Is the unbreaking enchantment beyond III effective?

Question

In my singleplayer world, I was testing the difference between an Unbreaking III and an Unbreaking IV enchanted armour (using TooManyItems), but I didn't see much of a difference. I would like some help from the mathematicians of this website to tell me what at what level of unbreaking the enchantment stops being more effective. Also, does this calculation apply to all versions past 1.5.2?

Answer 1

###Yes, but with rapidly diminishing returns

According to the wiki page on Enchanting¹, the effect of Unbreaking is:

Increases durability

[...]

For armor, (60 + (40/(Level+1)))% chance a use reduces durability. (In other words, each durability hit against “unbreaking” armor has a 20%/27%/30% chance of being ignored.) Thus, on average, armor lasts 25%/36%/43% longer.

You can see that the formula is different for armor, and even with the regular three levels, the chance to not lose durability when hit increases slower with higher levels. Extrapolating the "20%/27%/30%" using the formula given yields:

Unbreaking |  1 |  2 |  3 |  4 |  5 |  6 |  7 | ... | 10 | ... | 20
Chance     | 20 | 27 | 30 | 32 | 33 | 34 | 35 | ... | 36 | ... | 38
Increase   | 20 |  7 |  3 |  2 |  1 |  1 |  1 | ...

(Values are rounded in the table, but not in game). The chance converges to 40% for high values of Unbreaking, with diminishing returns the higher the enchantment level gets, to the point where is negligible.

If you directly compare an Unbreaking III Diamond Chestplate to an Unbreaking IV one, on average, the former breaks after 529100/(60+40/4)=756 hits, the latter after 529100/(60+40/5)=778. That is only a relative increase of 3% more hits.

^{1 While the page has changed, the game mechanics have been the the same since at least March 2013.}

Answer 2

According to the wiki, Unbreaking IV cannot be acquired in Survival Mode.

---

Armor enchanted with Unbreaking has a 40 * (1 - 1/(Level+1))% chance to ignore durability damage.

    Level   | Chance to ignore (%) | Durability increase (%)
 ===========================================================
      I     |          20          |           25         
     II     |         26.7         |           36
     III    |          30          |          42.9
     IV     |          32          |          47.1
      V     |         33.3         |           50
     VI     |         34.3         |          52.2 
     VII    |          35          |          53.8
     ...    |         ...          |          ...
  infinite  |          40          |          66.7

You can see that Unbreaking IV only gives a 3.6 % relative improvement over Unbreaking III, so of course you wouldn't notice the difference with quick tests.