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In this section, we learn about irrational numbers, defined as numbers that cannot be written in the form p/q, where p and q are integers. The discussion includes historical insights, key properties of irrational numbers, and examples to strengthen understanding, illustrating their placement on the number line.
In this section, we dive into the concept of irrational numbers, numbers that cannot be represented as a ratio of two integers (p/q, with q β 0). The term 'irrational' comes from their inability to be expressed in this form, a concept recognized since ancient Greece when the mathematician Pythagoras explored roots such as \( \sqrt{2} \), which challenged previously held beliefs about numbers.
Understanding irrational numbers is crucial for grasping concepts in higher mathematics, such as calculus, where such numbers frequently arise.
Rational Numbers: Numbers that can be written as fractions.
Irrational Numbers: Numbers that cannot be expressed in fractional form.
Real Numbers: All numbers on the number line, combining rational and irrational.
Historical Context: Understanding the historical implications of irrational numbers.
Irrationals can't be tamed, fractions can't be named.
Once upon a time, the Pythagoreans found a number that wouldn't behave. Each fraction told tales of simplicity, but \( \sqrt{2} \) whispered secrets of complexity.
Rational Rat, Irrational SnakeβRats can fraction, snakes just take!
Example 1: \( \sqrt{2} \) ~ 1.41421356237 (irrational number).
Example 2: Ο ~ 3.14159265359 (irrational number).
Term: Rational Number
Definition: A number that can be expressed as a fraction p/q, where p and q are integers and q β 0.
A number that can be expressed as a fraction p/q, where p and q are integers and q β 0.
Term: Irrational Number
Definition: A number that cannot be expressed as a fraction p/q with integers p and q, where q β 0.
A number that cannot be expressed as a fraction p/q with integers p and q, where q β 0.
Term: Real Number
Definition: The set of numbers that includes both rational and irrational numbers.
The set of numbers that includes both rational and irrational numbers.
Term: Nonterminating Decimal
Definition: A decimal expansion that continues infinitely without repeating.
A decimal expansion that continues infinitely without repeating.
Term: Nonrepeating Decimal
Definition: A decimal expansion that does not exhibit a repeating pattern.
A decimal expansion that does not exhibit a repeating pattern.