Questions tagged [number-comparison]

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

161 questions
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17 answers

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas…
Alec
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4 answers

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}\tag1$$ Now, $$f'(x)=\frac{e\pi(1-\ln…
mathlove
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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:…
lsr314
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Without using a calculator, is $\sqrt[8]{8!}$ or $\sqrt[9]{9!}$ greater?

Which is greater between $$\sqrt[8]{8!}$$ and $$\sqrt[9]{9!}$$? I want to know if my proof is correct... \begin{align} \sqrt[8]{8!} &< \sqrt[9]{9!} \\ (8!)^{(1/8)} &< (9!)^{(1/9)} \\ (8!)^{(1/8)} - (9!)^{(1/9)} &< 0 \\ (8!)^{(9/72)} -…
29
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4 answers

Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ can someone suggest any other nice method? Thank you everyone.
math110
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Which is greater, $98^{99} $ or $ 99^{98}$?

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an elementary way to do this. The best I could find was on…
24
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7 answers

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq (n+1)^8\left(8\left(\frac{9}{10}\right)^8\right)\\ &\gt (n+1)^8…
JasonMond
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Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$

How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$ Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 +…
Parth Thakkar
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7 answers

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…
19
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9 answers

How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$

How can you determine which one of these numbers is bigger (without calculating): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
user2637293
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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 steps that can be used in any situations where $x
Ma Joad
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Showing $\sqrt{2}\sqrt{3} $ is greater or less than $ \sqrt{2} + \sqrt{3} $ algebraically

How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$? I know I can plug the values into any calculator and compare the digits, but that is not very satisfying. I've tried to solve…
12
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3 answers

Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?

Which is greater among $300 !$ and $\sqrt {300^{300}}$ ? The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.
Apurv
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How can I prove that $2^{\sqrt 7}$ is bigger than $5$?

This is the one of my tries: $5=2^{\log_{\ 2} 5}$. Then I should prove that: ${\sqrt 7} > {\log_2 5}$. So, can you help me end this proof or suggest another?
False Promise
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How to prove that $7^{31} > 8^{29}$

How can I prove that $7^{31}$ is bigger than $8^{29}$? I tried to write exponents as multiplication, $2\cdot 15 + 1$, and $2\cdot 14+1$, then to write this inequality as $7^{2\cdot 15}\cdot 7 > 8^{2\cdot 14}\cdot 8$. I also tried to write the right…
mrJoe
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