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Question:I noticed tonight that the sum of the decimal parts of the square roots of integers between x^2 and (x+1)^2, when x is large, approaches x + 1/6.
For example, between perfect squares 9 and 16 you have 10,11,12,13,14,15. Add the decimal portions of the square roots of 10 through 15, and you get 3.163+. Not very close to sqrt(9) + 1/6, but the pattern begins:
Between 16 and 25: 4.164+
Between 100 and 121: 10.1662+
Between 961 and 1024: 31.16662+
Between 10,000 and 10,201: 100.166662+
Between 1,000,000 and 1,002,001: 1000.16666662+
Between 100,000,000 and 100,020,001: 10000.1666666662+
Seems I've stumbled upon an odd property. I've done no analysis yet, and there is probably a straightforward reason for it, but I really didn't expect a sum of irrational numbers to render a rational number.
Anybody have any literature on this property?
Answers:Interesting! First, let's think about the graph of the function y = sqrt(x). And let's imagine a second graph, with straight lines between each of the integer square roots. i.e. the graph contains the points (0,0), (1,1), (4,2), (9,3), (16,4), (25,5) etc, with straight lines connecting these points. Now if the square root function really did follow the straight line function, then all the sums you compute would have remainder 0. For instance, between 4 and 5, the sum of the remainders would be 1/9 + 2/9 + 3/9 + 4/9 + 5/9 + 6/9 + 7/9 + 8/9 = 36/9 = 4.0 exactly. What this suggests is that the extra remainder you get (1/6) is the difference between the square root function and the straight line function. It is possible to calculate the area underneath the two curves and confirm this is the case. Between the points (a^2, a) and ((a+1)^2, a+1) (for some arbitary number a), the area under the square root curve can be calculated using calculus: area = (integate from a^2 to (a+1)^2) x^1/2 dx = [from a^2 to (a+1)^2] 2/3 x^3/2 = 2/3 ((a+1)^3  a^3) = 2a^2 + 2a + 2/3 And the area under the straight line section is (using * for multiply): = width * (height of rectangle + 1/2 height of triangle) ((a+1)^2  a) * (a + 1/2) = (2a + 1) * (a + 1/2) = 2a^2 + 2a + 1/2 So the difference between the two curves is 1/6. That's really quite an interesting result: even when a is very large (and the area between the curves is very long in the x direction and very small in the y direction) the difference in area is always the same. When you add up the remainders of each square root, you are in effect approximating the area of this difference curve. As the number a gets larger, this approximation gets more and more accurate.
Answers:Interesting! First, let's think about the graph of the function y = sqrt(x). And let's imagine a second graph, with straight lines between each of the integer square roots. i.e. the graph contains the points (0,0), (1,1), (4,2), (9,3), (16,4), (25,5) etc, with straight lines connecting these points. Now if the square root function really did follow the straight line function, then all the sums you compute would have remainder 0. For instance, between 4 and 5, the sum of the remainders would be 1/9 + 2/9 + 3/9 + 4/9 + 5/9 + 6/9 + 7/9 + 8/9 = 36/9 = 4.0 exactly. What this suggests is that the extra remainder you get (1/6) is the difference between the square root function and the straight line function. It is possible to calculate the area underneath the two curves and confirm this is the case. Between the points (a^2, a) and ((a+1)^2, a+1) (for some arbitary number a), the area under the square root curve can be calculated using calculus: area = (integate from a^2 to (a+1)^2) x^1/2 dx = [from a^2 to (a+1)^2] 2/3 x^3/2 = 2/3 ((a+1)^3  a^3) = 2a^2 + 2a + 2/3 And the area under the straight line section is (using * for multiply): = width * (height of rectangle + 1/2 height of triangle) ((a+1)^2  a) * (a + 1/2) = (2a + 1) * (a + 1/2) = 2a^2 + 2a + 1/2 So the difference between the two curves is 1/6. That's really quite an interesting result: even when a is very large (and the area between the curves is very long in the x direction and very small in the y direction) the difference in area is always the same. When you add up the remainders of each square root, you are in effect approximating the area of this difference curve. As the number a gets larger, this approximation gets more and more accurate.
Question:i don't understand. i need someone to explain?
Answers:Find a pattern: Even integers 8, 6, 4, 2, 0, 2, etc... Difference between integers is two. So if "i" was any integer, the next integer would be "i+2" right? and the next next one would be "i+4" right?
Answers:Find a pattern: Even integers 8, 6, 4, 2, 0, 2, etc... Difference between integers is two. So if "i" was any integer, the next integer would be "i+2" right? and the next next one would be "i+4" right?
Question:I have 2 math questions that I'm stuck on. could someone please help to explain?
1A given positive number has principal square root 2.1. the other square root of the given positive number is ?WHAT?.
the given positive number is ?WHAT?
Im pretty sure the first answer is 2.1, lol. but can't get the rest.
2 the number 3(put check mark for square root here) 30 (on the inside of the check mark) lies between two consecutive integers. Find the consecutive integers.
if someone could please help with this Id appreaciate it. :) Thank you all for your help. :)
Answers:The "principal" square root of a number is the positive square root. The negative is also a square root. So the answers are: The other square root is 2.1 (you had that). The given positive number is 2.1 ^ 2 (where ^ = squared) or 2.1 ^ 2 = 4.41 so the Given Positive Number is 4.41 2. I'm going to use ^.5 to mean square root here. What you have is 3(30^.5) I think. Or, in english, 3 times the square root of 30. The square root of 30 is 5.477, so 3 * 5.477 = 16.43 So, 16.43 lies between what 2 consecutive integers? The answer is 16 and 17 since 16.43 is greater than 16 and less than 17.
Answers:The "principal" square root of a number is the positive square root. The negative is also a square root. So the answers are: The other square root is 2.1 (you had that). The given positive number is 2.1 ^ 2 (where ^ = squared) or 2.1 ^ 2 = 4.41 so the Given Positive Number is 4.41 2. I'm going to use ^.5 to mean square root here. What you have is 3(30^.5) I think. Or, in english, 3 times the square root of 30. The square root of 30 is 5.477, so 3 * 5.477 = 16.43 So, 16.43 lies between what 2 consecutive integers? The answer is 16 and 17 since 16.43 is greater than 16 and less than 17.
Question:the problem says "An integer can be represented by the expression 2n. Find three consecutive even integers that have a sum of 54. " i reallly need help, so please give me anything you got . the problem says "An integer can be represented by the expression 2n. Find three consecutive even integers that have a sum of 54. " i reallly need help, so please give me anything you got and explain how you did every step
Answers:2n is an even integer. 2n + 2 and 2n + 4 are the next two consecutive even integers. 2n + (2n + 2) + (2n + 4) = 54 6n + 6 = 54 6n = 48 n = 8 The integers are 16, 18, and 20.
Answers:2n is an even integer. 2n + 2 and 2n + 4 are the next two consecutive even integers. 2n + (2n + 2) + (2n + 4) = 54 6n + 6 = 54 6n = 48 n = 8 The integers are 16, 18, and 20.
From Youtube
Consecutive Integer Word Problems :How to use all five steps to solve a consecutive integer word problem, using only ONE variable.Using all five parts to solve a word problem with two unknowns, using only ONE variable. This is the 5th out of 12 in the solving word problems series. The other three series are "Solving Equations", "Factoring" and "Graphing".