Math 8 Unit 2 The Number System by Ms. Phoebe Saltzstein

c. 1800 BCE: Ancient Babylonians and Fractions

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

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

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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!

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

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

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

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

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

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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

1202 CE: Fibonacci and the Hindu-Arabic Number System

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

1761 CE: Lambert Proves π is Irrational

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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

1872 CE: Richard Dedekind's Cuts

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

1874 CE: Cantor's Contributions

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

1900 CE: Hilbert's Problems

phoebesaltzstein — 2025-07-25 21:40:36 UTC

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.

Approximating Irrational Numbers with Rational Numbers

phoebesaltzstein — 2025-07-25 21:41:20 UTC

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.

phoebesaltzstein — 2025-07-25 21:58:05 UTC

phoebesaltzstein — 2025-07-25 22:03:25 UTC

phoebesaltzstein — 2025-08-11 17:13:56 UTC

phoebesaltzstein — 2025-08-18 20:38:06 UTC

Convert a repeating decimal to a fraction in 3 steps.

phoebesaltzstein — 2025-08-18 20:46:23 UTC

Let x equal the number.
Multiply by 10, 100, 1,000, etc based on how many digits are repeating. 10 is for 1 digit, 100 is for 2 digits...
Subtract "x" on both sides, but on the right subtract the repeating digits, since they equal x.
Solve the equation.

Approximate a rational number in 3 steps.

phoebesaltzstein — 2025-08-18 20:57:31 UTC

phoebesaltzstein — 2025-08-18 21:02:04 UTC

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 :)