Questions tagged [number-comparison]

Tag for problems about comparing explicitly given numbers, often by hand calculation only.

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How do you find by what percent a number is more or less than another number. [closed]

Lets say by what percent 50 is less than 100? (100-50/100)*100 (100-50/50)*100 over here which of the following options are right 1 or ...
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1 vote
0 answers
75 views

Is this number, N, greater than Graham’s Number?

So, using Knuth’s up-arrow notation, if 3 ↑ 3 = 3^3 and 3 ↑ ↑ 3 = 3 ↑ (3 ↑ 3) = 3 ↑ 27 = 3^27 Then consider number N defined by Knuth's up-arrow notation: $$N = googolplex\uparrow^{googolplex} ...
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48 views

How to evaluate the difference/distance between 2 values positive and negative on a scale

Problem 1 : The input is 2 values, that can be in a scale between [-3.89, 10.66] And i need to compare the difference between an oldValue (A) and a newValue (B). So i want to create a variable that ...
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  • 1
1 vote
1 answer
92 views

Which one is greater: $\log_{13}160$ or $\log_{17}291$?

Which one is greater: $\log_{13}160$ or $\log_{17}291$? Comparing logarithms with equal bases is fairly easy. Here they aren't equal, though. In similar problems I have seen that we can compare each ...
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  • 1,903
5 votes
4 answers
126 views

Determine the greatest of the numbers $\sqrt2,\sqrt[3]3,\sqrt[4]4,\sqrt[5]5,\sqrt[6]6$

Determine the greatest of the numbers $$\sqrt2,\sqrt[3]3,\sqrt[4]4,\sqrt[5]5,\sqrt[6]6$$ The least common multiple of $2,3,4,5$ and $6$ is $LCM(2,3,4,5,6)=60$, so $$\sqrt2=\sqrt[60]{2^{30}}\\\sqrt[3]3=...
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  • 1,903
4 votes
5 answers
240 views

Which of the two quantities $\sin 28^{\circ}$ and $\tan 21^{\circ}$ is bigger .

I have been asked that which of the two quantities $\sin 28^{\circ}$ and $\tan 21^{\circ}$ is bigger without resorting to calculator. My Attempt: I tried taking $f(x)$ to be $f(x)=\sin 4x-\tan 3x$ $f'(...
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Formula to calculate the chance of one number being higher then another when taken from two different ranges

I'm working on a program and I need to calculate the chance of a number being bigger or equal to a second number. The numbers are selected from a range of +-10% an original number for each. For ...
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16 views

Calculate similarity in time and duration of two people?

I have an example problem Person A starts playing soccer at 01:00 and plays for 40mins Person B starts playing soccer at 01:20 and plays for 80mins Person C starts playing soccer at 05:00 and plays ...
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1 vote
1 answer
43 views

Pairwise comparing two sequences (notation)

Lets assume we have to sequences with equal number of variables $A = \{a_1, a_2,..., a_n\}$ and $B = \{b_1, b_2,..., b_n\}$. I need to compare each value pairwise: $a_1$ to $b_1$, $\cdots$, $a_n$ to $...
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-7 votes
2 answers
120 views

How to show $2\sin 1 < \log_3 (7)$? [closed]

How to show $2\sin 1 < \log_3 (7)$? $1$ is in radians - tried numerically and found it's true. But is there another way to prove it, either geometrically or via algebra? Can't seem to find a ...
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0 answers
24 views

Apples to Apples comparison?

I calculate a scores of different sets of data like this: $$scores_j = avg(x_{j,i} * a_i)$$ where $a_i$ are constant and $x_{j,i}$ change between data sets. Let say I have as a result: $$scores_1 = (...
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  • 139
0 votes
0 answers
9 views

Statistic that captures efficiency and attempts

I'm looking for a statistic to compare entities based on 1) their efficiency and 2) their number of attempts. Concretely, I want to compare fantasy basketball players so that I can determine whether ...
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21 votes
7 answers
990 views

Proving that $3^{(3^4)}>4^{(4^3)}$ without a calculator

Is there a slick elementary way of proving that $3^{(3^4)}>4^{(4^3)}$ without using a calculator? Here is what I was thinking: $$4^4=256>243=3^5,$$ hence $$4^{4^3}=4^{64}=(4^4)^{16}=(3^5)^{16}\...
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  • 9,602
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1 answer
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What is the meaning of $1$ in a relative error?

If we measure a length and is measured as $12.5$ meters long, accurate to $0.1$ of a meter this means the absolute error is $0.05$m. The relative error is: $\frac{0.05}{12.5} = 0.004$. This means that ...
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  • 1,507
2 votes
1 answer
216 views

Correct comparison of real number for n digits precision (absolute vs relative difference)

To compare if $2$ real numbers are equal, we define a desirable precision e.g. $n$ digits and then check if the following condition holds: $-\frac{1}{10^n} \lt x - y \lt \frac{1}{10^n}$ Now I was ...
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  • 1,507
5 votes
7 answers
393 views

Comparing $2^{317}$ and $81^{50}$ by hand

How to compare these two numbers without calculator: $2^{317}$ and $81^{50}$ (Pen & paper test) I thought about using logarithms and doing Taylor approximation, but these numbers are close to one ...
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  • 321
1 vote
2 answers
30 views

Invariance of number properties under different bases

Are in number theory always numbers with basis 10 considered? I‘m asking which role the basis plays in number theoretic properties or notions like prime numbers for examples. For example: the number 3 ...
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  • 540
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1 answer
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How to solve this problem on comparison and use the value of average? [closed]

Question: " 6 articles A,B,C,D,E and F are sold at a different price. B is costlier than only 2 items and C is not one of them. D is cheaper than A and is costlier than F, which is costlier than ...
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3 votes
1 answer
111 views

Numbers with 1000 digits which are the sum of the 1000th powers of their digits

In the book$^1$ that I am reading, the author dubs an $n$-digit positive number a Smallbrain number if it is equal to the sum of the $n$th powers of its digits, with $371 = 3^3 + 7^3 + 1^3$ given as ...
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1 vote
6 answers
165 views

Comparing $\sqrt{5} + \sqrt{6} + \sqrt{11}$ and $8$ without calculating the values [closed]

I want to compare $\sqrt{5} + \sqrt{6} + \sqrt{11}$ and $8$ without calculating the actual value of square roots. I tried to apply square on both side but it still carries the root terms. Any trick I ...
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  • 321
2 votes
1 answer
108 views

What does <> mean?

What do the less-than and greater-than symbols right next to each other mean? Does it mean either less than or greater than? In other words, not equal? I am trying to understand a book that says this: ...
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  • 21
10 votes
5 answers
366 views

$2^\sqrt{10}$ vs $3^2$

Is there a neat way to show that $2^\sqrt{10} < 3^2$? I have tried raising to larger powers, like $(2^\sqrt{10})^{100}$ vs $3^{200}$ but the problem is the two functions $2^{x\sqrt{10}}$ and $3^{2x}...
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  • 325
3 votes
6 answers
143 views

Prove $\sqrt[3]{4} - \sqrt[3]{3} < \sqrt[3]{3} - \sqrt[3]{2}$

I am a student in Germany, and I prepare for Math Olympiad by solving math problems. I have been solving the following question, which took about 4 hours to solve. Prove the following inequality ...
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4 votes
3 answers
189 views

Comparing numbers of the form $c+\sqrt{b}$ (eg, $3+3\sqrt{3}$ and $4+2\sqrt{5}$) without a calculator

It is easy to compare to numbers of the form $a\sqrt{b}$, simply by comparing their squares, for example $3\sqrt{3}$ and $2\sqrt{5}$. But what if we have $a=3+3\sqrt{3}$ and $b=4+2\sqrt{5}$ for ...
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0 votes
3 answers
67 views

How to show properly $ 10n^4 + 3n^3 -n \in O(n^4)$?

Given: $$ 10n^4 + 3n^3 -n \in O(n^4)$$ So i do understand Big O and what it does. I am however not quite if my approach for the solution is correct, so anyone who could help me out or show me an ...
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  • 191
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0 answers
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Comparison of two constants of a general form

In a lecture, our sir gave us a question comparing 2 constant values that were all of the form ​​​​​​​​$α^β$ and $β^α$. Example ($π^e$, $e^π$) and ($2008^{2009}$, $2009^{2008}$). He used variable ...
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  • 101
3 votes
3 answers
174 views

Problem about convergent series

I'm trying to prove this series converges by using some sort of comparison test. $$\sum_{n=1}^{\infty}\frac{1}{n^{0.51}}-\sin\left(\frac{1}{n^{0.51}}\right)$$ I know by $\sin n\le n$ that the series ...
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0 votes
1 answer
67 views

Can we give a precise definition to say "$b$ is $m$- times greater than $a$?"

Let $a,b\in R$ and $a<b.$ Can we give a precise definition to say "$b$ is $m$- times greater than $a$?" in general. For the positive $0<a<b$ it is clear. "We say that $b$ is $m$...
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  • 23
0 votes
3 answers
58 views

Compare $\sqrt{6}-\sqrt{3}$ and $\sqrt{3}-\sqrt{2}$

Compare the numbers $a=\sqrt{5-2\sqrt{6}},b=\sqrt{6}-\sqrt{3}$ and $c=\sqrt{3}-\sqrt{2}$. We have $$a=\sqrt{5-2\sqrt{2}\sqrt{3}}=\sqrt{\left(\sqrt{2}\right)^2-2\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^2}...
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  • 1,903
2 votes
3 answers
69 views

Represent $x_1=\frac{1-\sqrt{37}}{3}$ and $x_2=\frac{1+\sqrt{37}}{3}$ on a number line

In which case the numbers $x_1=\dfrac{1-\sqrt{37}}{3}$ and $x_2=\dfrac{1+\sqrt{37}}{3}$ are correctly represented on the number line? I did the calculations with $\sqrt{36}=6$ and then we have $x_1\...
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  • 1,903
3 votes
6 answers
81 views

In which of the intervals is $\sqrt{12}$

In which of the intervals is $\sqrt{12}:$ a) $(2.5;3);$ b) $(3;3.5);$ c) $(3.5;4);$ d) $(4;4.5)$? We can use a calculator and find that $\sqrt{12}\approx3.46$ so the correct answer is actually b. How ...
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  • 1,903
4 votes
2 answers
97 views

Comparing powers of 13 and 19

There is a beginner olympiad question asking us to compare $13^{99}$ and $19^{93}$. This is easy enough, but I toyed around with it and got (via a calculator) that $13^{99}$ lies between $19^{86}$ and ...
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2 votes
2 answers
27 views

Finding a faulty value in a dataset

I have prices from wholesalers for a product. But sometimes the wholesalers doing mistakes and list the product to a unrealistic price. I want to filter the unrealistic prices out. This is an example: ...
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35 votes
7 answers
2k views

Prove $7^{71}>75^{32}$

My math teacher left two questions last week, prove (1) $6^9>10^7$ and (2) $7^{71}>75^{32}.$ I did the first question: \begin{align}\frac{6^9}{10^7}&=\frac{4}{5}\times\frac{27^3}{25^3}\\&...
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  • 15.5k
1 vote
1 answer
22 views

Minimal number of comparisons to determine larger set

There are $2n+1$ balls in a row, on each one printed either $1$ or $0$, but we can not see what is written - we can only see the position in which they are placed. I need to take out a ball that ...
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  • 75
0 votes
5 answers
87 views

Prove that $e^{\frac{x+y}{2}} \le \frac{e^x + e^y}{2}$

To prove: $e^{\frac{x+y}{2}} \le \dfrac{e^x + e^y}{2}$ We observe: $e^{\frac{x+y}{2}} \le \dfrac{e^x + e^y}{2} $ $\Leftrightarrow \sum_{n=0}^\infty \dfrac{(\dfrac{x+y}{2})^n}{n!} \le 1/2\sum_{n=0}^\...
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3 votes
4 answers
94 views

Number comparison: $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$

Is it possible to prove that $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$ without using a calculator or logarithms (middle school math only recommended)? My idea for the first one was to use ...
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  • 588
1 vote
1 answer
65 views

Polygon / Any shape invariant for comparison or fiting

For my personal curiosity, I was wondering which would be simplest algorithmic way to compare two shapes to say whether they are the same or not. After some researches, I found out that there are many ...
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  • 23
0 votes
1 answer
25 views

How do I sort a number of surd expressions?

Suppose I have the a number of expressions, some of which resolve to rational number and others to irrational numbers. $\sqrt{56}$, $7.5$, $5 + \sqrt{6}$, $10-\sqrt{6}$, $11-\sqrt{12}$ and $\sqrt{12} +...
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5 votes
2 answers
130 views

Comparing power towers of $2$s and $3s$

Let $x=[x_1,x_2,...,x_n]$ be a finite list of positive real numbers, and define $\tau x$ as the power tower formed by these numbers. The function $\tau$ can be recursively defined by the following two ...
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3 votes
5 answers
245 views

Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\cdots+\frac{1}{3001}<\frac{4}{3}$ [duplicate]

Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\cdots+\frac{1}{3001}<\frac{4}{3}$ Using AM- HM inequality, $\left(\sum_{k=1001}^{3001} k\right)\left(\sum_{k=1001}^{3001} \frac{1}{k} \right) \geq(...
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  • 171
5 votes
2 answers
89 views

Find the greatest integer less than $\frac{1}{\sin^2(\sin(1))}$ without calculator.

Find the greatest integer less than $$\frac{1}{\sin^2(\sin(1))}$$ This was on one of my tests. All angles in radians. Here's my work: $$0<1<\frac{\pi}{3}<\frac{\pi}{2}$$ Since $\sin(x)$ is ...
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2 votes
1 answer
60 views

Inqualitiy with exponent

I was trying to prove this inequality $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq 2 \exp(\frac{1}{3})$ My attempt was using AM-GM mean $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq2 \exp(\frac{1}{2\pi e})$...
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0 votes
0 answers
28 views

What are good graphs to compare two sets of data of network delays?

I have two sets of numbers, each set contains 100 numbers, which are delays that I have measured from a network. I want to visualize the comparison between these two sets, instead of just provide two &...
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2 votes
3 answers
161 views

What Is Bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$

Hello every what is bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$? I tried to square up and I got $100^{200}$ or $99^{99} \cdot 101^{101}$ and I don't have an idea how to continue.
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  • 352
11 votes
6 answers
422 views

Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$

Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$ My try: we have $$\frac{13}{32}=\frac{2^2+3^2}{2^5}=\frac{1}{8}\left(1+(1.5)^2)\right)$$ Let $x=1.5$ Now consider the function $$...
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0 votes
0 answers
57 views

Comparing the trigonometric functions of any two angles in the same quarter of the unit circle

I'd like to talk about the comparison of the trigonometric functions of angles. For example, both angles in degrees, different from one another, are known and guaranteed to be in the same quarter of ...
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0 votes
1 answer
35 views

how to quantify or compare the proportion of 1: 2: 3

Problem description:I'm working on a machine learning project, and one of the features is represented by the proportion of three levels' sample numbers. When I was doing preprocessing normalization, I ...
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1 vote
0 answers
25 views

Is is appropriate to mix inequalities and equalities in a mathematical statement? [duplicate]

This question is extremely basic but I can't find any information online and it has never been mentioned at university: Is the use of equal signs within comparisons allowed/mathematically correct? ...
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1 vote
1 answer
94 views

Prove that $\frac{1}{2020} < \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times ... \times \frac{2019}{2020} < \frac{1}{44}$

Prove that $\frac{1}{2020} < \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times ... \times \frac{2019}{2020} < \frac{1}{44}$ I have proven the first half of the inequality, which is the ...
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