Why don't they teach simple visual logarithms (and hyperbolic trig)?
Mathologer・2 minutes read
When squishing and stretching a shape by the same factor, the area remains the same, but there are anti-shapeshifters like 1/x that are unaffected. The concept of natural logarithm, e, and hyperbolic trigonometric functions play essential roles in mathematics and have practical applications in physics and engineering.
Insights
- Squishing and stretching a shape by the same factor maintains its original area, a fundamental concept in mathematics.
- The Rubik’s cube shape 1/x is an anti-shapeshifter unaffected by squish-and-stretch transformations, showcasing unique properties in mathematical analysis and integration.
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Recent questions
What is the relationship between squishing and stretching shapes?
Squishing and stretching shapes by the same factor maintains the original area.
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Summary
00:00
"Shapes, Squish, Stretch, and Mathematical Infinity"
- Squish a square down by a factor of 2 and then stretch the resulting rectangle horizontally by the same factor.
- The rectangle and the original square have the same area due to the squishing and stretching process.
- Any shape squished and stretched by the same factor retains its original area.
- Anti-shapeshifters exist, shapes immune to squish-and-stretch transformations, crucial in mathematics.
- The Rubik’s cube shape 1/x is an anti-shapeshifter, unaffected by squish-and-stretch transformations.
- Algebraically, the 1/x shape remains unchanged under squish-and-stretch transformations.
- The area of the 1/x shape can be determined by focusing on specific spikes within the shape.
- The blue spike in the 1/x shape has infinite area, leading to the entire shape having infinite area.
- The infinite sum of 1 plus 1/2 plus 1/3 and so on equals infinity, a fundamental mathematical truth.
- The area function under the curve between 1 and x behaves like a logarithm, following logarithmic rules and formulas.
15:30
Exploring the Mathematics of Natural Logarithm
- Evaluating (1+1/3)^3 results in approximately 2.37.
- The special number being approached is approximately 2.71828, known as e, the base of natural logarithm.
- The relationship between 1/x and the area function is expressed through a basic integral.
- The natural logarithm is defined as the logarithm to the base e.
- Different logarithms are derived from scaling the original anti-shapeshifter.
- The base of the anti-shapeshifter corresponds to the logarithm to the base e squared.
- The derivative of the natural logarithm is 1/x, and the derivative of e^x is e^x.
- Hyperbolic trigonometric functions, such as cosh and shine, are introduced as counterparts to circular trigonometric functions.
- Hyperbolic functions have applications in physics and engineering, like describing hanging chains and Lorentz transformations in special relativity.
