Decimal numbers provide a precise and flexible way to represent values that are not whole numbers, making calculations easier in fields like finance, engineering, and mathematical analysis. This article explores decimal expansion, multiplication, division, exponentiation, fractions, negative exponents, exponential equations, greatest common divisor (GCD), and least common multiple (LCM).
1. Decimal Expansion and Representation
Decimal expansion expresses numbers in decimal form, simplifying mathematical operations:
1.25 = 1 + 0.2 + 0.05
Decimal numbers often arise from fraction-to-decimal conversions:
1/2 = 0.5 1/4 = 0.25
Decimal expansions can be finite (e.g., 0.75) or infinite (e.g., 0.333...).
2. Multiplication of Decimal Numbers
Steps for Decimal Multiplication
Ignore decimal points and multiply normally.
Count total decimal places in the numbers and apply them in the final result.
1.2 × 0.03 = (12 × 3) ÷ 100 = 0.036
3. Division of Decimal Numbers
Steps for Decimal Division
Convert the divisor into a whole number.
Apply the same decimal shift in the dividend.
Perform division normally.
2.4 ÷ 0.4 → (24 ÷ 4) ÷ 10 = 6
4. Exponents in Decimal Numbers
General Formula for Decimal Exponents
(0.2)² = 0.2 × 0.2 = 0.04
For negative exponents, take the reciprocal:
(0.5)⁻² = 1 / (0.5 × 0.5) = 4
5. Decimal Expansion in Fractions and Negative Exponents
In fractions and negative exponents, calculation follows reciprocal rules:
1 / (0.1)² = 1 / 0.01 = 100
This approach is essential in numerical analysis and real-world applications.
6. Exponential Equations
General Formula for Exponential Equations
aˣ = b
To solve exponential equations, use logarithms:
2ˣ = 8 → x = log₂(8) = 3
Exponential equations are widely used in population growth, compound interest, and physics.
7. Greatest Common Divisor (GCD)
Steps to Compute GCD
Find the factors of each number.
Select the largest common factor.
GCD(12, 18) = 6
Using Euclidean algorithm for GCD:
GCD(48, 18): 48 ÷ 18 = 12 (remainder) 18 ÷ 12 = 6 (remainder) 12 ÷ 6 = 0 → GCD = 6
8. Least Common Multiple (LCM)
Steps to Compute LCM
Find the multiples of each number.
Select the smallest common multiple.
LCM(12, 18) = 36
Using GCD to compute LCM:
LCM = (Number₁ × Number₂) ÷ GCD LCM(12, 18) = (12 × 18) ÷ 6 = 36
