Numbers and Expressions

【Math I】Numbers and Expressions: Your First Step to Mastery

Hello! Let’s embark on an adventure into the world of "Numbers and Expressions." This chapter—the very beginning of Math I & A—forms the foundation for all the mathematics you will learn from here on out. Think of it as the "foundation work" for building a house.
At first, you might feel intimidated by all the letters and symbols, but don't worry. If we tackle the rules one by one, you'll find that solving these is as fun as completing a puzzle!

1. Simplifying Polynomials

Let’s start by organizing expressions neatly. When an expression is well-organized, calculation errors drop significantly.

Monomials and Polynomials

Let’s review the definitions of the "language" used in mathematics:

• Monomial: An expression consisting only of the product of numbers and variables, such as \(3x^2\) or \(-5abc\).
• Polynomial: An expression made by adding monomials together. Each individual monomial is called a term.
• Degree: The number of variables multiplied together. For \(x^3\), the degree is 3.
• Constant term: A term consisting only of a number, containing no variables.

Descending Order

Rearranging a jumbled expression so that the terms are in order of highest degree to lowest is called organizing in descending order.
Example: \(x + 5 + 3x^2\) → \(3x^2 + x + 5\)
Just doing this makes the expression much easier to read!

【Key Tip】

When you are told to "focus on a specific variable," you treat all other variables as if they were just numbers (constants). For example, if asked to "focus on \(x\)," treat \(y\) as if it were just a number.

◎Summary: Organizing expressions in descending order helps you avoid mistakes and makes solving problems easier!

2. Expansion (Multiplication Formulas)

Expansion is the process of removing parentheses to break an expression apart. If you think of it like cooking, it’s just like "taking ingredients out of their packaging."

Basic Formulas (You must know these!)

1. \( (a+b)^2 = a^2 + 2ab + b^2 \)
2. \( (a-b)^2 = a^2 - 2ab + b^2 \)
3. \( (a+b)(a-b) = a^2 - b^2 \) (The product of a sum and difference is the difference of their squares!)
4. \( (x+a)(x+b) = x^2 + (a+b)x + ab \)

【Common Mistakes】

A very frequent mistake is writing \( (x+3)^2 \) as \( x^2 + 9 \)! Don't forget the middle \( 2ab \) (in this case, \( 6x \)).
It’s helpful to memorize the rhythm: "(first term squared) + (2 × first × last) + (last term squared)."

【Pro Tip】

The formula \( (a+b)(a-b) = a^2 - b^2 \) is beloved in the math world for being incredibly elegant and useful.

◎Summary: Practice these expansion formulas until you can use them as unconsciously as your multiplication tables!

3. Factorization

Factorization is the reverse of expansion. It’s the process of "grouping scattered expressions into parentheses." This is a crucial topic that appears frequently in common tests.

Steps to Factorization (Follow this if you're stuck!)

1. Factor out common terms: If there is a variable or number common to every term, pull it out first.
Example: \(2x^2 + 4x = 2x(x + 2)\)
2. Use formulas: Apply the expansion formulas in reverse.
3. Cross-multiplication method (Tasuki-gake): Use this when there is a coefficient in front of \(x^2\), such as in \(2x^2 + 5x + 3\).

Tips for Cross-multiplication

You look for numbers by following these steps: "Find factors for the front," "Find factors for the back," and "Multiply diagonally and check if the sum equals the middle term." It takes time at first, but if you treat it like a puzzle and keep practicing, you will get faster!

【Key Tip: Complex Factorization】

When an expression looks long and difficult, try organizing it by the "variable with the lowest degree." This often clears the path to the solution.

◎Summary: The first step to factorization is always finding the common factor (the part that's the same)!

4. Real Numbers and Square Roots

Here, we will learn about types of numbers and how to handle square roots (\(\sqrt{\quad}\)).

Classification of Numbers

• Rational Numbers: Numbers that can be expressed as fractions (integers, terminating decimals, repeating decimals).
• Irrational Numbers: Numbers that cannot be expressed as fractions (like \(\pi\) or \(\sqrt{2}\)).
• Real Numbers: The set of all numbers that can be plotted on a number line, including both rational and irrational numbers.

Absolute Value

The absolute value \(|a|\) is defined as the "distance from the origin (0)." Since it represents a distance, it can never be negative.
\(|3| = 3\), and \(|-3| = 3\).

Watch out! When removing absolute value bars \(|a|\), if \(a\) is a negative number, you must flip its sign to \(-a\).
Think of it like: \(|-5| = -(-5) = 5\).

Rationalizing the Denominator

In the math world, leaving a square root in the denominator is considered "improper manners."
By doing something like \(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}\), we multiply the numerator and denominator by the same root to clear the root from the bottom.

◎Summary: Absolute value is the "distance from 0"! And always kick roots out of the denominator (rationalization)!

5. Linear Inequalities

These are calculations involving expressions with "\(>\)" or "\(<\)". They are almost identical to equations, but there is one super important rule.

The Golden Rule of Inequalities

"When you multiply or divide by a negative number, the direction of the inequality sign flips!"
Never forget this.
Example: Solving \(-2x < 6\) requires dividing both sides by \(-2\), which gives you \(x > -3\).

【Analogy】

"Debt of 100 yen < Debt of 50 yen," but if you remove the debt (the minus sign) and look only at the numbers, "100 yen > 50 yen." The relationship between sizes flips when you move between the negative and positive worlds.

◎Summary: If you divide by a negative, "flip" that inequality sign around!

Closing

How did you find the "Numbers and Expressions" section? At first, the calculations might feel tedious, but the formulas and rules introduced here will become the "weapons" you use to solve every problem that comes after.
"It might feel difficult at first, but you'll be fine." If you practice your calculations a little bit every day, you will eventually be able to solve these as easily as breathing. Let’s do our best together!