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<< Much Less Than Symbol

by Scott
(San Antonio)










































what is the cut off for the term much greater than. I took a test where i assumed .3 was ">>" much greater than .05. was this wrong to assume? it seems like its a relative answer to me. so when is it ok it use this symbol. i do understand that 100000>> 1 but when is it not ok to use this what is the "cutoff" limit?

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Dec 01, 2011
<< Much Less Than Symbol
by: Staff


Question:

by Scott
(San Antonio)


what is the cut off for the term much greater than. I took a test where i assumed .3 was ">>" much greater than .05. was this wrong to assume? it seems like its a relative answer to me. so when is it ok it use this symbol. i do understand that 100000>> 1 but when is it not ok to use this what is the "cutoff" limit?


Answer:

1. Numbers

The cutoff is subjective.


Generally x >> y would mean that x is a probably several orders of magnitude larger than y.

This can be very important in practical mathematics. The symbol >> can be used to show that certain numbers, or parts of an equation can be safely ignored.

This has lots of applications in daily life.

For example, if you are standing on the deck of a ship how far can you see? If you climb a mast you can see a greater distance than you can standing on the desk. How far is that?

The equation generally used for calculating the distance to the horizon is accurate for many purposes. However, it is an approximation based on the concept that one number is >> (much greater than) another number.

The approximate equation is especially useful because it is simple. You can estimate the distance in your head quite easily. It is shown below:

d = distance to horizon in miles

h = the height of your eye in feet

d ≈ 1.32*√(h)


If this widely used equation is only an approximation based on the concept that one number is >> (much greater than) another number, what is the real thing?

R = the radius of the earth = about 3,963 miles

d = √(2Rh + h²)


Looking at the equation, the number 2Rh is so large (2 times the radius of the earth times h) compared to h², the h² can be completely ignored. For a sailor, h² will not significantly change the calculation no matter what h is. (If you were looking at the horizon from a satellite orbiting the earth, that would be different. h² would be significant in that case.)


For our purposes, however:

2RH is much greater than h²

2Rh >> h²

2Rh is so much greater than h² that h² can be dropped from the equation, leaving:

d = √(2Rh)

converting between feet and miles

d ≈ 1.32*√(h)

d = distance to horizon in miles

h = the height of your eye in feet


2. Functions

So far we have been talking about numbers, but >> symbol can also apply to functions:

f(x) >> g(x)

This is used as an asymptotic comparison. The growth of g(x) is asymptotically bounded by f(x).





Thanks for writing.

Staff
www.solving-math-problems.com


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