Let's take sqrt(15) - sqrt(14) vs sqrt(14) - sqrt(13) Multiply the expressions by their conjugates in the numerator and denominator, and you get: 1/(sqrt(15) + sqrt(14)) vs 1/(sqrt(14) + sqrt(13)) . Here we can see obviously that the first fraction is the smaller one; so: sqrt(15) - sqrt(14) < sqrt(14) - sqrt(13), and you conclude that sqrt(14) - sqrt(13) is the larger one.
@yoav613
3 күн бұрын
Old but gold
@mystychief
2 күн бұрын
sqrt(2)-sqrt(1) = 1.4...-1 = 0.4... < 1 = 1-0 = sqrt(1)-sqrt(0). While for positive numbers sqrt(n) is increasing and the derivative is decreasing, you have the same situation at higher x values so the inequality is valid for sqrt(n+2)-sqrt(n-1) and sqrt(n-1)-sqrt(n) as well.
@cdiesch7000
3 күн бұрын
Alternative: ratio of the two expressions and then multiply by the two complements both the numerator and the deniminator: (sqrt(15)-sqrt(14))(/sqrt(14)-sqrt(13))=………, =(sqrt(14)+sqrt(13))/(sqrt(15)+sqrt(14))
@yurenchu
2 күн бұрын
This has to be the quickest and most elegant algebraic solution!
@DergaZuul
2 күн бұрын
@@cdiesch7000 no need to make a fraction but yea idea is to get to easier comparison sqrt(13)
@forcelifeforce
2 күн бұрын
*@ SyberMath Shorts* -- Here is a method to handle any real number x greater than or equal to 1: √(x + 1) - √(x) vs. √(x) - √(x - 1) √(x + 1) + √(x - 1) vs. √(x) + √(x) [√(x + 1) + √(x - 1)]^2 vs. [2√(x)]^2 x + 1 + x - 1 + 2√(x + 1)√(x - 1) vs. 4x 2x + 2√(x + 1)√(x - 1) vs. 4x x + √(x + 1)√(x - 1) vs. 2x √(x + 1)√(x - 1) vs. x [√(x + 1)√(x - 1)]^2 vs. (x)^2 (x + 1)(x - 1) vs. x^2 x^2 - 1 < x^2 Therefore, √(x + 1) - √(x) < √(x) - √(x - 1). In your example, x = 14.
@ghstmn7320
2 күн бұрын
Orrrr Mean Value Theorem
@pwmiles56
3 күн бұрын
Alternatively sqrt(14) - sqrt(13) vs sqrt(15) - sqrt(14) 2 sqrt(14) vs sqrt(15) + sqrt(13) Square both sides 4 x 14 vs 15 + 13 + 2 sqrt((14+1)(14-1)) Simplifying and dividing by 2 14 vs sqrt(14^2 - 1) 14^2 vs 14^2 - 1 LHS is larger i.e. sqrt(14) - sqrt(13) > sqrt(15) - sqrt(14)
@dan-florinchereches4892
3 күн бұрын
Let f(x)=√x . As an elementary function it is continuous and derivable on intervals [13,14] and [14,15]. According to theorem of Lagrange there exists c inside interval (13,14) and d inside (14,15) such that f'(c)=(f(14)-f(13))/(14-13)= √14-√13 and f'(d)=(f(15)-f(14))/(15-14)=√15-√14 f'(x)=1/(2√x) and it is a monotonous decreasing function from +inf to 0 From interval conditions we have 13√15-√14 A bit convoluted but it is a fun method. Edit the property of f'(x)
@DonRedmond-jk6hj
3 күн бұрын
Using the AG inequality will save you the trouble of having to drag calculus in.
@Don-Ensley
3 күн бұрын
problem √15 - √14 or √14 - √13 Square both sides. 29 - 2√210 or 27 - 2√182 Subtract 27. 2 - 2√210 or - 2√182 Multiply by - 1 both sides. (This will invert the comparison operator at the end) 2√210 - 2 or 2√182 Divide by 2 √210 - 1 or √182 Square both sides. 211 - 2√210 or 182 Subtract 182. 29 - 2√210 or 0 Take the 2 inside the radical to become 29 - √840 or 0 √841 - √840 or 0 The left is > the right but since we multiplied by - 1, the sense inverts and we have the right is > than the left. Left side is smaller, right side wins.
@DergaZuul
2 күн бұрын
Easier multiply by both conjugates: (sqrt(15)-sqrt(14))(sqrt(15)+sqrt(14))(sqrt(14)+sqrt(13)) and (sqrt(14)-sqrt(13))(sqrt(15)+sqrt(14))(sqrt(14)+sqrt(13)) turns into sqrt(14)+sqrt(13) and sqrt(15)+sqrt(14) which is obviously less(
@yurenchu
2 күн бұрын
Sorry to say, but your first step is performed incorrectly. The square of (√15 - √14) equals (29 - 2√210) instead of (29 - √210); and likewise for the square of (√14 - √13). Furthermore, there is a mistake towards the end: 4√210 does _not_ equal √840 .
@Don-Ensley
2 күн бұрын
@@yurenchu i have corrected my comment. Thanks for pointing out my math error.
@yurenchu
2 күн бұрын
@@Don-Ensley You're welcome! Correction went largely well, but there are still mistakes: the subtraction of 182 at the end went wrong. Also keep in mind that you multiplied by (-1) at some point.
@Don-Ensley
2 күн бұрын
@@yurenchu thank you for the correction. I have made corrections and all should now be ok.👍🏼
@danielpraise2146
3 күн бұрын
It's not sweet 😢 You should have speak nah
@tygrataps
3 күн бұрын
Sorry to hear about your voice. I would have enjoyed you doing the same wonderfully corny joke about 'happy birthday 2 u'. Music choice had lots of squealing violins. Painful :(
@yurenchu
2 күн бұрын
Obviously, (√15 - √14) < (√14 - √13) , because (√C - √B) represents the difference in side lengths between a square with area = B and a square with area = C ; and if we subtract a square with area = 14 from a square with area = 15 , we get a _thinner_ L shape than when we subtract a square with area = 13 from a square with area = 14 . (The L shapes will have the same area = 1, but the first L shape will be thinner because the horizontal and vertical lengths are greater.) However, to prove it algebraically: √15 - √14 = = √( 14 + 1 ) - √14 = √( 14 + 2*(√14)*(1/(2√14)) ) - √14 < √( 14 + 2*(√14)*(1/(2√14)) + 1/(2√14)² ) - √14 = √( [√14 + 1/(2√14)]² ) - √14 = ( √14 + 1/(2√14) ) - √14 = 1/(2√14) = √13 + 1/(2√14) - √13 = √( [√13 + 1/(2√14)]² ) - √13 = √( (√13)² + 2(√13)/(2√14) + 1/(2√14)² ) - √13 = √( 13 + 2(√13)(2√14)/(2√14)² + 1/(2√14)² ) - √13 = √( 13 + (√52)(√56)/(√56)² + 1/(√56)² ) - √13 = √( 13 + √(52*56)/(56) + 1/(56) ) - √13 = √( 13 + [ √(52*56) + 1 ]/56 ) - √13 = √( 13 + [ √( (54-2)*(54+2) ) + 1 ]/56 ) - √13 = √( 13 + [ √( 54² - 2² ) + 1 ]/56 ) - √13 < √( 13 + [ √( 54² ) + 1 ]/56 ) - √13 = √( 13 + [ 54 + 1 ]/56 ) - √13 = √( 13 + 55/56 ) - √13 < √( 13 + 1 ) - √13 = √14 - √13 ==> (√15 - √14) < (√14 - √13)
@robertveith6383
Күн бұрын
*Thumbs-down!* That is *not* "obvious" what you described. You are misusing the word "obvious."
@yurenchu
Күн бұрын
@@robertveith6383 For me, it indeed was obvious because I immediately saw the geometrical implication of the expressions, which I explained in the first paragraph. It's comparable to how (for example) 1/29 < 1/27 is obvious, even though it may not be immediately obvious to a person who has never worked with fractions before.
@martinphipps2
2 күн бұрын
Square both of the you get 14 + 15 - 2sqrt(14x15) and 13 + 14 - 2sqrt(13x14) which is 29 - ? and 27 - ?. Let me think. What is sqrt(x2+x)? It is x times sqrt(1+1/x). Using the binomial theorem (1 + x)^(1/2) = 1 + (1/2)x - (1/8) x^2 + ... + 𝛤(3/2)/(n! 𝛤(3/2 - n)) x^n + ... sqrt(1+1/x)= 1 + (1/2x) plus neglible terms so 29 - ? and 27 - ? become 29 - 2x14 - 1 and 27 - 2x13 - 1 which are both zero so we have to include the second order term (1/8x^2} which when multiplied by 2x becomes (1/4x) which beomes smaller as x becomes larger so the second number is bigger.
@robertveith6383
Күн бұрын
You need grouping symbols! In the fourth line, it needs to be 1/(2x). In the fifth line, it needs to be 1/(8x^2). In the sixth line, it needs to be 1/(4x).
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