it is \( < \), pick the negative (-) intervals.

6. Absolute Value

The absolute value of \( x \), written as \( |x| \), represents the "distance from \( 0 \) to \( x \)." Therefore, the result is never negative.

Shortcut Formulas:

1. \( |x| < a \) means \( -a < x < a \) (the values are "between").
2. \( |x| > a \) means \( x < -a \) or \( x > a \) (the values "split" outwards).

Key Point: Always remember to check your solutions, especially when there are variables outside the absolute value sign!

Final Thoughts

The real number system isn't just about digits; it’s about understanding relationships and constraints.

• Don't forget: The denominator cannot be \( 0 \), and the value inside an even root (square root) cannot be negative.
• Technique: Practice factoring regularly; it will make solving problems much faster.
• Encouragement: Math is just like sports—the more you train (solve problems), the better you get!

I hope you have fun learning about real numbers. If you get stuck, try going back and reading each part again. You've got this! ✌️