Half Comes From Where? Exploring the Origins of 1/2

  • Thread starter waqarrashid33
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In summary, the conversation discusses solving an integral and simplifying a limit expression to get a form of the answer. There is a disagreement about whether the limit expression equals 1/2 or not, but it is eventually resolved.
  • #1
waqarrashid33
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Let me know how 1/2 comes from it.see attachemet
 

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  • #2
Have you tried anything for yourself?

Solve the integral and see if you can simplify the limit to transform the expression into a form that you know the answer to.
 
  • #3
i tried a lot but answer goes wrong..
i didn't touched with calculus since long time...May be it is because of this...
For the integral of 1+cos(2t) gives 2T+sin(2T).
 
  • #4
waqarrashid33 said:
i tried a lot but answer goes wrong..
i didn't touched with calculus since long time...May be it is because of this...
For the integral of 1+cos(2t) gives 2T+sin(2T).

Yes that would probably be the problem then.
[tex]\int1+\cos(2t)dt=t+\frac{1}{2}\sin(2t)[/tex]

Start from there.
 
  • #5
waqarrashid33 said:
i tried a lot but answer goes wrong..
i didn't touched with calculus since long time...May be it is because of this...
For the integral of 1+cos(2t) gives 2T+sin(2T).

Try solving your improper integral (for 1 + cos(2t) you will get anti-derivative t + 1/2sin(2t)) and then expand into F(T) - F(-T), collect your constant outside of the integral (1/2T) bring it together and you will get a limit expression in terms of T where T goes to infinity.

There are limit theorems you can use to solve this also.
 
  • #6
Thanks...
 
  • #7
waqarrashid33 said:
Thanks...



Interesting: you don't even need [itex]\,\,T\to\infty\,\,[/itex]. It is 1/2 for any [itex]\,\,T\neq 0\,[/itex].

DonAntonio
 
  • #8
DonAntonio said:
Interesting: you don't even need [itex]\,\,T\to\infty\,\,[/itex]. It is 1/2 for any [itex]\,\,T\neq 0\,[/itex].

DonAntonio

Not quite, the final steps of the solution are to simplify [tex]\frac{1}{2}\left(1+\lim_{T\to a}\frac{\sin(2T)}{2T}\right)[/tex]

and that expression is only equal to 1/2 if [tex]\lim_{T\to a}\frac{\sin(2T)}{2T}=0[/tex] which only happens for [itex]a=\infty[/itex]
 
  • #9
Mentallic said:
Not quite, the final steps of the solution are to simplify [tex]\frac{1}{2}\left(1+\lim_{T\to a}\frac{\sin(2T)}{2T}\right)[/tex]

and that expression is only equal to 1/2 if [tex]\lim_{T\to a}\frac{\sin(2T)}{2T}=0[/tex] which only happens for [itex]a=\infty[/itex]



I don't know how you got that. I get
[tex]\int_{-T}^T \cos^2(t)dt=\left[\frac{t+\cos t\sin t}{2}\right]_{-T}^T=\frac{1}{2}\left[T+\cos T\sin T-\left(-T-\cos(-T)\sin(-T)\right)\right]=\frac{2T}{2}=T[/tex]
as [itex]\,\,\cos(-T)\sin(-T)=-\cos T\sin T\,\,[/itex] , and then
[tex]\frac{1}{2T}\int^T_{-T}\cos^2 t\,dt=\frac{1}{2}[/tex]
like that, without limit...

DonAntonio
 
  • #10
DonAntonio said:
I don't know how you got that. I get
[tex]\int_{-T}^T \cos^2(t)dt=\left[\frac{t+\cos t\sin t}{2}\right]_{-T}^T[/tex]

How did you get that?

[tex]\cos^2t=\frac{1}{2}\left(1+\cos(2t)\right)[/tex]

Oh ok I see what you have, after integrating you converted sin(2t) to 2sin(t)cos(t)

DonAntonio said:
[tex]\left(-T-\cos(-T)\sin(-T)\right)[/tex]

This should be

[tex]\left(-T+\cos(-T)\sin(-T)\right)[/tex]
 
Last edited:
  • #11
Mentallic said:
How did you get that?

[tex]\cos^2t=\frac{1}{2}\left(1+\cos(2t)\right)[/tex]

Oh ok I see what you have, after integrating you converted sin(2t) to 2sin(t)cos(t)



This should be

[tex]\left(-T+\cos(-T)\sin(-T)\right)[/tex]


Yes indeed. So much worrying about the change of sign in the sine of -T that I forgot I had that minus sign out of the parentheses. Thanx.

DonAntonio
 

1. What does "half" refer to in this context?

In this context, "half" refers to the quantity or amount that is equivalent to 1/2, or one half.

2. Where does the concept of "half" come from?

The concept of "half" has been used in various cultures and civilizations throughout history. The Ancient Egyptians, Babylonians, and Chinese all had their own systems of representing fractions, including 1/2. However, the modern notation for fractions, including 1/2, was developed by the ancient Greeks.

3. How is 1/2 represented in different cultures?

Different cultures have different ways of representing 1/2. For example, in the decimal system, 1/2 is represented as 0.5. In the Egyptian system, 1/2 was represented as a hieroglyphic symbol of a mouth. In the Mayan system, 1/2 was represented as a dot. The Chinese used a character for "half" to represent 1/2.

4. What is the mathematical definition of 1/2?

The mathematical definition of 1/2 is a fraction that represents one part of a whole that has been divided into two equal parts. It can also be represented as a decimal (0.5) or a percentage (50%). In terms of division, 1/2 is the result of dividing a number by 2.

5. Can 1/2 be written in any other forms?

Yes, 1/2 can be written in various forms, such as a decimal (0.5), a percentage (50%), a ratio (1:2), or a fraction in lowest terms (1/2). It can also be expressed as a mixed number (1/2 = 0.5 = 1/2 = 2/4 = 3/6 = ...).

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