Maths is all about problem-solving, logic and reasoning. It’s about asking questions and then trying to find the answers.
Sometimes these questions are relatively easy to answer, such as what is “One plus one?”, which can be answered by anyone who is familiar with basic addition. But some maths problems are trickier to answer — either because no one has been able to solve them yet, or the question itself simply doesn’t make sense.
Questions like “Why can’t you divide by zero?” have been asked time and again, by students and mathematicians alike. If you’ve ever wondered what the answer is to this question, you might find this article helpful.
Division is the act of splitting something and zero means nothing, so you can’t divide by zero because nothing is being done, and the number (or object) in question remains whole.
The concept of dividing by zero gives the impression that an action is occurring when in fact nothing is happening and everything remains as it is.
Because the question itself is not defined, there’s no definitive answer. Read on to find out more about why you can’t divide by zero.
Along with multiplication, addition and subtraction, division is one of the four basic operations of arithmetic.
When children are introduced to division at school, the topic is explained by splitting an object — or multiple objects — into equal groups or parts. Imagine a pizza, for example. If a pizza is cut into ten slices and these slices are to be distributed equally between five people, each person would receive two slices. In mathematical terms, that’s 10 ÷ 5 = 2. Similarly, if the pizza is cut into ten slices, but only one person is eating it, they would receive ten slices. Or 10 ÷ 1 = 10.
However, things get more complicated when you try to divide the pizza between no one. No one is there to eat the pizza, so it can’t be distributed. It might help to consider the word “when”. This highlights the problem because it’s impossible to distribute ten slices of people to nobody.
Alternatively, it might help to ask yourself the question “How many people can I feed if I’m dividing my pizza by zero?”. The answer is that there’s no limit to the number of people you can feed because you’re not feeding them anything. An infinite number of people can come to you and ask for a slice of the pizza that you’re dividing by zero, and you can agree to it because their share of the pizza actually amounts to nothing. In other words, there’s no answer to the question “What’s ten divided by zero?”, because it is meaningless or undefined. Such a division doesn't make sense, because splitting something into zero parts doesn’t make sense.
You can also look at it like this: If you share the pizza between two people, each will tell you they ate five pieces each. If you give it to one person, they will say they had ten slices each. But if you don’t let anyone eat the pizza, you can’t ask how many slices have been eaten because there are zero people — or no receivers. And without receivers, there can’t be sharing, distributing or dividing.
Another way to look at it is to define division as how many times the divisor can go into the dividend. For example, the number five (the divisor) goes into 10 (the dividend) twice, so 10 ÷ 5 = 2. This means that with the “Why can’t you divide by zero?” question, you’re asking how many times zero can go into a dividend. But no matter how many zeros you add together, the dividend will never amount to anything.
If you want to know more about division in general, you can read our article ‘What is Division’.
As mentioned, zero means nothing. In everyday language, zero is also commonly referred to as nought, nil and zilch.
According to the Merriam-Webster dictionary, the arithmetic symbol ‘0’ denotes the absence of all magnitude or quantity, and, more specifically, is the number between the set of all positive numbers and the set of all negative numbers.
This means you can’t divide by zero because zero is nothing. You’re not dividing anything. You’re not doing anything, nothing is happening, you’re leaving everything as it is and a number divided by nothing remains the same. If something isn’t divided, the concept of division becomes irrelevant.
You may have noticed that when you divide a number by increasingly smaller numbers the answer you get becomes increasingly bigger. For example:
10 ÷ 5 = 2
10 ÷ 2.5 = 4
10 ÷ 1.25 = 8
10 ÷ 0.0000005 = 20,000,000
These numbers seem on their way to negative infinity and positive infinity, however, trending to infinity and being equal to infinity are not the same thing.
What’s more, infinity is an idea as opposed to a rigid number, only existing in abstraction.
If you attempt to divide by zero in computing, a program error may occur and, depending on the type of number and programming, could result in the following:
- A crash
- A special not-a-number value
- An error message
- Exception handling
- Program termination
- The generation of positive or negative infinity by the IEEE 754 floating-point standard
When solving division problems, students are normally encouraged to check their answers using multiplication. If, for example, you want to check that 10 ÷ 5 = 2, you would take the answer (two) and multiply it by the divisor (five) to get ten.
But again, the difficulties arise when dealing with zero. Like trying to divide by zero, in ordinary arithmetic, multiplying by zero doesn’t make sense because there’s no value attributed to it.
Any number multiplied by zero will generate get the same answer: Zero. This also applies to longer multiplication problems. For example, with the question:
10 x 5 × 2 × 0 × 5 × 4 × 20 = ?
The answer will still be zero because there is a zero in the calculation.
Maths is all about problem-solving, logic and reasoning. It’s about asking questions and then trying to find the answers. Along with multiplication, addition and subtraction, division is one of the four basic operations of arithmetic. With that in mind, you may have found yourself wondering why it’s not possible to divide by zero.
To answer this question, it might be helpful to consider dividing a pizza between no one. It’s impossible to distribute ten slices of people to nobody. Alternatively, it might help to ask yourself the question “How many people can I feed if I’m dividing my pizza by zero?”. The answer is that there’s no limit to the number of people you can feed because you’re not feeding them anything. Another way to look at it is to think of division as how many times the divisor can go into the dividend. No matter how many zeros you add together, the dividend will never amount to anything. Finally, you can also look at it like this: If you share the pizza between two people, each will tell you they ate five pieces each. If you give it to one person, they will say they had ten slices each. But if you don’t let anyone eat the pizza, you can’t ask how many slices have been eaten because there are zero people — or no receivers. And without receivers, there can’t be sharing, distributing or dividing.
