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c. 1800 BCE: Ancient Babylonians and Fractions

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c. 1800 BCE: Ancient Babylonians and Fractions

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The Babylonians developed a sophisticated system of fractions, working with rational numbers in their base-60 number system. They used these numbers for practical calculations in astronomy and everyday life.

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c. 500 BCE: Pythagoras and the Discovery of Irrational Numbers

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c. 500 BCE: Pythagoras and the Discovery of Irrational Numbers

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The Pythagoreans discovered that the square root of 2 cannot be expressed as a ratio of two whole numbers, leading to the first known proof of irrational numbers. This discovery shook their belief that 'all is number' and was initially kept secret!

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c. 380 BCE: Plato's Contribution to Number Theory

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c. 380 BCE: Plato's Contribution to Number Theory

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Plato and his academy further developed the understanding of irrational numbers, helping to establish them as legitimate mathematical concepts despite the initial resistance from some mathematicians.

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c. 300 BCE: Euclid's Elements and Irrational Numbers

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c. 300 BCE: Euclid's Elements and Irrational Numbers

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Euclid provided rigorous proofs about irrational numbers in his famous work 'Elements'. He showed that there are infinitely many irrational numbers and developed methods for constructing them geometrically.

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c. 825 CE: Al-Khwarizmi's Algebraic Approach

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c. 825 CE: Al-Khwarizmi's Algebraic Approach

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Persian mathematician Al-Khwarizmi developed algebraic methods for working with both rational and irrational numbers, helping to bridge the gap between arithmetic and geometry.

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1202 CE: Fibonacci and the Hindu-Arabic Number System

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1202 CE: Fibonacci and the Hindu-Arabic Number System

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Leonardo of Pisa (Fibonacci) introduced the Hindu-Arabic decimal system to Europe, making it easier to work with and represent both rational and irrational numbers.

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Johann Heinrich Lambert proved that π (pi) is irrational, showing that this fundamental mathematical constant cannot be expressed as a simple fraction.

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Dedekind developed a method called 'Dedekind cuts' to rigorously define real numbers, providing a formal way to understand the relationship between rational and irrational numbers.

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Georg Cantor proved that there are more real numbers than rational numbers, demonstrating that the set of real numbers is 'uncountable'. This revolutionized our understanding of infinity and number systems.

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David Hilbert included questions about real numbers in his famous list of 23 problems, setting the stage for 20th-century research in number theory and mathematical analysis.

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Approximating Irrational Numbers with Rational Numbers

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Approximating Irrational Numbers with Rational Numbers

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One common example of approximating irrational numbers with rational numbers is the approximation of π (pi). Pi is often approximated by the fraction 22/7, which results in a rational number close to the irrational value of π.
Another example is the square root of 2, which can be approximated by 1.414. Similarly, the number e (Euler's number) is approximately 2.718, which is used in calculations despite its irrationality.

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Convert a repeating decimal to a fraction in 3 steps.

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Convert a repeating decimal to a fraction in 3 steps.

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1. Let x equal the number.

2. Multiply by 10, 100, 1,000, etc based on how many digits are repeating. 10 is for 1 digit, 100 is for 2 digits...

3. Subtract "x" on both sides, but on the right subtract the repeating digits, since they equal x.

4. Solve the equation.

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Approximate a rational number in 3 steps.

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Approximate a rational number in 3 steps.

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If you are re-watching these to study don't forget you can click the gear to speed me up. Sometimes I talk a little slow :)

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Math 8A Unit 2 Playlist The Real Number System

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This playlist contains all videos from Unit 2 on Rational Numbers.