0xffffffffffffffffn: uint64 hack to decrement by one using addition


## Introduction

In the world of software development, there are often clever tricks and hacks that developers use to solve specific problems. One such trick involves using `uint64` (unsigned 64-bit integer) arithmetic to subtract `1` from a number without actually using subtraction. This might sound confusing at first, but with a simple analogy and some examples, it becomes clear!

## The Toy Cash Register Analogy

Imagine a child's toy cash register that can only count up to `9`. When it reaches `9`, and you try adding `1` more, it wraps around and returns to `0`. Now, if this toy didn't have a subtraction button, how could you subtract `1` from any number? The trick is to add `9`! When you add `9`, you essentially go full circle and land `1` number behind where you started.

## The Technical Explanation

In binary arithmetic, numbers are represented using bits. A `uint64` number uses 64 bits. When you add two numbers and the result is too big to fit in 64 bits, it wraps around, similar to our toy cash register.

The magic number `0xffffffffffffffffn` (which represents all `1`s in binary for a 64-bit number) acts like the number `9` in our analogy. When you add this number to any `uint64`, it's equivalent to subtracting `1`.

| Bit Position:        | 63-56    | 55-48    | 47-40    | 39-32    | 31-24    | 23-16    | 15-8     | 7-0      |
| -------------------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- |
| Previous Number: 3   | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000011 |
| Magic Number to Add: | 11111111 | 11111111 | 11111111 | 11111111 | 11111111 | 11111111 | 11111111 | 11111111 |
| Result: 2            | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000000 | 00000010 |

### Example Code

```javascript
// Convention for BigInts is to add an "n" at the end.
const previousNumber = 3n;

// Hexadecimal representation of 64-bit magic number.
const magicNumber = 0xffffffffffffffffn;

// Convert to 64-bit unsigned integer.
const result = BigInt.asUintN(64, previousNumber + magicNumber);

console.log(result); // This will be 2n.
```

## Why is this Useful?

In the realm of programming, there are instances where certain operations, like subtraction, might be restricted due to system constraints. This "magic number" trick offers a clever workaround for such situations.

Take, for example, [Deno KV](https://deno.com/kv). This key-value storage system in Deno provides an atomic [sum](https://docs.deno.com/kv/manual/operations#sum) operation, allowing you to add two [`Deno.KvU64`](https://deno.land/api@v1.37.0?s=Deno.KvU64&unstable=) values. This is particularly handy for incrementing numeric counters in an [ACID-compliant way](https://en.wikipedia.org/wiki/ACID). However, there's a catch: you can't directly decrement a value using this method since `Deno.KvU64` requires a positive number in the construction.

Here's how you'd typically increment a counter:

```typescript
await kv
  .atomic()
  .mutate({
    type: 'sum',
    key: ['users_count'],
    value: new Deno.KvU64(1n)
  })
  .commit();
```

But what if you wanted to decrease this counter? Enter our magic number trick. By using the magic number `0xffffffffffffffffn`, you can effectively subtract from the counter:

```typescript
await kv
  .atomic()
  .mutate({
    type: 'sum',
    key: ['users_count'],
    value: new Deno.KvU64(0xffffffffffffffffn)
  })
  .commit();
```

In essence, this approach leverages the properties of binary arithmetic to bypass system constraints. Pretty cool, right?

## Conclusion

Understanding the underlying principles of binary arithmetic can open the door to many clever tricks in programming. The `uint64` subtraction trick is just one example. Hopefully, you can find efficient solutions to complex problems by understanding and applying these concepts to your following projects.


Nap Joseph Calub

Posts with the "Binary Arithmetic" tag