What is 1 + 2 + 5 × 0? On the order of operations
Every now and then a puzzle like this one goes round on social media. What number is this expression equal to?
Let’s work it out together:
- One plus two is three.
- Three plus five is eight.
- And finally, eight times zero is zero.
Did you work it out the same way as me? Unfortunately, we were both wrong ‼️‼️‼️ We forgot the classic school rule: multiplication takes priority over addition. What does that mean?
If an expression contains both addition and multiplication, we first multiply everything that can be multiplied, and only then go back to the addition. In our case that means we first multiply 5 · 0 and ignore everything else in the expression. Five times zero is zero. We write this zero back into the expression:
Now we have an expression containing only addition, which is easy: 1 + 2 + 0 is equal to three. So
Another example. What is this equal to?
Again, the rule says we deal with the multiplication first and leave the addition alone. So we first work out 2 · 3, which is 6:
Now the expression contains only addition, which is easy:
Why does a calculator give a different answer?
If you try the same examples on a calculator, you may get the same wrong result as when we first worked out 1 + 2 + 5 · 0. How is that possible? There are basically two options: either the calculator doesn’t know the order of operations, or you press the equals button in the middle of the calculation. If you press the buttons on a calculator like this:
- [One] [plus] [two] [equals] -> the calculator shows 3,
- [plus] [five] [equals] -> the calculator shows 8,
- [times] [zero] [equals] -> the calculator shows 0.
then you haven’t actually worked out the question set above, because by pressing “equals” you worked out three different, unrelated expressions instead of one. In the last step the calculator simply works out the expression “eight times zero”, which is just zero, and the calculator shows that correctly. At that moment the calculator doesn’t know that there was a multiplication earlier that should have taken priority.
If you don’t press “equals”, but enter the whole calculation as
- [One] [plus] [two] [plus] [five] [times] [zero]
and the calculator still shows zero, then your calculator doesn’t know the order of operations, which is quite possible. In that case you can check the calculation on Google, because besides searching, it is also good at calculating:
- 1 + 2 + 5 · 0 on Google
- 1 + 2 + 5 · 0 on WolframAlpha (WolframAlpha is a well-known online tool for working out all sorts of problems; you can solve equations and work out derivatives, integrals…)
How to change the natural order of operations: brackets
If we want to change the order of operations, we can use brackets. If we had the expression
and wanted to add 1 + 2 first and only then multiply the result by five, we can use brackets like this:
Expressions in brackets take priority over the usual order of operations. So we would work out 1 + 2 first and leave the multiplication as it is. We would get:
which is equal to fifteen. So
What about division and subtraction?
Subtraction works the same way as addition, and division works the same way as multiplication. So multiplication and division take priority over addition and subtraction. Multiplication and division have the same priority as each other, and so do addition and subtraction.
For example, take a look at this expression
This expression is the same as the following one, in which we have added brackets so that the order of operations doesn’t change:
And so
which equals
If multiplication and division come straight after each other, we work them out from left to right as they come:
First we work out 6 · 4, which is 24. We don’t touch the division yet:
And now we just divide 24 by eight:
Why multiplication takes priority over addition
You might ask why multiplication takes priority over addition in the first place. And that is a good question! It can be said that there is nothing behind it other than a convention agreed by mathematicians. There is no rule from which it would be obvious that multiplication must take priority. All of maths would work just as well if addition had higher priority than multiplication. Or if they had the same priority and we always worked from left to right, as when we were pressing buttons on the calculator.
On the other hand, if addition took priority, we would need more brackets. Let’s show this with a slightly more complicated expression, a quadratic function:
In current mathematics this expression means “three times x squared”, which we add to “four times x”, which we then add to “five”. If addition took priority over multiplication, we would have to use brackets to get the same expression:
Because we use this expression and others like it in maths quite often, it is more practical for mathematicians if multiplication takes priority over addition. But there is no reason why it clearly has to be this way and not another.
If you are interested in more, you can read the article Why Multiplication Has Higher Priority than Addition: A Pedagogical Remark by Olga Kosheleva and Vladik Kreinovich.
Isn’t it better to use brackets?
Often it is. Many people don’t know how the priority of multiplication and addition works, and even a simple expression such as
can be misunderstood. If you want to increase the chances that people understand what you are trying to say in mathematical notation, simply add brackets to the expression:
