Basic tetration introduction

As we have learned, multiplication is repeated addition, and exponentiation is repeated multiplication, but what if we have repeated exponentiation? The answer for that is called tetration (also known as "hyperpower"). This fascinating operation is unfortunately not taught at school so let's learn it now!

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#math #algebra #mathbasics

[erniesmith]

The dual black & red pen handling is awesome!

[Ninja]

The final question is clearly no. The LHS is 16*65536 which is just over 1 million. The RHS is clearly way bigger because just 5 2’s in the stack already gives a number with 19 729 digits.

On another point, another way to write tetration is to use the up-arrow notation. a tetrated to b is written as a↑↑b (2 arrows) meaning we stack a in a power tower b times. I like this notation more personally as we can actually generalise this to more up arrows. Because exponentiation can also be written as a^b = a↑b (1 arrow). So to have more up arrows, we just repeat the previous level.

[creepermods]

Wow, it's so fast to understand tetration, thank you!

[chaosinsurgency]

For tetrations to be added, like when multiplying two powers of the same base, ex: (3^5)×(3^4)=(3^9)
To achieve this in tetratiin we would have to raise the tetratiin to the power of the tetratiin, ex: (2^^3)^(2^^4)=(2^^7)
This can be further generalized for the nth-tration(don't know the general term for tetratiins or pentrations etc) because when you use the nth-tration operation with itself the power/titration multiplies.
Ex: ((3^4)^5)=(3^20)
And also can be generalized for when you add the power/titration by using the (n-1)th-tration
Ex: (4^2)×(4^5)=(4^7)
I think this can be further generalized for (n+/-x)th-tration being used simultaenously, such as powers used with pentration, or powers(tritration, I think) used with multiplication.
Please excuse the likely incomprehensible jargon I've said as I am neither an expert in this nor am I awake enough to be typing this.

[davidknight]

I am now more confused. 2 to the third tetration is 16, but to fourth is 65536? Shouldn’t that be third tetration is 256?
First 2x2=4
Second 4x4=16
Third 16x16=256
Fourth 256x256=65536
I am 50 and never needed more than basic algebra since I left high school so I have forgotten everything😢

[DavyanHatch]

Pentation comes next, then hexation, then heptation, than octation and so one.

[Lordmewtwo]

"Which if I remember correctly is 65, 536." let me double check that real quick: 2^^4 is 2^2^2^2 so 2^16. I don't know off hand what that is, but I can use exponent properties to break it down into something more manageable because unless I'm mistaken, (2^4)^4 is equal to 16^4, which by using exponent properties is equal to 16^2*16^2. Now for the hard part: calculating 256^2. That would be 256*6+256*50+256*200, so 1536+12, 800+51, 200=65, 536. Checks out.

[beautie]

Master of using two markers at a time! Beautiful.

[TechDethMetalHed]

I never seen someone teach that so easily thx

[MrVanillaCaramel]

On what occasion would anyone use tetrations?

[TGears]

Do it simply with 2 and 2^2. Just the first and second tetrations. 2*4=8 and that does not equal what the third retraction is. 2^2^2 is 2^4 which is 16. 16 does not equal 8. Therefore a^2 * b^2 does not equal a+b^2. I wish I could notate that better but I’m unsure how to do that in comments on my phone.

[tobybartels]

I'd never guess that ³2 × ⁴2 = ⁷2; there's no pattern to suggest that. Sure, we have 2³ × 2⁴ = 2⁷, but that's just one level. We don't have 3•2 × 4•2 = 7•2, so clearly this rule only works for exponentiation, not multiplication or tetration.

What I _would_ guess is ³2 ^ ⁴2 = ⁷2. This fits a pattern: 3•2 + 4•2 = 7•2, then 2³ × 2⁴ = 2⁷, so why not ³2 ^ ⁴2 = ⁷2? If you write both sides out as power towers, they even have the same number of 2s in the tower.

But this isn't true either! Ultimately, this is because exponentiation (unlike addition and multiplication) isn't associative. So (2^2^2)^(2^2^2^2) isn't the same as 2^2^2^2^2^2^2; the parentheses matter.

[Zyrkoon]

Should WE use the "left exponent" notation, or Knuth arrows notation ? Like 2 ↑↑4 = 2^2^2^2=65536

[puroocio]

just asking, what's the inverse function of tetration? like log to exp, or how would a "root" would work?

[williamwilting]

I didn't come further than high school math, but I find this to be very interesting. However, I've seen some videos in which it is said that notations are done differently with upward arrows, because there are higher levels of hyperoperations.

About your question at the end of the video: To me it's obvious that ³2•⁴2 can't be equal to ⁷2, because you're multiplying two 2's somewhere inbetween, which means that you're not exponentiating 2 seven times consecutively. The answer to this question is 1, 048, 576, while the number ⁷2 is much, much bigger. I suspect that you'll have to calculate (³2) to the power of (⁴2) to get to ⁷2, because you're adding 4 more times of exponentiation with 2 to the first 3 times (I think that the parentheses are very important here, because the whole number must be involved, and otherwise you would probably just involve the base without the exponent directly). But I could be wrong, because the problem with exponentiation is that you can't just swap the base and the exponent to get the same result the same way as you could swap numbers with multiplication to get the same result. However, I honestly don't know if you can just add the hyperpowers to get ⁷2 if you calculate (³2) to the power of (⁴2), because (like I said) using (⁴2) as the base and (³2) as the exponent instead would probably give you another answer. In fact, I think you can't even get ⁷2 with either calculations, because you must probably calculate everything between parentheses first, even if it is before hyperoperations. So I'm not sure how all of this could work in a way similar to multiplying exponentiations (probably not at all). And now that I think of it while typing all of this, ⁷2 is calculated exponent by exponent from the top down, so even parentheses don't work in this case. Anyway, multiplying ³2 by ⁴2 doesn't get you even close to ⁷2.
And as for ⁷2:

⁷2=2^(2^(2^(2^(2⁴)
⁷2=2^(2^(2^(2¹⁶)
⁷2=2^(2^(2⁶⁵⁵³⁶)
And from here it doesn't make any sense to go on, because I can't even calculate this with any divice available to me. This number is just insanely high already, so '³2•⁴2 is nowhere near equal to ⁷2' is a very firm understatement. 😄

[SuperEMT]

2 to the hyper power of 3 time 2 to the hyper power of 4
2^4 * 2^16=2^20=1, 048, 576
Is that correct?
This is so much fun! 😀

[Sdoon]

Thx for this information I haven't studied the tetartion in school ❤😊

[Why_Fred]

Is tetration defined for all real numbers? Like does 2^^(√2) have a value? ( "^^" means hyperpower )

[ronjones]

2x2 is 4, 4x4 is 16, 16x16 is 256.
Can someone explain why 2 hyperpower 4 (pardon the jargon) isn't 256?

[MichaelRothwell]

Next we could use Knuth up arrow notation and generalise these binary operations a whole lot more...