Division by Zero Explained

Overview

This lecture explores why division by zero is undefined, the mathematical reasoning behind it, and what happens if we try to redefine it.

Dividing by smaller numbers gives larger results (e.g., 10 ÷ 0.000001 = 10,000,000).
As the divisor approaches zero, the result approaches infinity, but this is not the same as dividing by zero.
Division is the reverse of multiplication; dividing by a number means finding its multiplicative inverse.

The multiplicative inverse of x is a number that multiplied by x equals one.
Zero has no multiplicative inverse because any number times zero is still zero, not one.
Thus, division by zero is undefined since we cannot "undo" multiplication by zero.

If we define infinity as 1 ÷ 0, then zero times infinity must equal one.
Using algebraic properties, this leads to logical contradictions (e.g., one equals two).
Allowing such contradictions breaks the consistency of numbers as we know them.

Mathematicians have previously invented new numbers (e.g., imaginary numbers).
Trying to redefine division by zero in the obvious way leads to unhelpful or inconsistent systems.
Specialized constructs (like the Riemann sphere) handle "division by zero" differently, but not in the standard sense.

Division by Zero — Attempting to divide a number by zero, an operation without meaning in standard arithmetic.
Multiplicative Inverse — A number which, when multiplied by a given number, yields one.
Distributive Property — A property stating that a(b + c) = ab + ac.
Infinity — A concept describing unbounded growth, not a specific number.
Riemann Sphere — An advanced mathematical structure used to extend complex numbers, not covered in detail here.

Reflect on why mathematical rules exist and how breaking them leads to new mathematical ideas.
Research the Riemann sphere for alternative takes on division by zero (optional).