Why Powers, Roots, and Logarithms Are Really the Same Pattern

A researcher in Amsterdam was checking his son's maths homework. The assignment covered powers, roots, and logarithms. He looked at the formulas and thought:

"Why do they make something so simple so difficult?"

That question turned into a book. This interactive article is based on it. We are going to show you a pattern that is already in your head — you just haven't noticed it yet — and then show how a small change in notation makes that pattern visible.

A pattern you already know

Let's start with something simple. Addition and subtraction belong together:

Look: 3 + 5 = 8. If you know 8 and 5, then 8 − 5 gives you 3.

Candy example: you had 3 candies and you get 5 more, so 3 + 5 = 8. Then you give 5 candies away: 8 − 5 = 3.

This pattern is important in maths. You can also see it in the signs: + and − belong together.

Now go one level up. Multiplication and division are partners in exactly the same way:

Again: if you know the result (12) and one input (4), the inverse (division) gets the other input back. And the symbols × and ÷ look like relatives — both based on a cross-shape.

So we have a nice pattern:

Each level has an operation and its inverse, and you can tell they belong together because they look alike.

Where school notation breaks the pattern

There is a third level. Multiplication is repeated addition (3 × 4 = 3 + 3 + 3 + 3). In the same way, there is an operation that means "repeated multiplication":

So far so good. Now: what is the inverse? If you know the answer is 8 and the exponent is 3, how do you get the base (2) back? In school, this is written with a radical sign:

And what if you know the answer is 8 and the base is 2, and you want to find the exponent (3)? That is called a logarithm:

Do these three look related to you? Do they look like they belong to the same family?

They don't. The first uses a tiny superscript. The second uses a √ sign with a little number in the crook. The third spells out the word "log" and puts the base as a subscript. Three completely different visual systems for three operations that are supposed to be inverses of each other.

This is the problem: the maths still follows the same pattern, but these three very different symbols make that pattern much harder to see.

What if we fix it?

The fix is surprisingly simple. Instead of three different visual systems, use three symbols that look like variations of each other:

And the table now has a pattern on every level:

Three levels. Same structure everywhere: the inverse undoes the operation, and the symbols tell you they belong together. That's the whole idea.

But wait — why two inverses?

At the first two levels, you only need one inverse: − for +, ÷ for ×. That's because order doesn't matter there: 3 + 5 = 5 + 3 and 3 × 4 = 4 × 3.

But for powers, order matters: 2 ↑ 3 = 8, while 3 ↑ 2 = 9. So there are two numbers that can go missing — the one left of ↑ and the one right of ↑ — and you need a different inverse for each:

That's the only rule you need. Missing the left number? Use ↓. Missing the right number? Use ⇓.

The power of inverses: they cancel each other

There's something beautiful that follows from this. You already know that + and − cancel each other:

And × and ÷ cancel each other:

With the new notation, exactly the same works for ↑ and ↓:

And for ↑ and ⇓:

In school notation, that first cancellation would be written as ³√(5³) = 5, and the second as log₅(5³) = 3. But look at them: can you see that ³√ and ³ cancel? Or that log₅ and 5³ cancel? Not really — they look completely different. With ↓ and ↑, the cancellation is visible: they're the same symbol pointing in opposite directions. And ⇓ and ↑ work the same way. That's the whole point of the notation: it doesn't just label the operations — it shows you how they relate.

Your turn

The calculator on this page supports ↑, ↓, and ⇓. Here are some things to explore:

Four examples, one pattern

Theory is one thing. Let's see if it holds up in the real world — with examples from finance, music, geology, and public health. In school, each would require a different technique. With our notation, they all follow the same pattern: set up an equation with ↑, figure out which number is missing, and pick the matching inverse (↓ or ⇓).

Where this comes from

This notation was developed by Steven Pemberton, a computer scientist at CWI Amsterdam. In his book "Numbers," he starts from the very beginning — counting with sticks — and builds up through addition, multiplication, and powers, showing that each level follows the same pattern. The notation isn't arbitrary: it was designed to make that pattern visible.

If this notation made something click for you — or if you want to share it with a student or teacher — the full book by Steven Pemberton is available as a PDF.