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Let x be an odd number. Then x is the difference of two square numbers
proof if the statement is right or wrong it's right or wrong
if x is on even then we have x=2n+1
help!
The statement is a fallacy.
Any odd number can be expressed as (2n+1) for some integer n; likewise an even number can be expressed as 2n for integer n
The difference of two odd numbers is always a positive number p-q is even if p and q are odd
proof: let p=(2n+1) and q=(2k+1) p-q= 2n-2q=2(n-q) say n-q=z; 2z is always even
The square of an odd number O is odd for O>1 and the square of an even number E is even.
proof: O=2n+1 O^2=4n^2+4n+1; E=2n E^2=4n^2
since x is odd x can also be expressed as 2m+1
Since only one of p or q is an odd number
2m+1=(2n+1)^2-(2k)^2
2m+1=(4n^2+4n+1)-4k^2
2m=(4n^2+4n+1)-4k^2-1
2m=(4n^2+4n)-4k^2
m=2n^2+2n-2k^2
since n and k are integers multiplied by 2 the R.H.S. must be positive
similarly you can perform the same operations on q is odd (ll leave this up to you)
or 2m+1=(2n)^2-(2k+1)^2
x being an odd number does not imply that is is the difference between two square integers
144 = 12^2
16 = 4^2
144 – 16 = 128, which is not an odd number
What is true however, is that, the sum of odd numbers from 1 to 2n-1 equates to a square number.
Proof:
the sum of an arithmetic series S(n) of n terms with first term a and common difference d
is given by the formula
S(n) = (n/2)*(2a + (n-1)d)
With first term a = 1 and common difference d = 2, the sum of the first n odd numbers is
S(n) = (n/2)*(2*1 + (n-1)*2)
= (n/2)*(2 + 2n – 2)
= (n/2)*(2n)
= n^2
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I learned something new today, and something surprising. I’ve opened up my fair share of bellies and seen intestines doing their slow peristaltic dance in there, and I knew in an abstract way that guts were very long and had to coil to fit into the confined space of the abdominal cavity, but I’d always just assumed it was simply a random packing — that as the gut tube elongated, it slopped and slithered about and fit in whatever way it could. But no! I was reading this new paper today, and that’s not the case at all: there is a generally predictable pattern of coiling in the developing gut, and it’s species-specific.
The midgut forms as a simple linear tube of circular cross-section running down the midline of the embryo, and grows at a greater rate than the surrounding tissue, eventually becoming significantly longer than the trunk. As the size of the developing mid- and hindgut exceeds the capacity of the embryonic body cavity, a primary loop is forced ventrally into the umbilicus (in mammals) or yolk stalk (in birds). this loop first rotates anticlockwise by 90° and then by another 180° during the subsequent retraction into the body cavity. Eventually, the rostral half of the loop forms the midgut (small intestine) and the caudal half forms the upper half of the hindgut (the ascending colon).
The chirality of this gut rotation is directed by left-right asymmetries in cellular architecture that arise within the dorsal mesentery, an initially thick and short structure along the dorsal-ventral axis through which the gut tube is attached to the abdominal wall. this leads the mesentery to tilt the gut tube leftwards with a resulting anticlockwise corkscrewing of the gut as it herniates. however, the gut rotation is insufficient to pack the entire small intestine into the body cavity, and additional loops are formed as the intestine bends and twists even as it elongates. Once the gut attains its final form, which is highly stereotypical in a given species, the loops retract into the body cavity. during further growth of the juvenile, no additional loops are formed, as they are tacked down by fascia, which restrict movement and additional morphogenesis without inhibiting globally uniform growth.
That is just plain awesome. now I want to open up a zebrafish and look at the curling of its intestines, or better yet, peer into a larva and see if there are any predictable rules of formation. oh, jeez, I want to look inside my own belly, although that would be a kind of self-defeating experiment.
Morphology of loops in the chick gut. a, Chick gut at embryonic day 5 (E5), E8, E12 and E16 shows stereotypical looping pattern.b, Proliferation in the E5 (left) and E12 (right) gut tubes (blue) and mesentery (red). each blue bar represents the average number of phospho-H3-positive cells per unit surface in 40 (E5) or 50 (E12) 10-mm sections. each red bar represents the average number of phospho-H3-positive cells per unit surface over six 10-mm sections (E5) or in specific regions demarcated by vasculature along the mesentery (E12). The inset images of the chick guts align the proliferation data with the locations of loops (all measurements were made in three or more chick samples). Ant., anterior; post., posterior. Error bars, s.d. c, The gut and mesentery before and after surgical separation at E14 show that the mesentery shrinks while the gut tube straightens out almost completely. d, The E12 chick gut under normal development with the mesentery (left) and after in ovo surgical separation of the mesentery at E4 (right). The gut and mesentery repair their attachment, leading to some regions of normal looping (green). however, a portion of the gut lacks normal loops as a result of disrupting the gut-mesentery interaction over the time these loops would otherwise have developed.
So how do species-specific coiling patterns emerge? A naive expectation might be that there are specific genes associated with the process that selectively impose bends at specific locations along the length of the intestine — that there is genetically determined spatial information along the tube that defines how it should coil. this is not the case. Instead, the reproducible pattern of coiling is an emergent property of some general parameters of the tissues.
You do need to know some very elementary anatomy to know what’s going on here. The gut begins embryonically as a simple, straight tube, fixed at both ends at the mouth and anus. Initially, the gut is the same length as the body, and is suspended from the back of the body cavity by a continuous sheet of tissue, the mesentery, that is also the same length as the gut. But then what happens is that the gut elongates, while the mesentery grows much more slowly. this difference in growth rates means that the gut is under compression along its length, restrained by the mesenteries, which causes it to periodically buckle.
One way to test the role of the mesentery is to remove it. If you carefully cut it away from the gut, as is shown in (c) and (d) of the figure above, it straightens out — in a fully relaxed state, without the compression of the mesenteries, the gut is straight and linear. you can do partial cuts, too, and wherever a stretch of gut is released from the mesentery constraint, it uncoils.
Take it another step. is this how generic tubes and sheets interact? The authors took a rubber tube of length Lt, and a rubber sheet of length Lm, where Lm is less than Lt. they stretched the rubber sheet to length Lt, stitched it to the rubber tube, and then let it go. Voila, it spontaneously coiled into a configuration (b) that closely resembles the chicken gut (c).
Rubber simulacrum of gut looping morphogenesis. a, To construct the rubber model of looping, a thin rubber sheet (mesentery) was stretched uniformly along its length and then stitched to a straight, unstretched rubber tube (gut) along its boundary; the differential strain mimics the differential growth of the two tissues. The system was then allowed to relax, free of any external forces. b, on relaxation, the composite rubber model deformed into a structure very similar to the chick gut (here the thickness of the sheet is 1.3 mm and its Young’s modulus is 1.3 MPa, and the radius of the tube is4.8 mm, its thickness is 2.4 mm and its Young’s modulus is 1.1 MPa. c, Chick gut at E12. The superior mesenteric artery has been cut out (but not the mesentery), allowing the gut to be displayed aligned without altering its loop pattern.
This is qualitatively convincing — they do look very similar, and at this point I’m willing to believe that mechanical forces are sufficient to explain the coiling pattern. The authors take another step, though: they bring out the math and get all quantitative. this is a reasonable idea; from the model above, it does look like the shape is reducible to a small number of parameters, so it’s a manageable problem. so brace yourself: a little math coming right up.
We now quantify the simple physical picture for looping sketched above to derive expressions for the size of a loop, characterized by the contour length, λ, and mean radius of curvature, R, of a single period. The geometry of the growing gut is characterized by the gut’s inner and outer radii, ri and ro, which are much smaller than its increasing length, whereas that of the mesentery is described by its homogeneous thickness, h, which is much smaller than its other two dimensions. Because the gut tube and mesentery relax to nearly straight, flat states once they are surgically separated, we can model the gut as a one-dimensional elastic filament growing relative to a thin two-dimensional elastic sheet (the mesentery). As the gut length becomes longer than the perimeter of the mesentery to which it is attached, there is a differential strain, ε, that compresses the tube axially while extending the periphery of the sheet. when the growth strain is larger than a critical value, ε* the straight tube buckles, taking on a wavy shape of characteristic amplitude A and period λ>A. At the onset of buckling, the extensional strain energy of the sheet per wave- length of the pattern is Um∝Emε2hλ2, where Em is the Young’s modulus of the mesentery sheet. The bending energy of the tube per wavelength is Ut∝EtItκ2λ, where κ ∝ A/λ2 is the tube curvature, It ∝ ro4-ri4 is the moment of inertia of the tube and Et is the Young’s modulus of the tube. using the condition that the in-plane strain in the sheet is ε* ∝ A/λ and minimizing the sum of the two energies with respect to λ then yields a scaling law for the wavelength of the loop:
Did you get all that? If not, don’t worry about it. What it all means is that we can measure general properties of gut tissues, plug the parameters into these formulas, and ask a computer to predict what the gut should look like in a numerical simulation. And it works!
Predictions for loop shape, size and number at three stages in chick gut development. a, Comparisons of the chick gut at E16 (top) with its simulated counterpart (bottom). b, Scaled loop contour length, λ/ro, plotted versus the equivalently scaled expression from equation (3) for the chick gut (black squares), the rubber model (green triangles) and numerical simulations (blue circles). The results are consistent with the scaling law in equation (1). c, Scaled loop radius, R/ro, plotted versus the equivalently scaled expression from equation (4) for the chick gut, the rubber model, and numerical simulations (symbols are as in b). The results are consistent with the scaling law in equation (2). Error bars, s.d.
At this point, you should be saying enough — that’s more than enough awesome to convince you that they’ve determined the rules that shape the gut. But no, they go further: all the above work is in chickens, so they reach out and start disemboweling other species, and ask if their formulas work to describe their gut coiling, too. Would you be surprised to learn that it does?
Comparative predictions for looping parameters across species. a, Gut looping patterns in the chick, quail, finch and mouse (to scale) show qualitative similarities in the shape of the loops, although the size and number of loops vary substantially. b, Comparison of the scaled loop contour length, λ/ro, with the equivalently scaled expression from equation (3) shows that our results are consistent with the scaling law in equation (1) across species. Black symbols are for the animals shown in a, other symbols are the same as in Fig. 4b. c, Comparison of the scaled loop radius, R/ro, with the equivalently scaled expression from equation (4) shows that our results are consistent with the scaling law in equation (2) across species (symbols are as in b). in b and c, points are reported for chick at E8, E12 and E16; quail at E12 and E15; finch at E10 and E13; and mouse at E14.5 and E16.5. Error bars, s.d.
What makes this a beautiful result is that it’s a perfect illustration of the principles D’Arcy Wentworth Thompson laid out in his book, On Growth and Form (and even the title of the paper is a nod to that classic of developmental biology). sometimes, simple mathematical rules govern the patterns we see in developing systems, whether it’s the Fibonacci spirals we see in the head of a sunflower or the coils of a nautilus shell, or tangled loops of our intestines. The form is not laid out in tightly-coded, case-by-case specification in the genome, but by the genetic definition of only a few parameters, in this case the relative rates of growth of two adherent tissues and the compression they impose on an elongating tube, from which a lovely arrangement flowers elegantly.
Savin T, Kurpios NA, Shyer AE, Florescu P, Liang H, Mahadevan L, Tabin CJ (2011) on the growth and form of the gut. Nature 476:57-63.
(Also on FtB)
25, 12, 31, 26, 17, 15, 24, 10, 16, 12
So the question has 3 parts,
a= Square of the sums of the angles,
b= Sample size,
c= Sum of the squares of then angles.
In the form c – (a/b)
For a, you add up all the angles and then square the final answer.
b is the sample size, so b=10
For c, you square each angle separately, and add up all the squares.
The final answer is 3996-(35344/10)= 461.6
Result so far–(x^2 – 2x + 1) +2(y^2+4y+4) = 11 + 1+ 8
and now you apply your perfect square factoring rule:
(x – 1)^2 + 2(y + 2)^2 = 10–How do you get this?
I am so confused! my book doesn't explain much.
since your book doesn't explain much go to answers.com and ask the same equation
well um if u were to do (x^2-2x+1) you would gt (x-1) (x-1) because -1x-1 is 1 and -1+-1 is -2. You need the sum of by and the multiplication of c.
well the factoring is right… here is the general formula:
(a+b)^2 = a^2 + b^2 +2AB ok so if we apply the factoring to the problem that is correct but I am confused by the numbers… where did they get the 11+1+8 and the 10? that makes no sense.
khake.com/page47
mathworld.wolfram.com
flashcardexchange.com
These will help with any math problem. They explain it all! I cite them because I think that you will use them.
hyperphysics is a good site, too.
Be an engineer. You can!
