Division by Zero: Math That Breaks Calculators

Estimated reading time: 6 minutes

For a long time, people have believed that mathematics is a universal language. A beautiful creation that validates logic and structure in everything from space to land.  But, every communication method has some limitations.  Division by zero is an example of such a limitation of mathematics. 

Division by zero is an almost neurotic problem in mathematics that is both easy and complex.  However, even the ancient philosophers have been baffled by basic premises, such as “What do you get after a number is divided by zero?”  The question, “What is a number divided by zero?” has perplexed even the most brilliant individuals, including modern computers.  Let’s check practical implications of this impossibility.

Understanding Division in Simple Terms

Let’s review what division is to understand why dividing by zero is so problematic.

Division is repeated subtraction or equal sharing. For example:

• 15 ÷ 3 = 5 means splitting 15 objects into 3 equal groups of 5.
• 25 ÷ 5 = 5 means asking how many times 5 fits into 25.

This works smoothly and is logical—until the divisor is zero. Imagine:

• Splitting 15 chocolates among 0 friends—what does that even mean?
• Asking how many times 0 fits into 25—does 0 fit infinitely many times or not at all?

These two perspectives logically fall apart.

Why Division by Zero Fails Mathematically

The definition of division is the inverse of multiplication:

a ÷ b = c ↔ a = b × c

Consider the example:

10 ÷ 0 = x

According to the above property, we should have:

10 = 0 × x

But 0 times any number is always 0.  The equation cannot be satisfied by any number “x.”  Division by zero is undefinable because of this paradox.

Also Read: Infinity minus Infinity.

The Infinity Illusion

“Well, can’t the answer just be infinity?” is a common thought. Let’s put your intuition to the test.

Take 10 ÷ n:

• For n = 1, result = 10
• For n = 0.1, result = 100
• For n = 0.01, result = 1000

As n decreases, the quotient increases. This implies that infinity could result from dividing by zero.

The catch is that:

• Positive infinity results from approaching zero from the positive side.
• Negative infinity is the result of approaching zero from the negative side.

We are left with two contradictory infinities, depending on the approach orientation. The operation is kept undefined since mathematics requires accuracy, and there is no consistent single value.

Also Read: The Infinity Paradox.

Historical Attempts to Tackle Division by Zero

The idea that one could divide by zero is nothing new.  

• Ancient Greeks like Aristotle did not understand how one could divide something by nothing because they believed that numbers referred to countable, and hence definable, entities. 
• The ancient Indian mathematician Brahmagupta is said to have added zero to the numeration system during the seventh century. He, however, struggled with the division of something by zero. Sometimes he referred to it as ‘infinity’, at other times he chose to leave it as ‘undefined’. 
• During the European Renaissance, as more and more people began practicing algebra, they realized that division by zero formulated paradoxes ascribed to it and began to call it ‘undefined’. 

The historical attempts to explain division by zero illustrate how the very fabric of math is steeped in division by zero.

Why Calculators and Computers Refuse Zero

When you enter 1 ÷ 0 in a calculator, you would normally come across an error message. Why? Devices like calculators and computers operate under rigid step-by-step procedures. Division by zero will not give a valid answer. They stop working to preserve the integrity of the results. Runtime error in computer science. 

Amazingly enough, dividing by zero is so disruptive that it has caused real-life disasters. For instance, the loss of the 1996 NASA Ariane 5 rocket. There was a software glitch that, over the course of the Ariane 5’s rocket’s 40-second flight, caused it to explode on lift-off. The problem happened because someone handled a math mistake badly.   

The error of dividing by zero in a computer spreadsheet, especially in the field of finance, which uses trading models, will spread throughout the entire system and will almost guarantee a breakdown in functionality.

Division by Zero: Real-World Analogies

To illustrate the reason why the division of a number by zero does not exist, let’s look at a couple of analogies: 

• Sharing Analogy: You have 12 apples, and you have 3 friends—everyone can have 4 apples, and you can share each with everyone. If you have 0 friends, to whom do you share? The act of sharing becomes nonsensical. 
• Speed Example: Speed equals distance divided by the interval of time. If time equals 0, the formula implies boundless speed. There are no physical constructs that pertain to boundless speed. 
• Programming Error: In code, the division of a number by zero causes exceptions that can close the entire program. That is one thing that computer engineers need to secure.

Division by Zero in Advanced Mathematics

In elementary arithmetic, division by zero is considered ‘impossible’. However, the borderline is crossed in Advanced Mathematics: 

• The Limit Concept in Calculus – Dividing is not the center of attention. Limits aid in understanding the behavior when a value nears ‘0’. Helps in understanding asymptotes and approaches to infinity in the graph. 
• Complex Analysis – The introduction of the ‘Riemann Sphere’, which extends the complex plane, treats infinity as a point. We are very careful when we divide by zero. It’s not just a single mistake. 
• Abstract Algebra – The study of complex systems where alternative rules bind structure. These are ‘division by zero’ systems. It is still an advanced branch of mathematics. Still a borderline division by zero.

Division by Zero: Conclusion

No one is quite certain at what point division by zero became a rule, but during the course of history, it has created a paradox that has hampered mathematicians, scientists, and engineers ever since calculators became commonplace. From old ideas to modern computers, we must set limits and give meanings, specifically with the division of zero.

From here on, the next time you see a calculator displaying an Error, remember it is doing more than malfunctioning. It maintains the essential reasoning that underlines numerical systems.

Additionally, to stay updated with the latest developments in STEM research, visit ENTECH Online. Basically, this is our digital magazine for science, technology, engineering, and mathematics. Further, at ENTECH Online, you’ll find a wealth of information.

References

1. Wangui Patrick Mwangi. (2018). Mathematics Journal: Division of Zero by Itself – Division of Zero by Itself Has Unique Solution. Pure and Applied Mathematics Journal, 7(3), 20-36. https://doi.org/10.11648/j.pamj.20180703.11