#How to approach this question in your head in a realistic fashion?

501 messages · Page 1 of 1 (latest)

simple skiffBOT
regal thicket
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you realize that a second is 1,000,000 microseconds and all you should expect from this problem are big numbers

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it would help if you posted the numbers you did get instead of a blank table

regal thicket
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its a baseline

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part of the purpose of this problem is to get you to realize what these various complexities end up meaning

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they tell you how much slower a program gets if you increase the output

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this is roughly like big O, yea

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yea

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wdym the first one

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f(n) = n is the easiest one, not the first one

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you dont begin from the top-down unless youre a computer

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do you think f(n) = lg n is easier than f(n) = n?

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be a human here

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you say this like only maths guys do easy things before hard things

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look at the table

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you need a calculator

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you use a calculator

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and also, you can write big digits as 10^#

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that is not a problem either

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...was that your real concern

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thats good

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did you solve f(n) = n yet?

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just the first column?

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what about the rest of them?

regal thicket
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dont critique the question, youre saying a lot when you havent shown evidence of completing even the easiest row

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please do

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youll notice here that the size n matches with the time taken

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now the next step is 2^n

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youre not making connections yet, just go through the table

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2^n is chosen because its easy to put in the calculator, and the numbers get big quick

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bro

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Im not making you see anything

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youre not making connections yet, just go through the table

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2^n is not the same thing as 2 * n

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and that would be wrong if it was 2 * n as well

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please

regal thicket
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alright?

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for reference, you should get n = 19 for the 2^n row, first column

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Ill brb
once you fill in the table, tell me if you used a trick or not for the 2^n row, I suspect either you wont because I told you or you adamantly did because I told you

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there is a trick, and it involves the f(n) = n row, but make sure its solid beforehand

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(its not a very good one)

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Ill be back

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gl

regal thicket
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the numbers you have in the f(n) = n row tell you how many microseconds are in each amount of time

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now its relatively simple to start with 1, press * 2 = on the calculator, then continually press = to multiply by 2 many times

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its not as simple to count how many times you do so

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but by doing this, you can go through all the powers of 2 and compare them with the f(n) = n row

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its a very big hint now that I realize it, but youve been at this for 20 minutes

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to help with this, 2^10 = 1024 is similar to 1000

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so 2^20 = 1048576 is similar to 1 million

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if you lose track, you can use powers of 1024 to get back on track

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I wouldve seen that pattern at 4am too

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it doesnt help that the problem is convoluted

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oh right theres also that

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but you came in looking for a shortcut

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the problem is asking you,

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f(biggest n) that is smaller than t

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so 2^n < 1 million for the first column to get n = 19

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right

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yea

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3.6 * 10^9, not "with 9 zeroes"

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this has 8 zeroes

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yea

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yea, the e stands for exponent

regal thicket
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so Ill show a quick way instead

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you could do 2^n = 3.6e9

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then solve for n

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to find the n that gets you that time of 1 hour

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(round down if n isnt an integer)

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have you heard of "solve for x" before

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then you mean "how do I solve for n" right

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do you know what a logarithm is

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do you know they are the reverse of exponents

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2^n = 3.6e9

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then you need to reverse 2^ both sides

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similar to how for 2 + n = 3.6e9, youd "reverse 2 + both sides" which is to "- 2 both sides"

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,align2^n&=3.6\times10^9
\{\color{yellow}\log_2({\color{black}2^n})}&={\color{yellow}\log_2({\color{black}3.6\times10^9})}
\n&=\log_2(3.6\times10^9)

brazen onyxBOT
regal thicket
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youre not saying that corresponding_column_of_f(n)=n only just means t_in_microseconds, the only reason Im having you use the f(n)=n was because it already calculated t_in_microseconds in advance to be used for the braindead approach earlier, the row has no other meaning

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wait how did you get 3,1104

regal thicket
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instead of 12 months, use 365 days
so your times should look like

1 second: t = 1e6
1 minute: t = 6e7
1 hour: t = 3,6e9
1 day: t = 8,64e10
1 month: t = 2,592e12
1 year: t = 3,1536e13
1 century: t = 3,1536e15

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now to solve the table, youre supposed to use the equation f(n) = t to solve for n, then round down to the nearest integer because you cant have a non-integer size

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so for the 2^n row, you use 2^n = t to get that n = log2(t) then round down to the nearest integer to get:

1 second: n = 19
1 minute: n = 25
1 hour: n = 31
1 day: n = 36
1 month: n = 41
1 year: n = 44
1 century: n = 51

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again, the n row mathes the t row because of n = t, but remember that the general process is "f(n) = t" not "f(n) = f(n) = n" (scroll up to see why):

1 second: n = 1e6
1 minute: n = 6e7
1 hour: n = 3,6e9
1 day: n = 8,64e10
1 month: n = 2,592e12
1 year: n = 3,1536e13
1 century: n = 3,1536e15

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@teal shale are you with me so far

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you chose to type twice as much for a method you deliberately chose to not use

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yes, really

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at least you werent serious about it

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now what time is it on your end

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I can tell

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now you know that to fill in the cells you begin with f(n) = t then you end with finding n

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now to ensure you dont get any wrong ideas, youll still just try to solve f(n) = t normally

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yes I did, but as long as all of your ideas are right it'll be worth it

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you havent even begun

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cant believe this guy

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in real life

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is it discord where your assignment is?

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I already told you what youre supposed to be doing and I can see youre still not doing it

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do you just want to talk? 🤨

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if I were you, Id solve f(n) = t in advance for each row before going through the calculations, so that the hard part is done first

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except for n lg n = t and n! = t - those cannot be solved

regal thicket
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how do you reverse a lg()?

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if you had an "reverse lg", then you could do

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wait

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@teal shale you here? I need to ask you a qusetion

regal thicket
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thats not the question

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I need to ask you a different question

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how good are you at your algebra

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do you understand the basics of algebra?

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do you understand why its valid to "do the same thing to both sides" of an equation?

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reread what you typed a second time

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(you already showed what x is by saying **x = **5 + 4y)

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dont bother

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what

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just dont bother

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different teachers have different ways of describing algebra steps

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so Ill need to see what language you use instead to solve an algebra problem

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say you have 2x + 5 = 10

regal thicket
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@teal shale no?

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were you aware that 2 * 2.5 = 5 at the time?

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did it occur to you that you couldve written x = 5/2?

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keep in mind you can leave fractions in the answer

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you are allowed to just let division be in there

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this indicates you dont know a proper method to go through this

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so far its "this ought to work" or "this should make sense"

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you can solidify this by putting down what you know will always work

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do you want to see what's valid for an algebra step?

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first, some terms out of the way

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in 2x + 5 = 10, the 2x + 5 and 10 are both called "expressions"

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this is compared to 2x + 5 = 10, which is called an "equation" for having an = sign

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so you have two expressions that can both be a value

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for example if x = 3, then the expression 2x + 5 is 11
and the expression 10 is just 10

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are you with me so far

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youll need to define what an expression is but it should be closed

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now in an equation, the left side of the = is usually named the "LHS" for "left-hand side"
the right side of the = is usually named the "RHS" for "right-hand side"

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so in 2x + 5 = 10, the LHS is 2x + 5 and the RHS is 10

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now let's consider what's valid

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an equation of expression = expression should boil down to setting two values equal

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for example, 5 = 5 or 7 = 7

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do you mean expression1 + expression2 = expression3 or expression1 = expression2 = expression3?

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I asked an "or" question

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that is also valid

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an expression can contain other expressions, just like in programming

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its ""recursive""

regal thicket
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imagine you embed this expression inside another expression, like so:

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outer_expression = inner_expression + 4

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so 5 = 5 turns into 5 + 4 = 5 + 4

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you can see that with the same inner_expression (5), you should have an outer_expression with the same value (9)

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this is the general rule

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you should always be able to do this

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no

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I didnt say that

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its an assignment

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well if its an assignment it should be like expression = expression + 4

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so applying the same assignment to both sides should retain that both sides are equal

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so the equation 5 = 5 is just as true as 5 + 4 = 5 + 4

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yes

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now being this general lets you do more general ideas

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such as expression = expression * expression + expression

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so that 5 = 5 is just as true as 5 * 5 + 5 = 5 * 5 + 5

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or in general, L = R is just as true as L * L + L = R * R + R

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by beginning with L = R, you assume that they evaluate to the same

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so going from L = R to L * L + L = R * R + R is to say
"if L = R, then L * L + L = R * R + R"

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does that make sense

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now this general idea can be used to convert 2x + 5 = 10 into x = #

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the idea is that you layer on moves that "unwrap" the left side

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for example, I can expression - 5 both sides

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2x + 5 = 10
2x + 5 - 5 = 10 - 5
2x = 5

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and then I can expression / 2 both sides

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2x = 5
2x / 2 = 5 / 2
x = 5 / 2

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this is what I mean by "doing the same thing to both sides"

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you are embedding both expressions into an outer expression
a.k.a. using the same assignment to both expressions
a.k.a. calculating the same moves to both expressions

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and since the expressions are equal before, they should be equal after

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this lets you choose steps that can simplify an expression to be valid and is the intended way to simplify 2x + 5 = 10 to be x = 5/2

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are you with me so far

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yes

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no one ever sees it coming

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or they think they do but they dont solidify it

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this is why they teach you algebra in school, theres no other way you'd know about this

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now lets get to some basics of algebra

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its easier to think about if you be specific with some particular valid moves

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first, you can

  • add the same expression to both sides
  • subtract the same expression from both sides
  • multiply both sides by the same expression
  • divide both sides by the same expression
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second, when you have an expression like
2x + 5 = 10, theres only one x to solve for
this lets you go through a method called "isolating x"
2x + 5 is doing "x times 2 plus 5"

you know that - 5ing both sides is valid
2x + 5 = 10
2x = 5
its chosen because - 5 is the reverse of + 5

you also know that / 2ing both sides is valid
x * 2 = 5
x = 5 / 2
its chosen because / 2 is the reverse of * 2

you can see here Im explicitly choosing an expression that reverses the outside, it uses the general rule to unwrap x

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(2x is a shorthand for 2 * x, and * is a symbol for multiplication because using x would be confusing with the variable x)

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typo, it uses the general rule to unwrap the expression until its only x

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now to isolate x, you need to know the reverse of what's on the outside

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the reverse of adding 5 is subtracting 5 for instance

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and the reverse of multiplying by 2 is dividing by 2

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now we can get to solving f(n) = t

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how much do you know about inverse functions?

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for exponents, there are two different ways to apply a change

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the first way is expression^#

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the second way is #^expression

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theyre both different, they each have an inverse

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the inverse of expression^2 is sqrt(expression), also written as √(expression)

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the inverse of 2^expression is log base 2(expression)

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now previously, you had 2^n = t

regal thicket
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"solving for n" means "get the equation to look like n = (an expression not based on n)"

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so in general,

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,align2^n&=t
\{\color{yellow}\log_2({\color{black}2^n})}&={\color{yellow}\log_2({\color{black}t})}
\n&=\log_2(t)

brazen onyxBOT
regal thicket
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you can see that "logging base 2 both sides" is a valid move

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and its done as part of the method to isolate n

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which is how you solve for n

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youre supposed to learn this in school over a couple of years

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you know most people are capable of remembering like thousands of words

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thats a lot of letters you know

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but its not flabbergasting to

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youre now seeing the scale of how much you now need to know

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if you want to learn algebra, you can go on khan academy and begin from the algebra part

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remember to prioritize knowing whats happening beyond knowing how to solve the problesms

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yes

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this entire problem is algebra

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its 8th grade material

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none of this is university material

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I hope youre aware of that

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you've barely begun

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you cant say that yet

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no Im in college

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you need to catch up, you need to go on khan academy as soon as you're done

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those problems are easier

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and theres a lot of them

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for now though, lets consider how to solve for n

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as Ive said earlier, you should begin by solving for n in f(n) = t for the rows in advance before entering in the times and calculating them

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entering in the times is called "plugging in" or "substituting"

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so "plugging in 1.000.000 for t" or "substituting t = 1.000.000" means to do f(n) = 1.000.000

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you replace all ts with 1.000.000

regal thicket
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do you understand both of these

regal thicket
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back

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you can ignore that for now

regal thicket
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khan academy has some videos on this

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for now though, you have these functions

image.png
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yea I figured, thats for later on then

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in a way you have part of the idea down when you saw 2x + 5 = 10 and you "thought backwards" to get x = 2.5

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and similarly when you had 2^n = t and you got the correct values by doing n = log2(t)

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so ignoring those,

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you need to solve these equations for n:

  • lg₂(n) = t
  • √n = t
  • n = t (already solved)
  • n lg₂(n) = t (not solvable)
  • n² = t
  • n³ = t
  • 2ⁿ = t (you solved this, n = lg₂(t) is correct)
  • n! (not solvable)
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remember that solving for n means you need to do the right things on both sides to change the equation to **n = **(something)

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I gave you the rhyme and also the reason

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well really all I gave you was the "rhyme" part, the general idea of what you need to to do

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unfortunately none of that is present here

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that list I gave earlier is just an example list

regal thicket
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unfortunately none of that is present here
the list I gave earlier is just an example list
but your problems are harder than that

regal thicket
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its one angle

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no

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one specific angle

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idk how you havent seen it yet so dont mention angles for now

regal thicket
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I gave you a more general version of the idea since just doing + - * / wont work

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youre allowed to do anything to both sides

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no idea what youre talking about

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you didnt remember what inverse functions were 🤷‍♂️ so dont mention angles for now

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if you let me finish,

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I never said trig you absolute wallnut

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youre doing a whole lot of not letting me finish right now

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disgusting

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so earlier you were able to solve 2^n = t

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do you remember how you were able to solve it

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thats pretty correct

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however it isnt based on algebra

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remember that algebra is not only about letters but also about that general idea of doing the same thing to both sides

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so taking a look at 2^n = t, what can you do to both sides to turn the 2^n into an n

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its almost the only reason you can get anywhere with algebra

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do you know what log2(2^n) simplifies to

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do you know what logarithms are

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yea

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so whats log_5(5^2837) then

regal thicket
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yea

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so in the equation 2^n = t,

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can you write out what the equation looks like after you log_2() both sides

regal thicket
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what you essentially did was do something to one side but not the other

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so thats not correct

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I asked you to log_2 both sides

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thats both sides of 2^n = t

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you were supposed to do

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2^n = t
lg(2^n) = lg(t)
n = lg(t)

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yea thats the one

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so now you see the algebra way of doing this

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you began with 2^n = t

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you know that since both sides are equal, they should remain equal after you lg() both sides

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so if you lg() both sides, you get lg(2^n) = lg(t)

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or that n = lg(t)

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showing that 2^n = t must mean that n = lg(t)

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you realize how easy this sounds if I replace the 2^ and lg with just +3 and -3:

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you began with n + 3 = t
you know that since both sides are equal, they should remain equal after you - 3 both sides
so if you - 3 both sides, you get n + 3 - 3 = t - 3
or that n = t - 3
showing that n + 3 = t must mean that n = t - 3

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they both use the same idea

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arent you in college bro

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I said I was in college, Im a freshman

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but you dont need to be a college freshman to do this problem

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bruh

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yea

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what country are you if you dont know that word

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I have a feeling American english isnt your first language

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I dont know any dutch or german so unfortunately itll have to stay that way

regal thicket
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you have 2^n = t, and you choose to lg() both sides

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that choice is because lg() reverses 2^

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similar to how - 3 reverses + 3

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so now, you need to pick the reverse for each of those equations

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and youll be able to solve all of them

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(except for the unsolvable n lg n and n!)

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I want you to do the algebra first because you can use a braindead approach for factorial

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no

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remember earlier what the "braindead" approach turned out to be?

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youre not going to reverse the factorial with algebra

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bruh

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you reacted pretty strongly to it

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told me it was 4AM

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told me it didnt stick

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you reacted so strongly that when you woke up the next day and told me something, you used part of the approach

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you said f(n) = n instead of just t

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I corrected you

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remember?

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goddamn

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colander moment

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just pretend we've just begun

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lets try to solve lg(n) = t

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already numbering how many hints youll need? confident

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as an algebra problem, you are allowed to do the same action to both sides

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earlier you were able to solve 2^n = t by doing this:

  1. 2^n = t
  2. lg(2^n) = lg(t)
  3. n = lg(t)
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dont think that way

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no numbers

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trust me

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what can you do to both sides
to end up with n =

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the other side doesnt have a log2, so no

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you cant just put a log2 in there either, because that counts as you changing only one side of the equation

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what would reverse lg()?

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no

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so what can you do to both sides

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theres a lg() on the left side
if you can reverse that lg(), there wont be a lg() anymore

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and itll just be n = instead of lg(n) =

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you know that the reverse of lg() is 2^

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what

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do you know what 2^ means

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doesnt it just mean to 2^ something

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lg(n) = t
**2^**lg(n) = **2^**t
n = 2^t

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I refuse to believe you are this incapable of thinking for yourself

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you just said that 2^ and lg() are opposites

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they are reverses

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so that means doing lg() then 2^ should combine to do nothing

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tf does that mean man

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I dont want to see you solve a jigsaw puzzle man

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I dont even know how much youre getting from this

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unless you get whats happening here, you really wont know how this question works

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you need to know your algebra

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where is this going

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its nighttime

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bro do you know your timezones

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man look

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do you want to know whats happening in this problem or not

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can you tell me whats happening in the problem then

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already we are not on the right track

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take a hint

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do you want to know whats happening in the entire problem or not

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the general idea

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if the general idea could be said in a single messge, you'd be done by now

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its a yes or no question

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then can you fill out the table

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then you dont know enough

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how about this

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you dont sound like youre trying for shit right now

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Im trying hard to get you to do this and youre just "ehhhhhhhh"

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melting into the ground by this point

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Ill make this easy for you

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5:21AM man

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your Zs are on the line

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hurry it up

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I didnt say to give it up

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it sounds like you dont care about this problem anymore

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the problem is essentially useless, youll learn nothing from going through the work to complete it

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let me make this easy for you

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  • lg(n) = t means that n = 2^t
  • √n = t means that n = t²
  • n = t means that n = t
  • n² = t means that n = √t
  • n³ = t means that n = ∛t
  • 2^n = t means that n = lg(t)
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the f(n) = n rows has numbers that match with how many microseconds each t is

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put those in for t and calculate those 6 rows

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youve already calculated the 2^n row correctly

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just finish the numbers on this problem and learn your algebra properly

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once youre finished with filling those out, you should have seven rows completed
send a screenshot of the numbers you have, dont just type several dozen numbers as text in the chat

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one more thing, you were having the right idea with the factorials, but I was thinking if I got you to do the algebra first, you would have enough brainpower afterwards
if you can, also fill out the factorial row

regal thicket
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sure but you havent even done the table yet

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using a formula is pre-algebra buddy

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you need me to put in the numbers for you too?

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if you dont know whats happening in the problem enough to forget how you did your own work, you need to go down to pre-algebra on khan academy

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somewhere in there you can learn how formulas work and how to use them

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your problem is still unsolved

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do you want to see it solved?

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no

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no

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this is very wrong

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tell me

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what use are tricks if you dont even know whats happening

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and how are you going to know whats happening

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if you dont have any numerical data?

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theres nothing stopping you from just using the formulas and entering in the numbers in the table

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this will tell you the behavior of each function

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you do not need to wait for the intuition to come to you

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the purpose of the problem is to showcase the behavior of each function

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by literally having you enter in numbers to simulate how they behave

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what do you think tables are used for

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gatekeeping because you think you know better on a problem is a fundamental misconception, it gets everyone but you should know not to do it

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you havent even seen the completed table yet

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how about you see the completed table then say that?

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rs this isnt a complicated affair

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you just put in 2^(those times) in your calculator

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and thats how you fill in the first row

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can you do that for me?

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show me which calculator youre using

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rs you prick

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Im not going to even get into what youve just done here

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why did you wait until now to reveal to me that the numbers were too big

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use scientific notation

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and dont copy-paste all of the digits into the chat

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I know youre going to do that

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copy-paste it into the table

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all of them

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rs dont you have a google doc to just copy-paste in the numbers

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I find it hard to believe youre given a document with this table on it and you have to copy down the table on paper

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then you dont need to write a table down on paper

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get an online one

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use google docs

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dont tell me you dont know how to get one of those

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google it

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going to say this is how you havent gone far in math

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look at what you chose to say

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didnt even want to hear me out, youre not the arbiter of tabling over here

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jk really but math is all about doing things that were impossible before

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I only get to be the arbiter if you listen

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you dont get to be the arbiter because you never listen

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thats the only difference

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wait what the fuck

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no its online you make a google account

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you dont "download google docs"

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are you still on your phone?

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do you have a computer?

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youre not going to type 56 separate boxes with the ios keyboard are you?

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you tell me whether you want a shortcut or not because at this rate, the next 20 minutes are going to be you entering in numbers and typing them by hand

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youre not going to gain anything if you have to do that, its too tedious to see the picture while youre typing it

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you might as well just see the completed table that I can make

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@teal shale noticing youre not as adamant about finding shortcuts anymore

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you need to get the numbers down before you try to find tricks with anything

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you never needed to mess around with all of this
just having the numbers down in a table of any sort is enough

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(or you can just ask me to make it for you)

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alr

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lol thats gonna pale compared to how many numbers you gotta type in

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nah you dont

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there are shortcuts

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but I aint telling you them until after you complete it

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first off

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no

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second off

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you can just type that into the table

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the number is too big to really get a sense for it

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when you put it into wolfram alpha it sends you a 10^10^something

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so just 2^1.000.000 is good enough

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yea just keep leaving those in there

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reasonable

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the rows below arent going to go this hard

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so this top row will be an exception

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it really is

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you can see there that you can fit 2^ as many operations

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part of the shortcut is in what that really means

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its not just "getting a big number out of it," there really are patterns

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thats a basic idea but its not going to explain the difference in scale between 2^1.000.000 and 2^60.000.000

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thats not correct

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you dont do that for exponents

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"multiplying by 59.999.999 many 2s" is a beginning way to consider it

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the idea behind 2^ is "every time you go up by one, you double"

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so going from 1 second to 60 means that you have to double it 60 times

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and so logarithmic ones are really good

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thats the pattern involved

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thats part of the idea

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a lot of those kinds of examples are just to contain the growth of 2^ and the similar

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the general "y = a^x" are called "exponentials"

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they grow really fast, and they grow faster than the other rows in this sheet

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(keep in mind by rows I mean the formulas youre using to fill in the numbers, not the expressions on the left column)

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alr youre not going to see that in the next row, n = t²

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nah the inverse factorial and the inverse n lg n dont appear in the table, for those Ill just tell you how you can do those when all the other rows are filled in

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yea it is

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yea Im doing it too

regal thicket
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@teal shale finished my table

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how much do you have

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you can take as much time as you need

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I gtg for now

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if you want the complete table I can send you it

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alr

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second time's the charm

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lunch

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alr cya