# Can we add without using addition?

Is there a formula that applies $a + b = c$ without addition. I tried many times to make such equation.

Here's what I have in mind, we get 2 values, call them $a$ and $b$. The objective is to get the sum, without using addition at all. What does "no addition" mean here? Basically, you can't use the addition operator in the equation. In fact, subtracting the additive inverse of $b$ to $a$ still counts as using the addition operator. No $\sum$ either.

So what else? "I tried many times to make such equation." Here's an example I did with $2 + 3 = 5$:

$$2 + 3 = 5$$ $$2 \times 3 = 6 \ne 5$$ $$6 - (3 - 2) = 6 - 1 = 5$$

So I thought $(a \times b) - (b - a)$ was plausible, but then I tried $3 + 6 = 9$:

$$3 + 6 = 9$$ $$3 \times 6 = 18 \ne 9$$ $$18 - (6 - 3) = 18 - 3 = 15 \ne 9$$

I haven't tried division yet because it can result to decimals, and we all know how fuzzy they are.

If it is even possible, can you get 2 values to result to their sum without any form of addition?

## 3 answers

You could use the Peano axioms instead to define addition - using the successor function,

$$a + 0 = a $$

$$a + S(b) = S(a + b)$$

But I'd argue that this is not different - addition is defined this way to begin with...

In my opinion the question makes no sense - addition is addition, you can break it down to its definition, but that's still addition...

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Yes, we can.

log(10^x * 10^y)

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If you're allowed to use subtraction but not addition, then you could try something like

$$c = 1000 - (( 1000 - a ) - b )$$

This reminds me of the way nearly all electronic and mechanical calculators and computers use the exact same ALU to do both addition and subtraction, rather than a dedicated adder and a separate subtractor.

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