Mathematical Sets: Definitions, Formulas, and Applications

1. Decimal Expansion and Representation

Decimal expansion expresses numbers by place value:

1.25 = 1 + 0.2 + 0.05

Common conversions from fractions:

• 1/2 = 0.5
• 1/4 = 0.25

Decimals may be finite (e.g., 0.75) or infinite/repeating (e.g., 0.333...)

2. Multiplication of Decimal Numbers

1. Ignore decimal points and multiply normally.
2. Count total decimal places and apply to the result.

Example:

1.2 × 0.03 = (12 × 3) ÷ 100 = 0.036

3. Division of Decimal Numbers

1. Make the divisor a whole number.
2. Apply the same decimal shift to the dividend.
3. Divide as usual.

Example:

2.4 ÷ 0.4 → (24 ÷ 4) ÷ 10 = 6

4. Decimal Exponents

(0.2)² = 0.2 × 0.2 = 0.04

Negative exponents use reciprocals:

(0.5)⁻² = 1 / (0.5 × 0.5) = 4

5. Fractions and Negative Powers

1 / (0.1)² = 1 / 0.01 = 100

Reciprocal rules are key to working with fractional exponents.

6. Exponential Equations

General form:

aˣ = b

Use logarithms to solve:

2ˣ = 8 → x = log₂(8) = 3

Applications: compound interest, growth models, and physics.

7. Greatest Common Divisor (GCD)

• Find the factors of each number.
• Select the largest shared factor.

GCD(12, 18) = 6

Euclidean Algorithm Example:

GCD(48, 18):
48 ÷ 18 → remainder 12
18 ÷ 12 → remainder 6
12 ÷ 6 → remainder 0 → GCD = 6

8. Least Common Multiple (LCM)

• List the multiples of each number.
• Choose the smallest one they share.

LCM(12, 18) = 36

Using GCD to Find LCM:

LCM = (Number₁ × Number₂) ÷ GCD

LCM(12, 18) = (12 × 18) ÷ 6 = 36

Conclusion

Mastering decimal operations, exponents, and number theory tools like GCD and LCM forms a critical foundation in math. These principles are essential in school-level mathematics and real-world contexts ranging from finance to data analysis.