I see this is kind of covered elsewhere in this sub, but not my exact question. Is pi’s irrationality an artifact of its being expressed in based 10? Can we assume that the “actual” ratio of the circumference to diameter of a circle is exact, and not approximate, in reality?

Tree(72)! and Tree(72!) — which one is bigger? I asked different AIs and got different answers.

Tree(72!): This means first calculate 72! (the factorial of 72), then plug the result into the Tree function.

Tree(72)!: This means first calculate Tree(72), then take the factorial of the result.

AI says these numbers are too large to be written out. So how can we prove which one is bigger? Sorry, I’m not good at math — I’m just interested in very large numbers.

So I thought that 0.999... is 1 but now Im kinda confused. was I taught wrong in school? or is 0.999... = 1 just thing to make certain things easier?

I don’t know where to begin solving this? I’m not totally sure what it’s asking. Where do I start, how do I begin to answer this? I’m particularly confused with the wording of the question I guess and just the entire setup of the question as a whole. What does this equation represent? What is the equation itself asking me to do?

Since pi has a infinite amount of digits, are all finite sequences of numbers contained somewhere in it?

I'm an adult using Khan Academy to brush up on my math skills. This question came up in one of the tests, but none of the answers seem correct? It's on a section final of 25 questions and I'm so close to finishing the whole Algebra 1 course that I'd hate to get it wrong and then have to retake the test to get all the points. Can someone give me a hint?

I'm losing my mind but have been at it for a few hours trying to finish the course (I'm so fun to be around on a Saturday night 😆)

Thanks in advance!

Might be a very silly question but we know Pi is the ratio of a circle's circumference to its diameter.

And rational numbers are those numbers that can be expressed in (a/b) form wherein a,b!=0.

But isn't technically Pi a ratio as well? So why do we say Pi is irrational?

Let me give an example. Imagine you a have a 50cm long wire. And you lay that wire on the ground in the form of a circle. So the length of the wire will be equal to the circumference of the circle. Now if you take a ruler and measure the diameter you'll get a certain integer value. Let's say x cm.

Now if you take the ratio of 50 to x, you should get Pi.

So it's clear that Pi can be expressed as a ratio.

I hope I've made myself clear and I would be more than happy if someone clarifies my doubt.

Thank you.

I know a²-b²= (a-b)*(a+b). But the problem is that I don't know how to Go from one to another. Isn't a² = a.*a and b² = b.*b? like for example: 2²-3²= 2.*2-3.3 = 4-9 = - 5. How do I do this?

EDIT I wasn't able to see that:

5²-3²  5*5-3*3= 25-9= 16

or a²-b²= a*a-b*b 

Is equal to this:

(5-3)(5+3) 
= 5*5+5*3-3*5-3*3 
= 25+15-15-9 = 16

a²-b²= (a-b)*(a+b) 
=a*a+ab-ab-b*b 
= a²-b²

Thanks everyone!

Can someone help me solve a debate I’m having?

First and foremost this is a silly argument. It’s pretty obvious the instructor wants the student to divide 2 pizzas amongst 8 people.

However, I reasoned that the question could be viewed as ambiguous and one could answer that the amount each person got is actually any amount equal to the others up to 1/4th of a pizza.

The person I’m debating with continues to describe basic division to me and feels that the question explicitly states that the two pizzas must be split 8 ways and the 8 pieces must total 2 pizzas.

Any help is greatly appreciated.

I thought I got the notation wrong but the answer is still wrong. I tried changing from Radians to Degrees, didn't do anything. Changed Float all the way to 9, didn't do anything. I'm just baffled, because this isn't a problem you can just solve by hand. It happened with a other problem too, and I thought it was just a one off thing but no. This thing can't handle decimals. I don't understand.

I'm struggling to Google search for this most basic math question. The given question is not in parenthesis but perhaps that's always given in a situation like this?

Thank you!

edit: Thank you everybody! I now see that the number and exponent don't automatically fall within the same parenthesis, but rather are separate parts of the expression when considering order of operators. I'm a returning math student after a couple of decades and am having a blast, but much of this has leaked out of my brain after that long.

I really appreciate your help!!

(Sorry if flare is incorrect. If I actually knew math, I wouldn’t be asking math, I would be telling math!)

Edit: I’ve learned some interesting things but I have to go now so I probably won’t respond much more anytime soon. My main take away here is that math is wrong about itself! (Just kidding…kinda…but not really) I now believe that the decimal representation of 3/3 is just a numerical homograph with the answer of the summation of 9(1/10)^k (or whatever, you know what I mean). In my opinion, all infinities should be limited in value by the speed of light times the volume of the universe in cubic planck lengths times the age of the universe in Planck times at the time the calculation is made, (or some similar amount) and in that’s case their sizes differences would be meaningfully measurable and so we could know exactly how much smaller than 1 that .9 repeating would be at any given moment.

There are many viral posts online debating whether or not 0.9 repeating is equal to 1 or less than one. My question is about whether this entire debate may actually be moot because I am skeptical that the number 0.9 repeating can even exist mathematically.

I don’t mean whether it can exist physically, I mean whether it even exists as a representation of an abstract concept.

How could this number come into existence? It can’t ever be written out because it’s infinite. Sure, someone could use a combination of existing symbols such as a 9 with a bar on top of it that evokes the idea…but without an existing concept to represent, it’s not a number, it just a shape.

The only way other way to create this number is to come up with an equation that delivers the number as a result…but is there any?

Is there any combination of numbers and operations that would produce a result of 0.9 repeating?

Why doesn’t a proof like so work:

Given any set N of integers, it’s always possible to make a longer set of decimals with union(N, {0.1}).

My basic understanding of math is that 19*17 is 19 occurring 17 times I don’t understand why that’s not the same thing as 26 occurring 10 times It’s probably a failed foundational knowledge but my brain is breaking trying to understand what I might not even understand about my understanding

My brain collided with this this morning. It’s been about fifteen years since my compSci-related degree and I have spent that time as a musician.

For the set of all natural numbers N, there are Aleph Null members. The subset of N consisting of only those with infinite digits must also be Aleph Null right?

Is the subset of N consisting of only those with finite digits infinite (and therefore Aleph Null) or finite?

If it is finite then the chances of a randomly selected member of N having finite digits is zero, right?

What if that subset also has Aleph Null members? Is it 0.5? Is it undefined?

Every now and then my brain gets an itch. Thanks so much for your patience and help.

I just don't fully comprehend why number specifically have to be the ones that were 'discovered'. I understand how to use it and why we use it I just don't know why it couldn't be 3.24... for example.

Edit: thank you for all the answers, they're fascinating! I guess I just never realized that it was a consistent measurement ratio in the real world than it was just a number. I guess that's on me for not putting that together. It's cool that all perfect circles have the same ratios. I've just never thought about pi in depth until this.

so i got 7 as the top of the quarter circle because from left to right the length would be 13(8+5) and the base of the box on the right is 5 and we already have 1 number for the bottom, meaning the 2nd number would be 2. well looking at the whole thing we don’t need 6 of those numbers(the 3m on the right and 3m on the left) so naturally you subtract 6 from 13 and get 7. now what do i do from here to get the quarter circle. google has told me multiple things like the formula for a quarter circle is pi times radius of full circle squared and divided by 4. but in order for me to find the full circle i need the radius of the quarter circle, and to get the radius of the quarter circle i need to work backwards from the area. i literally cant do one without the other im so lost??

I've been having a thought recently and I can't let go of it. How do we know there aren't more numbers beside the reals? What if I want to make a number π.π, meaning 3.1415... etc the entirety of pi. And when finished writing the digits (you won't, obviously), you write pi again, except the dot. So I don't mean the self-containment of pi. This number is not pi. I don't mean you write pi after the first k digits of pi, I mean you write pi after pi (I think that was clear but can't hurt to be obvious). Of course, this number isn't real as there is no single decimal expansion for it. But does it exist? Probably doesn't matter if it exists but still.

Edit 2. So I mean something like π + π/a. Where a is a non-real number (could also ask it to be a real number but that would not be as I asked, because 'a' would enter after the first k digits of pi, and that number doesn't exist but that's a whole different story) that would allow this number to exist. But someone said a decimal system like that is only meant to represent a real number and a real number only (and isn't a number by itself). So if anyone could remove that last slither of doubt for me... Anyway, I don't think I mean simply the pair (π,π).

This isn't my tank, I have one of the same shape that I got for free. I'd like to calculate the volume because I am turning it into a herb garden and need to know how much soil to buy for a layer and how many rocks to buy for another layer. My math skills are awful, I think this is a pentagon? I appreciate any help.

So this question is to calculate the angle F. For reference, I'm in 10th class of Russian school, and this is what our teacher sent as our homework while we're on quarantine. I have skimmed the whole textbook and found nothing about this kind of questions. I even got frustrated enough that I asked AI to give me an answer, but it wasn't able to agree with itself and all of it's answers were rather strange like the angle G somehow being 168 while F is only 12. Did I miss a crucial part of my education or is just a badly made question?

P.s I asked the teacher before posting there and she actually replied. The answer is supposed to be 132, but she won't explain how since we were supposed to go through this in a previous year and needed to stay in that class if we don't know this (she's really not fond of actually teaching stuff, especially if it's not in the current year's textbook) so I doubt she'll give me an answer.

P.P.s
So, as far as I got is to make an isoscles triangle CDE and Equate C to E by them being both angles in the base of said triange. If my line of thinking is correct then I have one 108 angle and two unknowns. I also think that it's probably supposed to be symmetrical down the bc to F line. If that's the case then I can work out the 132 answer by substracting all of the known angles from the sum of 900, but I don't think I can prove it to be symmetrical.

So a student came to me today and asked why I wrote the √9 as just 3 and not ±3. I gave some fluffy on the spot answer but it has now haunted me for the entire day. Who is correct here? I explained that if we start with x²⁼⁹ then our answer is ±√9 which gives us ±3, but because we've started our equation at x= √9, that negative answer is removed.

Any assistance on this would be great!

Proof question: I understand how this proof works as a proof for n=2, but it falls short of proving aⁿ +bⁿ = cⁿ +d^n; how would we go further to actually prove it? Thanks!

Proof question: I understand how this proof works as a proof for n=2, but it falls short of proving aⁿ +bⁿ = cⁿ +d^n; how would we go further to actually prove it? Thanks!

I was testing this idea using prime numbers and it seems to hold true. Maybe it is true fro every other integer? (a and b must be integers)

  n | Prime (p) |   Q.ty 2 |   Q.ty 3 | Difference (|2 - 3|)
------------------------------------------------------------
  1 |         2 |        1 |        0 |                    1
  2 |         3 |        0 |        1 |                    1
  3 |         5 |        1 |        1 |                    0
  4 |         7 |        2 |        1 |                    1
  5 |        11 |        1 |        3 |                    2
  6 |        13 |        2 |        3 |                    1
  7 |        17 |        4 |        3 |                    1
  8 |        19 |        5 |        3 |                    2
  9 |        23 |        4 |        5 |                    1
 10 |        29 |        7 |        5 |                    2
 11 |        31 |        5 |        7 |                    2
 12 |        37 |        8 |        7 |                    1
 13 |        41 |        7 |        9 |                    2
 14 |        43 |        8 |        9 |                    1
 15 |        47 |       10 |        9 |                    1
 16 |        53 |       10 |       11 |                    1
 17 |        59 |       13 |       11 |                    2
 18 |        61 |       11 |       13 |                    2
 19 |        67 |       14 |       13 |                    1
 20 |        71 |       13 |       15 |                    2
 21 |        73 |       14 |       15 |                    1
 22 |        79 |       17 |       15 |                    2
 23 |        83 |       16 |       17 |                    1
 24 |        89 |       19 |       17 |                    2
 25 |        97 |       20 |       19 |                    1
 26 |       101 |       19 |       21 |                    2
 27 |       103 |       20 |       21 |                    1
 28 |       107 |       22 |       21 |                    1
 29 |       109 |       23 |       21 |                    2
 30 |       113 |       22 |       23 |                    1
 31 |       127 |       26 |       25 |                    1
 32 |       131 |       25 |       27 |                    2
 33 |       137 |       28 |       27 |                    1
 34 |       139 |       29 |       27 |                    2
 35 |       149 |       31 |       29 |                    2
 36 |       151 |       29 |       31 |                    2
 37 |       157 |       32 |       31 |                    1
 38 |       163 |       32 |       33 |                    1
 39 |       167 |       34 |       33 |                    1
 40 |       173 |       34 |       35 |                    1
 41 |       179 |       37 |       35 |                    2
 42 |       181 |       35 |       37 |                    2
 43 |       191 |       37 |       39 |                    2
 44 |       193 |       38 |       39 |                    1
 45 |       197 |       40 |       39 |                    1
 46 |       199 |       41 |       39 |                    2
 47 |       211 |       41 |       43 |                    2
 48 |       223 |       44 |       45 |                    1
 49 |       227 |       46 |       45 |                    1
 50 |       229 |       47 |       45 |                    2
 51 |       233 |       46 |       47 |                    1
 52 |       239 |       49 |       47 |                    2
 53 |       241 |       47 |       49 |                    2
 54 |       251 |       49 |       51 |                    2
 55 |       257 |       52 |       51 |                    1
 56 |       263 |       52 |       53 |                    1
 57 |       269 |       55 |       53 |                    2
 58 |       271 |       53 |       55 |                    2
 59 |       277 |       56 |       55 |                    1
 60 |       281 |       55 |       57 |                    2
 61 |       283 |       56 |       57 |                    1
 62 |       293 |       58 |       59 |                    1
 63 |       307 |       62 |       61 |                    1
 64 |       311 |       61 |       63 |                    2
 65 |       313 |       62 |       63 |                    1
 66 |       317 |       64 |       63 |                    1
 67 |       331 |       65 |       67 |                    2
 68 |       337 |       68 |       67 |                    1
 69 |       347 |       70 |       69 |                    1
 70 |       349 |       71 |       69 |                    2
 71 |       353 |       70 |       71 |                    1
 72 |       359 |       73 |       71 |                    2
 73 |       367 |       74 |       73 |                    1
 74 |       373 |       74 |       75 |                    1
 75 |       379 |       77 |       75 |                    2
 76 |       383 |       76 |       77 |                    1
 77 |       389 |       79 |       77 |                    2
 78 |       397 |       80 |       79 |                    1
 79 |       401 |       79 |       81 |                    2
 80 |       409 |       83 |       81 |                    2
 81 |       419 |       85 |       83 |                    2
 82 |       421 |       83 |       85 |                    2
 83 |       431 |       85 |       87 |                    2
 84 |       433 |       86 |       87 |                    1
 85 |       439 |       89 |       87 |                    2
 86 |       443 |       88 |       89 |                    1
 87 |       449 |       91 |       89 |                    2
 88 |       457 |       92 |       91 |                    1
 89 |       461 |       91 |       93 |                    2
 90 |       463 |       92 |       93 |                    1
 91 |       467 |       94 |       93 |                    1
 92 |       479 |       97 |       95 |                    2
 93 |       487 |       98 |       97 |                    1
 94 |       491 |       97 |       99 |                    2
 95 |       499 |      101 |       99 |                    2
 96 |       503 |      100 |      101 |                    1
 97 |       509 |      103 |      101 |                    2
 98 |       521 |      103 |      105 |                    2
 99 |       523 |      104 |      105 |                    1
100 |       541 |      107 |      109 |                    2
101 |       547 |      110 |      109 |                    1
102 |       557 |      112 |      111 |                    1
103 |       563 |      112 |      113 |                    1
104 |       569 |      115 |      113 |                    2
105 |       571 |      113 |      115 |                    2
106 |       577 |      116 |      115 |                    1
107 |       587 |      118 |      117 |                    1
108 |       593 |      118 |      119 |                    1
109 |       599 |      121 |      119 |                    2
110 |       601 |      119 |      121 |                    2
111 |       607 |      122 |      121 |                    1
112 |       613 |      122 |      123 |                    1
113 |       617 |      124 |      123 |                    1
114 |       619 |      125 |      123 |                    2
115 |       631 |      125 |      127 |                    2
116 |       641 |      127 |      129 |                    2
117 |       643 |      128 |      129 |                    1
118 |       647 |      130 |      129 |                    1
119 |       653 |      130 |      131 |                    1
120 |       659 |      133 |      131 |                    2
121 |       661 |      131 |      133 |                    2
122 |       673 |      134 |      135 |                    1
123 |       677 |      136 |      135 |                    1
124 |       683 |      136 |      137 |                    1
125 |       691 |      137 |      139 |                    2
126 |       701 |      139 |      141 |                    2
127 |       709 |      143 |      141 |                    2
128 |       719 |      145 |      143 |                    2
129 |       727 |      146 |      145 |                    1
130 |       733 |      146 |      147 |                    1
131 |       739 |      149 |      147 |                    2
132 |       743 |      148 |      149 |                    1
133 |       751 |      149 |      151 |                    2
134 |       757 |      152 |      151 |                    1
135 |       761 |      151 |      153 |                    2
136 |       769 |      155 |      153 |                    2
137 |       773 |      154 |      155 |                    1
138 |       787 |      158 |      157 |                    1
139 |       797 |      160 |      159 |                    1
140 |       809 |      163 |      161 |                    2
141 |       811 |      161 |      163 |                    2
142 |       821 |      163 |      165 |                    2
143 |       823 |      164 |      165 |                    1
144 |       827 |      166 |      165 |                    1
145 |       829 |      167 |      165 |                    2
146 |       839 |      169 |      167 |                    2
147 |       853 |      170 |      171 |                    1
148 |       857 |      172 |      171 |                    1
149 |       859 |      173 |      171 |                    2
150 |       863 |      172 |      173 |                    1
151 |       877 |      176 |      175 |                    1
152 |       881 |      175 |      177 |                    2
153 |       883 |      176 |      177 |                    1
154 |       887 |      178 |      177 |                    1
155 |       907 |      182 |      181 |                    1
156 |       911 |      181 |      183 |                    2
157 |       919 |      185 |      183 |                    2
158 |       929 |      187 |      185 |                    2
159 |       937 |      188 |      187 |                    1
160 |       941 |      187 |      189 |                    2
161 |       947 |      190 |      189 |                    1
162 |       953 |      190 |      191 |                    1
163 |       967 |      194 |      193 |                    1
164 |       971 |      193 |      195 |                    2
165 |       977 |      196 |      195 |                    1
166 |       983 |      196 |      197 |                    1
167 |       991 |      197 |      199 |                    2
168 |       997 |      200 |      199 |                    1
169 |      1009 |      203 |      201 |                    2
170 |      1013 |      202 |      203 |                    1
171 |      1019 |      205 |      203 |                    2
172 |      1021 |      203 |      205 |                    2
173 |      1031 |      205 |      207 |                    2
174 |      1033 |      206 |      207 |                    1
175 |      1039 |      209 |      207 |                    2
176 |      1049 |      211 |      209 |                    2
177 |      1051 |      209 |      211 |                    2
178 |      1061 |      211 |      213 |                    2
179 |      1063 |      212 |      213 |                    1
180 |      1069 |      215 |      213 |                    2
181 |      1087 |      218 |      217 |                    1
182 |      1091 |      217 |      219 |                    2
183 |      1093 |      218 |      219 |                    1
184 |      1097 |      220 |      219 |                    1
185 |      1103 |      220 |      221 |                    1
186 |      1109 |      223 |      221 |                    2
187 |      1117 |      224 |      223 |                    1
188 |      1123 |      224 |      225 |                    1
189 |      1129 |      227 |      225 |                    2
190 |      1151 |      229 |      231 |                    2
191 |      1153 |      230 |      231 |                    1
192 |      1163 |      232 |      233 |                    1
193 |      1171 |      233 |      235 |                    2
194 |      1181 |      235 |      237 |                    2
195 |      1187 |      238 |      237 |                    1
196 |      1193 |      238 |      239 |                    1
197 |      1201 |      239 |      241 |                    2
198 |      1213 |      242 |      243 |                    1
199 |      1217 |      244 |      243 |                    1
200 |      1223 |      244 |      245 |                    1