The comparability used is numeric

These are numbers not strings of characters. You’ll be able to see this by trying on the code within the 2009 important.cpp of the Bitcoin reference implementation:

        uint256 hashTarget = CBigNum().SetCompact(pblock->nBits).getuint256();
        uint256 hash;

[...]

           if (hash <= hashTarget)
            {
                pblock->nNonce = tmp.block.nNonce;
                assert(hash == pblock->GetHash());

                    //// debug print
                    printf("BitcoinMiner:n");
                    printf("proof-of-work discovered  n  hash: %s  ntarget: %sn", hash.GetHex().c_str(), hashTarget.GetHex().c_str());

Notice that if (hash <= hashTarget) is a numeric comparability. Each hash and hashTarget are kind uint256 – an unsigned integer.

Numbers expressed in hexadecimal are nonetheless numbers

There’s a alternative of visible representations however the alternative made doesn’t change the underlying nature of the quantity or the way in which by which numbers are in contrast arithmetically or at a machine degree in a pc.

Your instance, 00005fad, is a quantity expressed in hexadecimal (base 16), the identical quantity might be written in regular decimal (base 10) as 24493. Anybody unfamiliar with non-decimal representations reminiscent of hexadecimal, octal and binary can verify this utilizing one thing just like the Home windows 10 calculator, within the menu select “Programmer Mode” then click on on “hex” and enter 5fad – it reveals the similar worth in a number of totally different representations.

Perhaps this can make it clearer?

Despite the fact that the block hashes have the identical variety of main zeroes, one is a failure and the opposite a hit.

Main zeroes

The notion that Bitcoin cares in regards to the variety of main zeroes in, say, a hexadecimal illustration, is a generally repeated mistake (do not ask me how I do know this).

In case you insist on writing numbers with main zeroes it’s nonetheless clearly true that 000015 (fifteen) with 4 main zeroes is smaller than 000150 (100 and fifty) with solely three main zeroes. It could nevertheless be a mistake to assume that smaller numbers all the time have extra main zeroes. Each you and Bitcoin know that 000017 (seventeen) is smaller than 000019 (nineteen) though each have the identical variety of main zeroes.

It’s true that a is lower than b in precisely the identical method that 7 is lower than 8 or that 2 is lower than 3. However it’s in all probability a mistake to begin evaluating particular person digits in a selected visible illustration. The hash and hash targets are odd numbers (although massive) which can be in contrast in an odd method.

So the place does this discuss of main zeroes come from? In response to a outstanding contributor:

hashcash, the unique PoW system, had a “problem” that was truly the variety of zero bits up entrance within the hash. Bitcoin’s proof of labor relies on it, however generalized to an enormous integer comparability.

See

Examples

Lets have a look at some latest blocks (most up-to-date at prime, reverse chronological order)

Discover that the issue elevated after block 669311 however the variety of main zeroes within the hashes has not elevated (not in hexadecimal and never in binary).

Implementations

If you wish to see precise particulars you would have a look at early variations of the Bitcoin reference implementation in C++. Nonetheless I might counsel as a substitute trying on the present BTCD implementation in go-lang as a result of that’s effectively commented and, in my view, a better language to learn.

e.g. https://github.com/btcsuite/btcd/blob/grasp/chaincfg/params.go

    // TargetTimespan is the specified period of time that ought to elapse
    // earlier than the block problem requirement is examined to find out how
    // it must be modified with a view to preserve the specified block
    // technology price.
    TargetTimespan time.Period

    // TargetTimePerBlock is the specified period of time to generate every
    // block.
    TargetTimePerBlock time.Period

and https://github.com/btcsuite/btcd/blob/grasp/blockchain/problem.go

    // Calculate new goal problem as:
    //  currentDifficulty * (adjustedTimespan / targetTimespan)
    // The outcome makes use of integer division which suggests it will likely be barely
    // rounded down.  Bitcoind additionally makes use of integer division to calculate this
    // outcome.
    oldTarget := CompactToBig(lastNode.bits)
    newTarget := new(massive.Int).Mul(oldTarget, massive.NewInt(adjustedTimespan))
    targetTimeSpan := int64(b.chainParams.TargetTimespan / time.Second)
    newTarget.Div(newTarget, massive.NewInt(targetTimeSpan))

Calculating the hash goal

See