Questions tagged [number-comparison]

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

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3 votes
7 answers
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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....
Hussain-Alqatari's user avatar
0 votes
3 answers
126 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 ...
Hooman's user avatar
  • 43
5 votes
2 answers
146 views

Show $\root{-e}\of{e}<\ln2$ without a calculator

I tried manipulating the expression to come up with inequalities such as $1<e^{\frac{1}{e}}\ln2$. One idea I have is to show that $\lim_{x\to\infty}\left(\frac{1}{x}+\ln2\right)\left(\frac{1}{x}+1\...
Dylan Levine's user avatar
  • 1,586
5 votes
1 answer
250 views

Proving $3^{100} > 5\cdot10^{47}$ with integral representation

I want to prove that $3^{100} > 5\cdot10^{47}$ without using calculator or any approximation of the logarithm. I tought about finding an integral representation of $(3^{100} - 5\cdot10^{47})$ with ...
user967210's user avatar
26 votes
7 answers
1k views

Show by hand : $e^{e^2}>1000\phi$

Problem: Show by hand without any computer assistance: $$e^{e^2}>1000\phi,$$ where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$. I come across this limit showing: $$\...
Miss and Mister cassoulet char's user avatar
-2 votes
2 answers
73 views

compare the numbers: $A = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $B =\frac{b}{a} + \frac{a}{c }+\frac{c}{b}$ [duplicate]

question Let's compare the numbers: $A = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $B =\frac{b}{a} + \frac{a}{c }+\frac{c}{b}$, where $a, b, c$ are real numbers such that: $0<a<b<c$ idea ...
IONELA BUCIU's user avatar
  • 1,185
0 votes
0 answers
30 views

Percentage Similarity within Range

I am trying to calculate the similarity between two times. In C# I'd get these values using something like TimeStamp.Ticks; in general, I'm thinking of some ...
Bondolin's user avatar
  • 103
2 votes
2 answers
35 views

How would you go about finding the exponent for any given x that most closely matches or surpasses factorial growth?

While analyzing the factorial function and comparing it to basic exponentiation, I couldn't help but notice the obvious fact that exponentiation can eventually overtake factorialization if the ...
elusivestream1337's user avatar
2 votes
6 answers
184 views

How can I prove that $\sqrt{3}-\sqrt{2}-0.32<0$

How can I prove that $$\sqrt{3}-\sqrt{2}-0.32<0$$ This is what I did : We have : $$\sqrt3\simeq1.732 \; ; \; \sqrt2\simeq1.414$$ $$\sqrt3<1.733 \; ; \; \sqrt2>1.413$$ $$\sqrt3<1.733 \; ; \;...
user579102's user avatar
16 votes
7 answers
888 views

Show that $\sqrt{10}+\sqrt{26}+\sqrt{17}+\sqrt{37} \gt \sqrt{341}$

Show that $\sqrt{10}+\sqrt{26}+\sqrt{17}+\sqrt{37} \gt \sqrt{341}$. This is inspired by Showing $x+y>z$, where $x=\sqrt{10}+\sqrt{26}$, $y=\sqrt{17}+\sqrt{37}$, and $z=\sqrt{323}$. Is my idea ...
marty cohen's user avatar
1 vote
1 answer
64 views

Comparing log with rationals [closed]

So I came across with this problem: To prove $\log_27<2\sqrt3$. This is not difficult as LHS$<3<$RHS. However, I did want to know how to generalise the solution. In particular, given integers ...
Kurt's user avatar
  • 49
1 vote
0 answers
44 views

Inequality using order relation

$1^{2n} +2^{2n} +3^{2n} \ge 2\times 7^n$ This question is from An excursion in mathematics in the section of order relation I tried using induction,tried to prove $1^{2n} +2^{2n+1} \ge 2\times 7^n$ ...
Anubhav Panchal's user avatar
-2 votes
1 answer
74 views

Compare $A$ and $B$ [closed]

Compare $A$ and $B$ with: $$A = \sqrt{2017} + \sqrt{2019} + \sqrt{2023}$$ $$B = \sqrt{2018} + \sqrt{2020} + \sqrt{2021}$$ I tried to prove $A^4 < B^4$ but it's too hard to do that.
Nguyen Huy Gia Bao's user avatar
1 vote
2 answers
71 views

Problem with exponents and inequalities

Hello I tried to solve this problem, below: 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) } ...
hikaru jakafura's user avatar
-2 votes
1 answer
106 views

Prove by using convexity: $33\sin33^{\circ}+29\sin29^{\circ}+28\sin28^{\circ}>45$

Without using convexity (and without using calculator, of course) we can make the following. Since, $33+29+28=90$, we need to prove that: $$33(\sin33^{\circ}-\sin30^{\circ})>29(\sin30^{\circ}-\...
Michael Rozenberg's user avatar
1 vote
2 answers
98 views

How do we compare $8^{2700}$ and $3^{5500}$ without using the logarithm?

I am trying to explain this to children with no knowledge of logarithm. We know that $8^{2700} = 2^{8100}$ and $3^{5500}$ is already in prime base. However, how do we know which one is larger in value?...
Nighty's user avatar
  • 2,152
0 votes
2 answers
86 views

Prove which term is bigger in magnitude

I am struggling to prove which of these two terms is bigger in magnitude. Assumption: \begin{align} A < \theta \end{align} Term 1: \begin{align} \sqrt{\delta^2 \theta^4 + 4 \theta^2 A (\epsilon - 1 ...
NC520's user avatar
  • 229
1 vote
5 answers
343 views

Which is greater: $101^{99}\text{ or }100^{100}$?

So I was scrolling through the homepage of Youtube when I came across this math problem by Learn with Christian Expo which proposed the following question:$$\text{Which is greater: }101^{99}\text{ or }...
CrSb0001's user avatar
  • 2,534
3 votes
1 answer
68 views

Comparison power series

I have a doubt concerning concerning power series comparison. There are two power series $f$ and $g$ with the coefficients $(a_i)_{i \in \mathbb{N}}$ and $(b_i)_{i \in \mathbb{N}}$. That is saying : $$...
NancyBoy's user avatar
  • 325
1 vote
0 answers
57 views

What is the best algorithm to use for tournament ranking where every winner also "autowins" every win of the loser?

Let's say I have a tournament where I need to get a complete ranking of all players at the end of the tournament. The rule of the ranking is that if player A wins player B, then they also win every ...
Kristo Vaher's user avatar
10 votes
7 answers
884 views

How to prove that $\log_5(6)>\log_6(7)$?

I know that $\log_5(6)>\log_6(7)$ but I wanted to prove it without calculating the values. After generalizing it it turned this way (for $x>1$): $$\frac{\ln(x)}{\ln(x-1)}>\frac{\ln(x+1)}{\ln(...
Fardad Mazloumi's user avatar
2 votes
1 answer
66 views

Comparing two sequence via their exponential generating function

I am studying two sequence with their e.g.f. The first one are the Bell numbers (sequence $A000110$ on OEIS) defined as follows. $B_0=1$ and $$ B_n =\sum_{k=0}^{n-1} {n-1\choose k} B_k $$ They have a ...
Bric's user avatar
  • 133
12 votes
6 answers
894 views

Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$

Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$. My effort: I tried using the fact $9^{\sqrt{2}}<9^{1.5}=27.$ Also We have $512 <729 \Rightarrow 2^9<27^2 \Rightarrow 2^{\frac{9}{...
Ekaveera Gouribhatla's user avatar
0 votes
1 answer
24 views

What is the best-case number of array item comparisons in this algorithm, in terms of n?

The Smart Bubble Sort. Preconditions: x1, x2, ... , xn is in U, a set that is totally ordered by <, and n ≥ 2. Postconditions: x1 ≤ x2 ⋯ ≤ xn. ...
Cte2My's user avatar
  • 3
-1 votes
1 answer
93 views

Percentage question in GRE

Ashley's score was 20% higher than Bert's score. Bert's score was 20% lower than Charles Score. Is Ashley's score greater than Charles Score? This is essentially a GRE question however my mind can not ...
Zain's user avatar
  • 1
0 votes
1 answer
93 views

Comparing $(2n)^{\ln {n}}$ to $(\ln{n})^{2n}$ for $n > 1000$ [closed]

I want to compare these two numbers and prove which is bigger for $n > 1000$: first: $(2n)^{\ln {n}}$ second: $(\ln{n})^{2n}$ I tried to somehow simplify them to similiar form and do induction, but ...
Coltrane's user avatar
16 votes
6 answers
989 views

Prove that: $e-\ln(10)>\sqrt 2-1.$

The author's original inequality is as follows. Prove that: $$e-\ln(10)>\sqrt 2-1$$ Is there a good approximation for $$e-\ln 10?$$ Actually, I am also wondering that, Where does $\sqrt 2-1$ come ...
User's user avatar
  • 1,671
1 vote
0 answers
51 views

Method for finding the largest positive difference between two pairs of IEEE 754 double precision floating point numbers and fixed-point numbers

I have two pairs of IEEE 754 double precision (64-bit) floating-point numbers and unsigned fixed-point numbers, and I'm trying to find which pair has the greatest difference. The fixed-point numbers ...
Polynomial's user avatar
1 vote
1 answer
51 views

comparison with logic problems

I am a bit confused regarding the logic combined with size comparisons. For example, if there is a statement x > y -> x >= y I believe that this would be ...
Crayon Basket's user avatar
0 votes
2 answers
31 views

How do I determine how similar the angles of a triangle are to a given triangle? [closed]

I am trying to determine how close a triangle is to a given known triangle based on the angles. I know the 3 angles of a TRUTH triangle. I know the 3 angles of an INPUT triangle. I want to determine a ...
pbhuter's user avatar
  • 149
2 votes
2 answers
138 views

What is the easiest way to compare $\log_2 (72)$ with $2\pi$?

I need to compare $\log_2 (72)$ with $2\pi$. What is the easiest way to do it?
George Glebov's user avatar
2 votes
0 answers
66 views

Comparing $37^{38}$ and $38^{37}$ [duplicate]

Which one is larger: $$37^{38}\quad\text{or}\quad38^{37}$$ I solved it as follow: First I divided both numbers by $37^{37}$ to get $37$ and $(\frac{38}{37})^{37}$. Since $(1+\frac1{37})^{37}<e<...
Etemon's user avatar
  • 6,455
0 votes
1 answer
2k views

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 ...
Billjesh Baidya's user avatar
1 vote
0 answers
422 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} ...
Nikoloz Chichua's user avatar
0 votes
0 answers
71 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 ...
Matt's user avatar
  • 1
0 votes
1 answer
120 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 ...
kormoran's user avatar
  • 2,963
5 votes
5 answers
167 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=...
kormoran's user avatar
  • 2,963
4 votes
5 answers
268 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'(...
Maverick's user avatar
  • 9,206
1 vote
1 answer
202 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 $...
Jürgen K.'s user avatar
0 votes
0 answers
31 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 = (...
sten's user avatar
  • 149
21 votes
7 answers
1k 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}\...
A. Goodier's user avatar
0 votes
1 answer
568 views

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 ...
Jim's user avatar
  • 1,589
2 votes
1 answer
467 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 ...
Jim's user avatar
  • 1,589
6 votes
7 answers
461 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 ...
Alma Do's user avatar
  • 499
2 votes
2 answers
107 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 ...
Carlos's user avatar
  • 691
0 votes
1 answer
54 views

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 ...
Reluctant Metallurgist's user avatar
3 votes
1 answer
212 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 ...
legionwhale's user avatar
  • 2,411
1 vote
6 answers
185 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 ...
ChuNan's user avatar
  • 303
2 votes
1 answer
4k 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: ...
trapezy's user avatar
  • 31
10 votes
5 answers
387 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}...
Joseph's user avatar
  • 334