Questions tagged [number-comparison]

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

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57 views

Can we have an approximately unequal sign? [closed]

I may be wrong, but my intuition is that we actually don’t need such sign, because: 1/ Human nature of problem solving & question answering favored equilibria aspects of Mathematics more than its ...
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0answers
38 views

What makes one binary vector bigger than other? [closed]

Can we compare binary vectors ? What makes one binary vector bigger than other ? Do they form some form of lattice ! by which it can be compared. What about integer vectors ? I'm using large sparse ...
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1answer
20 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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5answers
80 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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3answers
63 views

Comparing powers of $2$ and $5$

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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0answers
7 views

Compare (x, y) trajectory plots with uneven samples.

I am not sure if this is the right *Exchange to be asking this, so correct me if I am wrong. Here's my concern: I have two plots, i.e. two sets of (x,y) coordinates: EstimateSet and GroundTruth. The ...
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1answer
24 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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1answer
20 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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2answers
76 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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5answers
157 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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2answers
82 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 ...
2
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1answer
55 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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0answers
17 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 &...
2
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3answers
142 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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6answers
340 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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0answers
28 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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1answer
33 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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0answers
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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0answers
39 views

Upper bound of class kappa infinity function

From the definition in Wiki, a continuous function $f: [0,a)\rightarrow[0,\infty$) is said to belong to the class kappa infinity if 1. it is strictly increasing; 2. it is s.t. $f(0) = 0$; 3. it is s....
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1answer
78 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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1answer
345 views

Is there a system of mathematics where $4>2$ is false?

A recent question on propositional logic posted on Philosophy Stack Exchange yielded an answer which states, in part, that, The fact that $4$ is greater than $2$ is not a "logical fact" but and [...
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0answers
19 views

What Kind Of Mathematics Is Comparing Numbers?

We can compare 2 numbers with >, <, >=, <=, or = . And get a "true" or "false" result. I believe this is some kind of mathematics. What kind is it?
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1answer
47 views

Comparing Binomial Probability to Poisson Random Variable Probability

A text file contains 6000 characters. When the file is sent by e-mail from one machine to another, each character (independently of all other characters) has probability 0.001 of being corrupted. Use ...
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3answers
392 views

Showing that $\frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47} < 1$ without brute force calculation

Consider the sum $$S = \frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47}$$. A brute-force calculation (okay, I just used Wolfram Alpha) shows that $$ S = \frac{442017301628992345493}{...
9
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4answers
826 views

Is there an easy way to see that ${1\over5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12} > 1$?

The sum $$\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12}$$ is just a bit larger than $1$. Is there some clever way to show this other ...
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3answers
77 views

Proving that $x^{2x}<(2x-1)^{2x-1}$

I came across a problem that asks me to prove that $\log_2(3) > \log_3(5)$ and I ended up needing to prove that : $$ x^{(2x)} < (2x-1)^{(2x-1)}$$ for $x > 1$. I tried to solve it but i ...
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0answers
35 views

Negative (minus sign) in the exponent

After perusing a GRE quantitative comparison question, I noticed the following statement: Given $0 < a < b < 1$, the product of $a$ and $b$ is less than $b$. For this practice question, the ...
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1answer
34 views

comparing averages versus comparing the averages of percentage

Let’s say there are three rooms with 40 identical boxes in each room. Each box contains some number of cards inside. The number of cards in each box is different. It may be as many as 60 cards in one ...
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1answer
241 views

What is bigger: $\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$?

As the title implies: what is bigger $\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$. Specifically I am interested in working this out without actually calculating the values. So far I have tried ...
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1answer
31 views

Subtract two normal cumulative distribution functions rather than plotting a normal one to compare a binomial with a normal variable?

In order to understand the Central Limit Theorem, I am comparing a $Binomial(n,p)$ variable with a large $n$ and a normal variable with mean $\mu p$ and a standard deviation $\sigma = \sqrt{np(1-p)}$. ...
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7answers
150 views

Which is bigger: a googolplex or $10^{100!}$

A googol is defined as $ 10^{100}$ Let x = $10^{100}$ A googolplex is defined as $10^{x}$ Which is bigger: a googolplex or $10^{100!}$ I only know that: $100! = 1×2×3×...×98×99×100$ $10^{100} = 10×...
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1answer
147 views

Compare power towers

Prove or disprove: $3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours. Taking logs is useless, and there seems to be no other way to compare. Thanks!
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3answers
131 views

Comparing $a^b$ and $b^a$ when $b < e < a$

If $0 < b < e < a$, how can I determine whether $a^b$ or $b^a$ is greater? I know this question has been asked before, but I want to solve this question by this method. It worked fine for ...
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0answers
23 views

Check two list of values statistically significantly different from each other

So I have two lists. l_2014=[22,27,58,62,75,122] l_2015=[23,25,60,62,74,124] Each number inside a list represents the average travel time of a road segment for a year. l_2014 for the year 2014, ...
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1answer
100 views

$2^{\frac 12}$, $3^{\frac 13}$,$6^{\frac 16}$ put it in an increasing order

$2^{\frac 12}$, $3^{\frac 13}$,$6^{\frac 16}$ put it in an increasing order. How to solve it without using any calculator. I was trying to understand the behavior of $f(x)=x^{\frac 1x}=e^{\frac {\log ...
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3answers
237 views

Comparing two close numbers

How to compare these two numbers without using a calculator ? $A=\left(\dfrac{11}{10}\right)^{\sqrt{5}}$ and $\;B=\left(\dfrac{12}{11}\right)^{\sqrt{6}}$. Thanks for your help ! Here is what I ...
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2answers
127 views

A 1968 AHSME problem with exponents

Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are: $\text{(A) } x,z,y\quad \text{(B) } x,y,z\quad \text{(C) } y,x,z\quad \text{(D) }...
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1answer
73 views

Simplifying the probability that a r.v. is greater than all others (independent) r.v.s

Let $X_1 .... X_K$ represent $K$ independent random variables. These var (of unknown distribution). I am trying to understand how to simplify the probability that $X_1 $ is the greatest of all. That ...
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1answer
131 views

Comparing a number with a set of numbers

Before start writting, I should declare that I'm not a mathematician at all. I know only the basics that taught us at school and hence what follows could be huge nonsense. In order one to compare two ...
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0answers
64 views

How to compare numbers in primoradic notation?

What is the simplest solution to comparing which of two numbers is greater than or less than the other when they are both in primoradic notation. The easiest way is to remove the common factors. But ...
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1answer
23 views

Comparing means with drastically different population size

I am trying to compare averages with different sample size. I want to find a way to 'normalize' this data. I have read that a weighted average might work, but assigning weights based on what?
3
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2answers
162 views

Compare different base powers-towers (of 'height' five)?

Let's say I want to compare two numbers that are stacked powers of different bases: $a^{b^{c^{d^e}}}$ compared to $f^{g^{h^{i^j}}}$ where all ten values will be integers in the range $[1,10]$. ...
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1answer
80 views

Limit comparison test wrong equality

I am given the following comparison test question on my homework. Based on just plugging in the numbers, the first condition in all three answers is wrong, since it should say the original function is ...
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1answer
62 views

Comparing complex numbers with angles

Is $2 + i > 1 - i$ true? First off, what does it even mean to be "bigger" or "smaller" as a number? If we say that $a>b$ means that $a$ is on the right of $b$ on the number line, could we also ...
3
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7answers
190 views

Show that $19^{31}>13^{33}$

How can i prove that $19^{31}>13^{33}$? What I tried $$\bigg(\frac{19}{13}\bigg)^2=\frac{361}{169}>2>1$$ then $19^{2}>13^{2}$ and $\displaystyle 19^{30}>13^{30}$ How do I show it. ...
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1answer
96 views

Proof lower bound of $\lceil{n/2}\rceil$ comparisons for finding smallest and second smallest element

For a college assignment we are supposed to prove that a lowerbound of $\lceil{n/2}\rceil$ comparisons holds for a selection algorithm that finds the smallest and second smallest element in an ...
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1answer
31 views

Find the least possible n for the power [closed]

While comparing the $123^{124}$ and $124^{123}$ (just for fun) I come up with an interesting question. Is it possible to find the least possible natural $n$, such that $a = b - k, k > 0 \quad and \...
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0answers
47 views

Comparing two percentages, that regard to data A(first percentage) that is subset of data B (second percentage)

I want to compare two percentages to see how much they differ one from another. For example: I have 1000 products - 100 of them are on promotion. (10% products on promotion) The total products are ...
13
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1answer
212 views

Is there any simple ways to compare $x^y$ and $y^x$ without a calculator?

There are plenty of discussion on MSE about how to compare $x^y$ and $y^x$. For $x,y>e$, it is sufficient to just compare $x$ and $y$ to reach a conclusion. But I wonder if there are some general ...
0
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1answer
170 views

How can I compare logarithm and the number? [duplicate]

There are two numbers $\log_3 4$ and $\sqrt[4]{2}$. How they can be compared without calculator?