I Learned How to Divide by Zero (Don't Tell Your Teacher)

Summary

This video challenges the conventional wisdom that division by zero is impossible. It explores historical mathematical breakthroughs, such as defining the square root of negative numbers and introducing irrational numbers, to argue for a new definition. The script delves into calculus to explain why 1/0 might be infinity, introduces stereographical projection to suggest unsigned infinity, and discusses 'nullity' as a solution to indeterminate forms like 0*∞. It concludes by questioning whether we should redefine division by zero despite the algebraic complications it introduces.

Takeaways

• 🚫 **Division by Zero Myth**: Traditionally, division by zero is considered undefined, but the video challenges this notion.
• 🔄 **Historical Context**: Just as square roots of negative numbers and irrational numbers were once thought impossible, division by zero was also seen as a barrier.
• 📈 **Approaching Zero**: Dividing 1 by numbers approaching zero from the positive side results in values approaching positive infinity.
• 📉 **Negative Approach**: Similarly, dividing 1 by numbers approaching zero from the negative side results in values approaching negative infinity.
• 🤔 **Limit Concept**: The video uses calculus to explain that the limit of 1/x as x approaches zero does not exist because it yields different results from the left and right.
• 🌐 **Stereographical Projection**: Introducing a sphere onto the Cartesian plane to redefine infinity as a single point, unsigned infinity.
• 🔄 **Infinite Absorbing Nature**: The concept of infinity is explored where adding or multiplying by infinity should still result in infinity.
• 🚸 **Indeterminate Forms**: The video discusses 'indeterminate forms' from calculus, such as 0/0, which are undefined and need to be resolved contextually.
• 🔄 **Wheel Theory**: Introducing a new element, 'nullity', to handle indeterminate forms, which acts as an absorbing element in algebraic operations.
• ⚖️ **Algebraic Consequences**: The introduction of 'nullity' affects fundamental algebraic principles, such as any number times zero equals zero and one over one equals one.
• 🤷‍♂️ **Philosophical Question**: The video ends with a question about whether we should redefine division by zero, leaving the decision to the viewer.

Q & A

What is the traditional mathematical belief about dividing by zero?

Traditionally, it is believed that you cannot divide by zero as it leads to undefined results.

How does the script challenge the conventional thinking about division by zero?

The script challenges conventional thinking by suggesting that division by zero could be defined as unsigned infinity and introducing the concept of 'nullity' to handle indeterminate forms.

What is meant by 'unsigned infinity' in the context of the script?

In the script, 'unsigned infinity' refers to infinity without a positive or negative sign, representing a single, unified concept of infinity.

What is stereographical projection and how does it relate to the discussion on division by zero?

Stereographical projection is a system that maps a two-dimensional plane onto a sphere, with the idea that it could help in defining division by zero by linking positive and negative infinities to a single point.

What is an indeterminate form in calculus?

An indeterminate form in calculus is a mathematical expression that cannot be simplified directly, such as 0/0, ∞ - ∞, 0 × ∞, or ∞/∞, which requires further analysis to determine its value.

How does the script propose to resolve the issue with indeterminate forms like 0 × ∞?

The script suggests defining indeterminate forms like 0 × ∞ as 'nullity', which is an absorbing element that becomes any element it interacts with.

What are the implications of defining division by zero as infinity on algebraic rules?

Defining division by zero as infinity can lead to contradictions in algebraic rules, such as 1 = 0 × ∞ leading to 1 = 2, which undermines the fundamental properties of numbers.

What is 'nullity' as introduced in the script?

In the script, 'nullity' is a new mathematical element that acts as an absorbing element, turning any operation involving it into itself.

How does the script suggest we should think about division by zero?

The script suggests that while we can define division by zero as unsigned infinity, we should also consider whether it is practical or useful to do so, given the complications it introduces.

What are some areas where stereographical projection is used?

Stereographical projection is used in certain areas of geometry and complex analysis to map points from a plane to a sphere.

What does the script imply about the nature of infinity?

The script implies that infinity is an abstract concept rather than a real number and suggests that it has an 'absorbing' nature where any number added to infinity remains infinity.