Questions tagged [number-comparison]
Tag for problems about comparing explicitly given numbers, often by hand calculation only.
213 questions
14 votes
3 answers
639 views
Compare $\log_5(\log_4 3)$ and $\log_6(\log_6 3)$
Let $a=\log_5(\log_4 3)$ and $b=\log_6(\log_6 3)$. How to compare the two numbers if numerical solution is not used? The following is my thoughts, but it did not work. From the question, I get $5^a=\...
- 149
2 votes
2 answers
271 views
Is $\varphi = \frac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\frac{1}{13}$?
Is $\varphi = \dfrac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\dfrac{1}{13}$? My calculator suggests that $$\ln(\varphi) < -\...
- 1,835
0 votes
1 answer
75 views
A comparison scheme for complex numbers (or vectors in general)
Below I am stating a possible ordering scheme for complex numbers and n-dimensional vectors in general. I am vaguely aware that such kind of thigs exist in Mathematics, however I have put it as a ...
- 59
-1 votes
1 answer
141 views
Prove that $\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\cdots+\frac{1}{3001}>1$ [duplicate]
Prove that $$\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\cdots+\frac{1}{3001}>1$$ I tried in this way: Since there are total $2001$ fractions, I divided them into three groups each having $667$ ...
- 309
1 vote
4 answers
216 views
Comparing $\frac{13}{10}$, $\frac{113}{101}$, $\frac{1013}{1001}$ in ten seconds
I have a so basic question that anyone can do. When it was a computer-only exam, the time given for this question was $10$ sec. So how can we handle this question in exactly $10$ sec? Compare $a$, $b$...
10 votes
2 answers
699 views
Prove that $3^{271}<7^{153}$
Prove that $3^{271}<7^{153}$. Approach: $3^{271}<7^{153}\leftrightarrow \frac{\ln7}{\ln3}>\frac{271}{153}$ $\frac{\ln7}{\ln3}=1.771243\ldots$ $\frac{271}{153}=1.771241\ldots$ Can the ...
- 5,520
0 votes
1 answer
134 views
Nontrivial numerical comparison question [closed]
By a numerical comparison question, we mean the following type of problems : Given distinct positive real numbers $a$ and $b$, prove by hand without using any computing requirements, that $a > b$ ...
- 3,694
1 vote
1 answer
415 views
Which of the following is bigger? $50^{50}$ or $49^{51}$ [duplicate]
Which of the following is bigger? $50^{50}$ or $49^{51}$ My attempt: $\displaystyle\frac{49^{51}}{50^{50}}=\frac{{49}^{50}\cdot 49}{50^{50}}=\left(\frac{49}{50}\right)^{50}\cdot 49=(0.98)^{50}\cdot 49=...
- 137
3 votes
2 answers
210 views
How can I compare $\sin 40^\circ$ and $\tan 35^\circ$ without using any program or calculator? [closed]
I know that for the same angle, $\tan$ is greater than $\sin$ in the first quarter of unit circle. But for different angles, there is uncertainty. For example, $\sin 40^\circ \gt \tan 32^\circ$, but $\...
12 votes
10 answers
972 views
Which is bigger, ${\sqrt2}^9$ or $9^{\sqrt2}$? (no calculator)
I was thinking about the classic question, "Which is bigger, $e^\pi$ or $\pi^e$?" (no calculator), and I tried to create a question that is similar, but resistant to the usual methods used ...
- 40.5k
0 votes
0 answers
31 views
Let $a_{n} > 0$ and $\sum a_{n}$ converges. Prove that $\forall p > \frac{1}{2}$ , $\sum \frac{\sqrt{a_{n}}}{n^{p}}$ converges. [duplicate]
Question Let $a_{n} > 0$ and $\sum a_{n}$ converges. Prove that $\forall p > \frac{1}{2}$, $\sum \frac{\sqrt{a_{n}}}{n^{p}}$ converges. Attempt. Since, $\lim_{n \to \infty}a_{n} = 0$. After ...
- 670
5 votes
3 answers
354 views
Prove that $\frac{1}{90} < \sqrt{2024} - \sqrt{2023} <\frac{1}{88}$
Question a) Prove that $$\frac{1}{90} < \sqrt{2024} - \sqrt{2023} <\frac{1}{88}$$ Question b) Is $$\sqrt{2024} - \sqrt{2023}$$ smaller or larger than $$\frac{1}{89}$$ No calculator is allowed, ...
- 169
4 votes
7 answers
276 views
Compare $3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$
Compare $$3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$$ Options: $\text{(A)} > \space\space\space\space\space \text{(B)} <\space\space\space\space\space \text{(C)} =$ Notes: $1....
- 5,894
0 votes
3 answers
333 views
How to prove $\sqrt{99} + \sqrt{101} < 2\sqrt{100}$ without a calculator
This is one of my extension sheet questions and I was really stumped on how to approach it. $\sqrt{99} + \sqrt{101} < 2\sqrt{100}$ First I had approached it by looking at smaller and larger square ...
- 51
3 votes
4 answers
153 views
Number of binary strings of length $56$ vs number of permutations of English alphabet
This is exercise $1.2$ in Nicholas Loehr's book "Combinatorics". Which is larger: the number of binary strings of length $56$, or the number of permutations of the English alphabet ($26$ ...
- 3,396