What is the smallest number that you can make with four 2s, using only exponents, brackets and numbers with 2s as their digits?

Guest May 11, 2015

#7**+8 **

Come on Badinage. You have to tell us what tht equals!

Use the forum calc like I did

Xerxes, I suppose it depends on your take.

Tell me, does 2 have a zero in it? It is exactly the same as 2.0 so by your logic 2 must also have a zero.

Zeros are often necessary in numbers because they are place holders. BUT 2 and .2 do not need any zeros as place holders because they are right up against the decimal point so their place value is indisputable.

And yes xy is x*y so if you are not allowed to use a multiply sign, does that mean that you cannot multiply OR does it mean that you cannot put a * in the display. I guess that is up to the individuals interpretation.

Melody
May 11, 2015

#2**+5 **

I don't know but I will just start an answer thread here. ( the calc puts the zeros in but the are not needed)

How about

$${{\mathtt{22}}}^{{\mathtt{0.22}}} = {\mathtt{1.973\: \!935\: \!646\: \!137\: \!177\: \!9}}$$

$${{\mathtt{2}}}^{{\mathtt{0.222}}} = {\mathtt{1.166\: \!349\: \!369\: \!788\: \!467\: \!3}}$$

$${{\mathtt{2}}}^{\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)\right)} = {\mathtt{1.005\: \!560\: \!580\: \!398\: \!468\: \!2}}$$

$${{\mathtt{0.2}}}^{\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)\right)} = {\mathtt{0.987\: \!207\: \!031\: \!388\: \!177\: \!8}}$$

$${{\left({\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)}^{{\mathtt{0.2}}}\right)}}^{{\mathtt{0.2}}} = {\mathtt{0.879\: \!189\: \!311\: \!540\: \!889\: \!8}}$$

Well my answers are getting smaller

Melody
May 11, 2015

#3**+4 **

I think you can not take the number 0.22, as the 0 itself is a digit, isn't it? And the 0 is not a 2, Even Rough You don't need write it

But correct me if I am wrong

xerxes
May 11, 2015

#4**+5 **

I did not write it the calc put it in there 0.2 = .2 the 0 is not needed.

It is usually added just so that the . is not mistaken for a speck of dirt.

Melody
May 11, 2015

#5**+4 **

I know

But the number itself still contains the digit 0, doesn't it?

xy contains a times sign as well, doesn't it? Even tough we don't have to write it?

xerxes
May 11, 2015

#6**+8 **# ⛵⛵⛵⛵⛵⛵

**$$\textcolor[rgb]{\mathbf{0,0,1}{.2^{2^{22}}}}$$**$$\textcolor[rgb]{0,0,1}{\mathbf{.2^{2^{22}}}}$$

**evaluates to ∼ 2x10 ^{-2931693}**

Badinage
May 11, 2015

#7**+8 **

Best Answer

Come on Badinage. You have to tell us what tht equals!

Use the forum calc like I did

Xerxes, I suppose it depends on your take.

Tell me, does 2 have a zero in it? It is exactly the same as 2.0 so by your logic 2 must also have a zero.

Zeros are often necessary in numbers because they are place holders. BUT 2 and .2 do not need any zeros as place holders because they are right up against the decimal point so their place value is indisputable.

And yes xy is x*y so if you are not allowed to use a multiply sign, does that mean that you cannot multiply OR does it mean that you cannot put a * in the display. I guess that is up to the individuals interpretation.

Melody
May 11, 2015

#9**+4 **

I think Badinage's answer is great, I am now able to accept the .2 as a proper number. Without the 0. This nice construction is quite fascinating 😁

xerxes
May 11, 2015

#10**+3 **

I am glad that you are happy xerxes, Badinage's answer is great and playing with numbers is always fund for mathematicians and mathematicans in the making :)

Melody
May 11, 2015

#11**+4 **

It is indeed, and you learn a lot by doing this. Mathematicians in the making is a nice word, btw.

xerxes
May 11, 2015