Why dividing by zero is undefined | Functions and their graphs | Algebra II | Khan Academy

TL;DR

Dividing a non-zero number by zero is undefined and does not have a consistent answer.

Transcript

Comedian Steven Wright-- and I guess we can credit him with being a bit of a philosopher-- once commented that "Black holes are where God divided by zero." And I won't get in to the physics of it, and obviously the metaphor breaks down in certain ways But it is strangely appropriate, because black holes are where our current understanding of physic... Read More

Key Insights

• 👋 Dividing by zero is undefined in mathematics because there is no good answer or value.
• 🍗 Trying to define a value for dividing by zero using small positive numbers suggests it could be positive infinity.
• ❎ Dividing by negative numbers close to zero suggests it could be negative infinity.
• 😣 Mathematicians have not defined a value for dividing by zero because it would not be consistent with the rest of mathematics.
• 🖤 Dividing by zero and understanding black holes both represent areas where our current understanding of physics and mathematics breaks down.
• 📏 The concept of dividing by zero challenges the principles and rules of mathematics.
• 0️⃣ Defining a value like 42 for dividing by zero would be arbitrary and inconsistent.

Questions & Answers

Q: Why is dividing by zero considered "undefined" in mathematics?

Dividing by zero is undefined because mathematicians have not been able to come up with a good answer or value for it. It breaks the rules of mathematics and does not align with the rest of the concepts in the field.

Q: Can we define a value for dividing by zero, such as 1 divided by 0?

While one might suggest that 1 divided by 0 could be positive infinity based on dividing by smaller and smaller positive numbers close to zero, dividing by negative numbers close to zero suggests it could be negative infinity. Thus, there is no consistent answer.

Q: Why didn't mathematicians define a value like 42 for dividing by zero?

Defining a value like 42 for dividing by zero would not make sense and would not be consistent with the principles of mathematics. It is neither positive infinity nor negative infinity and would not align with other mathematical concepts.

Q: How does dividing by zero relate to black holes in physics?

Dividing by zero and understanding black holes both involve our current understanding of physics breaking down. They represent areas where our knowledge is limited and where our current theories do not provide satisfactory explanations.

Summary & Key Takeaways

• Dividing by zero in mathematics and understanding black holes in physics share the concept of breaking down our current understanding of the respective fields.

• Mathematicians have never defined what it means to divide by zero because there is no good answer or value for it.

• Trying to define 1 divided by a very small positive number suggests that the result could be positive infinity, but dividing by negative numbers close to zero suggests it could be negative infinity.