BIDMAS is a Lie

You don't have to work exponentials before multiplication

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

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It’s with much gratitude to

Jemma

that I write on this topic.

At MathsConf38 she gave a talk on order of operations. I missed it, but she gave me a whirlwind tour of the core ideas later that day, and it blew my mind.

At Unstoppable Learning we were about to put together some materials on order of operations for a tutoring programme we’re piloting, and it changed everything for me.

At the core of Jemma’s thinking was that BIDMAS, PEMDAS etc. are all mnemonics that, if at all useful, perhaps imply that the order of operations is arbitrary.

Even if these mnemonics don’t imply an arbitrary order of operations, it’s certainly true that they do nothing to show the mathematical structures that mean the order of operations is unavoidable.

When we set out to make this both Unstoppable and meaningful, we combined Jemma’s ideas with our preference for using brackets to signify multiplication.👇

Why We Don't Use '×' for Multiplication

This made the reason for the order of operations instantly obvious; barely a question. Certainly no need for a mnemonic anymore. No need to recall anything from memory, really - the written structure did all the work.

But then the real surprise popped out when we realised that BIDMAS is a lie, and you don’t actually have to apply exponentiation before you multiply.

BIDMAS is a lie

For example, given this:

\(5\times10+3^2\)

BIDMAS says you must first work the exponentiation:

\(5\times10+9\)

Before the multiplication:

\(50+9\)

To make sure we calculate correctly:

\(59\)

And yet, if we work the multiplication first instead:

\(5\times10+3^2 \)

\(50+3^2\)

\(50+9\)

\(59\)

We get the same result.

So it’s not true that ‘we must first work out exponentials before multiplication.’

When we brackets Instead of the × symbol for multiplication, this pops out as obvious.

So from here I will show how the order of operations are revealed by this symbology, how to deal with individual terms that combine both exponentiation and multiplication, how to bring in grouping symbols like brackets, and then finally how to address the inverse operations.

Brackets reveal structure

For example, image you’re asked to evaluate this:

\(3+5\times10\)

You know the operations, and you can work the numbers.

You have internalised at this point that ‘we work left to right,’ thanks to your lessons on reading and writing.

And so it would be reasonable to assume you first work out three plus five, to get eight, and then times that by ten, to get eighty.

So we teach students explicitly: NO!

There is an order to the operations!

It’s not just left to right!

The order is given by BIDMAS.

Which tells us you must do M and multiply, before A for add.

But now compare that expression to this one: