Summary of "[개념 정리] 중2 수학 (상) 1단원. 수와 식의 계산 - [진격의홍쌤]"

Overview

The video (middle‑school level) reviews core concepts about numbers and expressions:

Types of numbers: rational vs. irrational.
Decimal representations: terminating, repeating (periodic), and non‑repeating.
How to tell whether a fraction produces a terminating decimal.
Converting repeating decimals to fractions (algebra method and a shortcut).
Basic laws of exponents.
Basic algebraic operations: combining like terms, distributing, simplifying fractions/expressions.

Repeated advice from the teacher: understand methods by practicing many problems rather than relying on rote memorization.

1) Rational vs. irrational numbers

Rational numbers: any number that can be written as a fraction a/b with b ≠ 0. This category includes integers, terminating decimals, and repeating (periodic) decimals.
Irrational numbers: numbers with non‑terminating, non‑repeating decimal expansions (example: π). These cannot be expressed as a fraction.

2) Decimal types and notation

Terminating (finite) decimals: decimals that end cleanly (example: 1/2 = 0.5).
Repeating (recurring/periodic) decimals: decimals with a repeating block of digits. Common notation: place a dot or bar above the repeating digit(s), e.g., 0.\dot{3} or 0.\overline{3} = 0.333....
Non‑repeating infinite decimals: decimals that never terminate and have no repeating pattern → these are irrational.
How to mark repeating decimals: place a dot or bar above the repeating digit(s) or the repeating block (e.g., 0.\overline{142857}).

3) Determining whether a fraction has a terminating decimal

Steps:

Reduce the fraction to lowest terms.
Factor the denominator:
- If the reduced denominator has only primes 2 and/or 5 as factors (i.e., denominator = 2^m * 5^n), the decimal expansion terminates.
- If the denominator has any other prime factor, the decimal expansion is repeating (non‑terminating periodic).

Example: 3/20. Since 20 = 2^2 * 5, the decimal terminates.

4) Converting repeating decimals to fractions

A. Classic algebra method

General idea: - Let x equal the repeating decimal. - Multiply x by a power of 10 so that one copy of the decimal lines up with another copy with the repeating block aligned. - Subtract to eliminate the repeating tail, then solve for x.

Examples: - x = 0.333... Multiply by 10: 10x = 3.333... Subtract: 10x − x = 3.333... − 0.333... → 9x = 3 → x = 1/3. - x = 1.24444... (written 1.2\dot{4}) Use 100x and 10x: 100x = 124.444..., 10x = 12.444... Subtract: 100x − 10x = 124.444... − 12.444... → 90x = 112 → x = 112/90 = 56/45.

B. Shortcut method (for decimals with a non‑repeating part + repeating part)

Steps: 1. Denominator: write as many 9s as there are repeating digits, then write as many 0s as there are non‑repeating digits after the decimal. - Example for 1.2\dot{4}: repeating digits = 1 → one 9; non‑repeating digits = 1 → one 0 → denominator = 90. 2. Numerator: take the integer formed by all digits up to the end of one repeating block (remove the decimal point), then subtract the integer formed by the non‑repeating part only. - Example: all digits = 124; non‑repeating part = 12 → numerator = 124 − 12 = 112. 3. Simplify the fraction.

This yields the same result as the algebra method but is faster once the rule is understood.

5) Laws of exponents (basic rules)

Product of same base: a^m * a^n = a^(m+n).
Power of a power: (a^m)^n = a^(m·n).
Quotient of same base: a^m / a^n = a^(m−n) (for a ≠ 0).
Power of a product: (ab)^n = a^n b^n.
Power of a quotient: (a/b)^n = a^n / b^n.

These rules are best learned by expanding examples and practicing rather than memorizing in isolation.

6) Calculations with algebraic expressions

Addition / subtraction: combine like terms (terms that have the same variable parts and powers).
- Example: add coefficients of identical variable powers.
Multiplication: multiply coefficients and variable factors; use the distributive law when multiplying over parentheses.
- Example: a(2 + b) = 2a + ab.
- Multiplying monomials: multiply coefficients and add exponents for identical bases.
Division: rewrite as multiplication by the reciprocal and cancel common factors between numerator and denominator.
- Example: simplify rational expressions by factoring and canceling common factors.
General emphasis: follow the order of operations, group like terms, apply the distributive law, and simplify by canceling factors.

Closing advice

Practice many problems (dozens or hundreds) to internalize these methods rather than only memorizing formulas. Repeated practice makes the rules feel natural.