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Questions tagged [number-comparison]

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

3
votes
2answers
67 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 $[1,10]$. ...
0
votes
0answers
12 views

Baseball: How are data sets with differing number of occurrences compared?

I have two data sets, each showing batting statistics for batters hitting against a specific pitcher. These tables show lifetime batting statistics of batters hitting against pitchers, therefore the ...
0
votes
1answer
19 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 ...
0
votes
1answer
44 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
votes
7answers
167 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. ...
0
votes
1answer
25 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 ...
0
votes
1answer
29 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 \...
0
votes
0answers
19 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 ...
12
votes
1answer
138 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
votes
1answer
48 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?
0
votes
1answer
13 views

Comparison of two self ratings

I have a small doubt regarding the comparison of two ratings. I have two candidates, I asked them to rate themselves out of 10 in 4 different subjects. Both have the same level of knowledge in both ...
2
votes
4answers
73 views

Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator

In my Pre-Calculus class we were given the following problem: Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
2
votes
2answers
202 views

Comparing logarithms with different bases

$\log_3 4$ and $\log_7 10$: which of these two logarithms is greater? I figured out that both are between $1$ and $2$, then between $1$ and $1.5$. And then $\log_34$ is greater than $1.25$, and $\...
1
vote
3answers
145 views

Which is greater, $\left(\frac{e}{2}\right)^\sqrt{3}$ or $(\sqrt{2})^{\pi/2}$? (no calculators)

From a math contest in 1985: Determine which of the following is greater: (no calculators) $$\left(\frac{e}{2}\right)^\sqrt{3} \, \hspace{3mm} \text{or} \hspace{3mm} \, (\sqrt{2})^{\pi/2}$$ Hints ...
0
votes
0answers
19 views

How to compare ratios of log values to obtain fold change differences?

I have some RNA expression data in the form of fold change. I have been trying to further analyze this data by making ratios of one value of another. These values have meaning in that a greater ratio ...
0
votes
0answers
30 views

Compare two expressions in term of one variable

I am trying to compare below two expressions: $$E_1 = \sum_{i}W^1_{i}\frac{e^{W^2_{i}x}-e^{-W^2_{i}x}}{e^{W^2_{i}x}+e^{-W^2_{i}x}} \quad E_2 = \sum_{i}W^1_{i}\frac{e^{W^3_{i}x}-e^{-W^3_{i}x}}{e^{W^3_{...
9
votes
7answers
651 views

Compare $\arcsin (1)$ and $\tan (1)$

Which one is greater: $\arcsin (1)$ or $\tan (1)$? How to find without using graph or calculator? I tried using $\sin(\tan1)\leq1$, but how do I eliminate the possibility of an equality without ...
1
vote
2answers
74 views

Comparing powers

Is there any way to check besides acctually calculating whether $n^x$ is $> = <$ then $m^y$? For example how to check wheter $440902^{532446} > = < 555151^{523163}$?
4
votes
4answers
399 views

Efficient way to compare roots (calculator doesn't count)

How can I know which one of the following numbers is the greatest: $$2^{1/2},3^{1/3},4^{1/4},5^{1/5},{6^{1/6}}$$ That can be also written as: $$\sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4},\sqrt[5]{5},\sqrt[6]{...
0
votes
1answer
57 views

Which one is bigger? Logarithms and trigonometric functions

There's the problem: given that $0<\alpha<\beta<\frac{\pi}{2}$ , what is bigger $$\frac{\ln(\cos\alpha)-\ln(\cos\beta)}{\beta-\alpha}$$ or $$\tan(\frac{\alpha+\beta}{2})$$ I tried to ...
0
votes
3answers
49 views

Which one is greater (A or B)?

Q. If $$A=(99)^{50} + (100)^{50}$$ and $$ B=(101)^{50}$$ then- $(a) A>B$ $(b) A<B$ $(c) A=B$ My attempt - I thought of using the binomial approximation to A and B , which gives - $$ A= (100)^...
4
votes
6answers
166 views

Which is bigger, $ \log_{1000} 1001$ or $\log_{999} 1000 $?

Which is bigger, $ \log_{1000} 1001$ or $\log_{999} 1000 $? I've tried using the identity of $\log_n x = \dfrac 1 {\log_x n} $, but didn't find a solution though. Any suggestions? or clues I can use?
-1
votes
5answers
74 views

If $x>y$, is $|x+y|$ or $|x-y|$ bigger, or neither? [closed]

Question: If $x>y$, is $|x+y|$ or $|x-y|$ bigger, or neither? I got this question wrong on a GRE practice test, and now I know the correct answer, but I am curious what your thought process is for ...
0
votes
0answers
30 views

Comparison of $\sum w_i \frac{A_i}{B_i}$ and $\frac{\sum w_iA_i}{\sum w_iB_i}$

Hi I got a task of discussing the comparison result of $U=\sum w_i \frac{A_i}{B_i}$ and $V=\frac{\sum w_iA_i}{\sum w_iB_i}$ under the condition of $w_i,B_i>0$ and $\sum w_i \leq 1$ . $i=1,2,3...,n$ ...
0
votes
0answers
11 views

Comparing (ordering) rate and sample size

Consider some rate, i.e. fail rate where f is the number of fails and n is the number of attempts. $$failrate = \frac{f}{n}$$ $$failrate = \frac{100}{100} = 1$$ $$failrate = \frac{99}{100} = .99$$ ...
0
votes
1answer
118 views

Is this :$\cos(\sin x) > \sin(\cos x)$ true and if it is how I can prove it? [duplicate]

I'm affraid that $\cos(\sin x) > \sin(\cos x)$ is not true for $x\in\mathbb{R}$ , but it's seems works for some known values as :$\frac \pi 4$ and $\cdots$ , I have used standard method to ...
2
votes
3answers
86 views

Comparing $\log_2 3$ to $\log_3 5$

Having been asked to compare $\log_2(3)$ and $\log_3(5)$, this is my proof: $$\log_2(3)>\log_3(5)$$ Then one uses the rule $\log_a(b)=\frac{1}{\log_b(a)}$, so $$\log_3(2)>\frac{1}{\log_3(5)}$$ ...
-2
votes
2answers
48 views

Which is greater in the following [closed]

Which of the following is greater than the other Log 3 to the base 2 or log 5 to the base 3 Well i have tried proving that log (2x-1) to the base x is a monotonically decreasing function but i am ...
3
votes
0answers
74 views

Notational Shorthand: $a$ is greater than both $b$ and $c$?

Suppose I have three real scalars $a$, $b$ and $c$. I know $a>b$ and $a>c$, but know nothing about the relationship between $b$ and $c$. Like this question I want to express this relationship ...
0
votes
2answers
65 views

Prove inequality using Positivity Axioms

Given an ordered field and $b,d>0$ and $\frac{a}{b}<\frac{c}{d}$, prove that $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$ using positivity axioms. Unfortunately, I'm stuck with this one. I ...
0
votes
5answers
62 views

Prove an inequality with sizable exponents

The Problem Show $2^{100} + 3^{100} \lt 4^{100}$ My Attempt $$2^{100} + 3^{100} \lt 4^{100}$$ Rewriting $4^{100}$ as $2^{200}$ $$\implies 3^{100} \lt 2^{200} - 2^{100}$$ $$\implies 3^{100} \lt 2^...
0
votes
1answer
49 views

GRE Question regarding two quantities

These are sample questions The question states this w > 1 ....Quantity A..............................Quantity B 7w - 4 .....
0
votes
0answers
28 views

Big O problem. Which one implies other?

Here $O$ means BIG-O. https://en.wikipedia.org/wiki/Big_O_notation If i am given with some data such that $$\|y_{\alpha}-x\| \leq O(\alpha^{\mu}) + O(\alpha^{\eta})$$ where $y$ is some variable ...
3
votes
5answers
288 views

Compare sum of radicals

I am stuck in a difficult question: Compare $18$ and $$ A=\sqrt{7}+\sqrt{11}+\sqrt{32}+\sqrt{40} $$ without using calculator. Thank you for all solution.
8
votes
5answers
200 views

How do I compare $\sqrt{2}$ and $\pi^{1/ \pi}$?

How do I compare $\sqrt{2}$ and $\pi^{1/ \pi}$? I applied in calculator, I got $\pi^{1/ \pi}=1.4396194958475907$ and $\sqrt{2}=1.414213562373095$. So, $\pi^{1/ \pi} > \sqrt{2}$. How to show ...
0
votes
1answer
53 views

Compare two expressions in term of two variables

Suppose that$a>0$ and $b>0$. Compare two numbers: $$ S_1=a^2b^2(a^2+b^2-2), \quad S_2=(a+b)(ab-1). $$ My attempt. If $a=b$, then we have $$ S_1-S_2=2a^4(a^2-1)-2a(a^2-1)=2a(a-1)^2(a+1)(a^2+a+1)\...
2
votes
3answers
269 views

Prove by contradiction (not using a calculator) that $\sqrt6 + \sqrt2 < \sqrt{15}$?

Prove by contradiction (not using a calculator) that $\sqrt6 + \sqrt2 < \sqrt{15}$. How do you approach such a problem? I need to admit that I'm completely new to proof writing and I have ...
4
votes
4answers
3k views

Prove $ \frac{1\cdot 3 \cdot 5 \cdot 7 \cdot \dots \cdot 2007}{2 \cdot 4 \cdot 6 \cdot \dots \cdot 2008}< \frac{1}{40} $

Prove: $$ \frac{1\cdot 3 \cdot 5 \cdot 7 \cdot \dots \cdot 2007}{2 \cdot 4 \cdot 6 \cdot \dots \cdot 2008}< \frac{1}{40} $$ i have tried to write $1/40$ as $(1/40^{1/2007})^{2007}$ and prove $...
-2
votes
1answer
51 views

if C equals three fractions, what is the value of C?

$$ c = \frac{30}{7}\cdot \frac{999}{300} \cdot \frac{699}{100}$$ What is the value of c ? Is it smaller than 10? between 100 and 1000 ? between 10 and 100? or smaller than 1000? How can i find that ...
0
votes
1answer
21 views

sorting sets of triangulation

I have two sets of triangles: $A = \{[(0,0),(1,0),(1,1)], [(1,0),(1,1),(2,1)], [(2,1),(1,0),(2,0)]\}$ and $B = \{[(1,0),(0,0),(1,1)], [(2,1),(1,0),(2,0)], [(2,1),(1,1),(1,0)]\}$ What algorithm can ...
0
votes
1answer
24 views

Confusion when comparing numbers with powers

I am confused about this: Compare $$7^{26} - 7^{25} = 7^{25}(6)$$ This is what I have calculated: \begin{align}&(1.0 \times 7^{26}) - (1.0 \times 7^{25}) \\&=(10\times 7^{25}) - (1.0 \...
9
votes
1answer
224 views

Prove that ${e\over {\pi}}\lt{\sqrt3\over{2}}$ without using a calculator.

I have been working on a known question for a long time (this is "Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator") during this time I realized the ${e\over {\pi}}\lt{\sqrt3\over{2}}$. ...
2
votes
3answers
554 views

How can I compare $\log_2 3$ and $\log_3 5$ without using a calculator [closed]

Compare $\log_2 3$ and $\log_3 5$ without using a calculator. I am not very good at math please explain it clearly
1
vote
5answers
96 views

Comparing big powers

Which of the following is the largest? A. $1^{200}$ B. $2^{400}$ C.$4^{80}$ D. $6^{300}$ E. $10^{250}$ I'm stuck trying to solve this. Obviously A and C are wrong ($4^{80}$ ...
3
votes
3answers
228 views

Proof: $\cot(7.5^\circ) = \sqrt2 +\sqrt3 + \sqrt 4 + \sqrt6$ [duplicate]

I am working on this problem, but I am not able to simplify the expression. So please help me. Thanks in advance
2
votes
3answers
73 views

Which of the following is bigger (logarithms)

I need to compare those two expressions and decide which is bigger. $2 \sqrt2$ or $\log_2(3)+\log_3(4) $. So I tried to simplify so the log expression so I know and so $$ \log_2(4) \times (\log_4(...
9
votes
5answers
2k views

$\sin(40^\circ)<\sqrt{\frac{3}7}$

Prove without using of calculator, that $\sin40^\circ<\sqrt{\frac{3}7}$. My attempt. Since $$\sin(40^\circ)=2\sin(20^\circ)\cos(20^\circ)<2\sin(20^\circ)$$ $$=2\sin(60^\circ-40^\circ)=\sqrt{3}...
3
votes
3answers
175 views

Prove $\frac{1}{\log_2\pi }+\frac{1}{\log_5\pi}>2$

Knowing that $\pi^2 < 10$. Prove that: $$\frac{1}{\log_2\pi}+\frac{1}{\log_5\pi}>2.$$ I have tried to do this the following way: $\log_2\pi+\log_5\pi>\frac{1}{2} \Leftrightarrow \log_2\pi+...
1
vote
6answers
966 views

Showing that $S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000} \gt 1$

If $$S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000}$$ then $$S\gt 1,$$ but how? I understood that there are $451$ pair of terms. So clubbed two terms together. $\frac{1}{100}+\frac{1}{...
8
votes
4answers
652 views

What is the meaning of the symbol $\not\geq$, and why would it be preferred to $<$?

I do not know if it is the right section to ask for it, but I wanted to ask some questions about math symbols that I often find on automation books. One of these is the following: $$\not\ge$$ This ...