Mathematics Foundations/4.1 Rational and Irrational Numbers

Rational and Irrational Numbers Introduction The real number system consists of two major subsets: rational numbers and irrational numbers. This section explores their definitions, properties, and the fundamental differences between them. Rational Numbers Definition A rational number is any number that can be expressed as a quotient or fraction of two integers, where . The set of all rational numbers is denoted by , which can be formally defined as: Properties of Rational Numbers Density: Between any two rational numbers, there exists another rational number. For example, between and , we can find . Decimal representation: Every rational number has either a terminating or repeating decimal expansion. Countability: The set of rational numbers is countably infinite, meaning they can be put in one-to-one correspondence with the natural numbers. Examples (integers are rational numbers) (terminating decimal) (repeating decimal) (negative rational number) Converting Repeating Decimals to Fractions Any repeating decimal can be converted to a fraction using algebraic methods. Example: Convert to a fraction. Let Then Subtracting: Therefore Thus Irrational Numbers Definition An irrational number is a real number that cannot be expressed as a ratio of integers. The set of irrational numbers is denoted by (the real numbers minus the rational numbers). Properties of Irrational Numbers Non-repeating decimals: Irrational numbers have decimal expansions that neither terminate nor repeat. Density: Between any two real numbers, there are infinitely many irrational numbers. Uncountability: The set of irrational numbers is uncountable. Famous Irrational Numbers (pi) = (Euler's number) = = (golden ratio) = Proof of Irrationality Theorem: is irrational. Proof: We will use proof by contradiction. Assume is rational. Then where and are integers with no common factors and . Squaring both sides: Multiplying by : This means is even, which implies is even. If is even, then for some integer . Substituting: Dividing by 2: This means is even, which implies is even. But this contradicts our assumption that and have no common factors (since both would be divisible by 2). Therefore, cannot be expressed as a fraction of integers, making it irrational. The Real Number Line The real number line consists of both rational and irrational numbers: An important property of the real numbers is that the irrational numbers "fill the gaps" between rational numbers, making the real number line continuous and complete. Approximating Irrational Numbers While irrational numbers cannot be expressed as exact fractions, they can be approximated by rational numbers to any desired precision. Example: Approximations of : (accurate to 2 decimal places) (accurate to 6 decimal places) Exercises 1. Prove that the sum of a rational number and an irrational number is always irrational. 2. Prove that the product of a non-zero rational number and an irrational number is always irrational. 3. Show that is irrational using the method similar to the proof for . 4. Convert to a fraction. 5. Is the number rational or irrational? Justify your answer.