Elementary my dear Watson.

[RoverP6B]

It is more often said than not that the number one raised to any power at all will always result in one. That is to say,..one squared is one. One cubed is one. One raised to the power of one billion is one. Lets not forget negative numbers,...one raised to the power of negative one is one,...to negative 1000 is one....how frustrating. I know...what about zero...one raised to the power of zero....you guessed it...is one.

Is this always the case? What do you think?

Ron.

[Milton477]

It is more often said than not that the number one raised to any power at all will always result in one. That is to say,..one squared is one. One cubed is one. One raised to the power of one billion is one. Lets not forget negative numbers,...one raised to the power of negative one is one,...to negative 1000 is one....how frustrating. I know...what about zero...one raised to the power of zero....you guessed it...is one.

Is this always the case? What do you think?

Ron.

I thinks it's past your bedtime

[RoverP6B]

Hello Milton,

No one of school age will know the answer. I left school over 20 years ago. Any question on mathematics that I ask you don't have to worry as I already know the answer..:p

Ron.

[**Discovery300Tdi**]

Who likes chicken?

[awabbit6]

It is more often said than not that the number one raised to any power at all will always result in one. That is to say,..one squared is one. One cubed is one. One raised to the power of one billion is one. Lets not forget negative numbers,...one raised to the power of negative one is one,...to negative 1000 is one....how frustrating. I know...what about zero...one raised to the power of zero....you guessed it...is one.

Is this always the case? What do you think?

Ron.


Perhaps 1^i would be an exception, however it can't be equated.

[slug_burner]

I take it that "i" is as in imaginary numbers?

[Disco44]

I'll have a half:wasntme:

[RoverP6B]

awabbit6 wrote,..
Perhaps 1^i would be an exception, however it can't be equated.

Hello Paul,

That is partly correct in that the exponent is the square root of minus one, that it is to say 'i'.

However it most certainly can be equated.

Ron.

[Ferret]

1 raised to the power of infinity does not equal 1. The correct answer is 'indeterminate'

[Landy Smurf]

your wrong series 1^o = upset but series 1^10 = happy but yes 1 to the power of anything = 1

[RoverP6B]

schmierer LR at singleton wrote,...
your wrong series 1^o = upset but series 1^10 = happy but yes 1 to the power of anything = 1

I am sorry schmierer LR, that statement is incorrect.

1 raised to the power of anything as you put it does not always equal 1.

As Ferret stated, 1 raised to the power if infinity does not equal 1, the answer rather is defined as indeterminite, so no actual value can thus be attributed to that expression.

In the case of 1^i there are many values that it equals including 1.

Ron.

[ramblingboy42]

I think its about time some of you got out into the bush.........

[Hardchina]

As Ferret stated, 1 raised to the power if infinity does not equal 1, the answer rather is defined as indeterminite, so no actual value can thus be attributed to that expression.


Ron.

mmmm

if 1^1 = 1
and 1^1000000 = 1
and 1 ^ kazillion = 1
and we plot a graph...... the line does not deviate from 1 at any time
all the way to infinity and beyond.

I call bull****. 1 ^ infinity equals 1 :)

[RoverP6B]

Hardchina wrote,..
if 1^1 = 1
and 1^1000000 = 1
and 1 ^ kazillion = 1
and we plot a graph...... the line does not deviate from 1 at any time
all the way to infinity and beyond.

I call bull****. 1 ^ infinity equals 1

Hello Hardchina,

My understanding,...and I may have misunderstood...;) is that unlike the integers that you have given in your example, infinity is not an numerical quantity, that is it is without bound, hence the reason the answer was given as indeterminite. I can see your logic though.

What about 1^i? What does it equal?

Ron.

[Landy Smurf]

also isnt anything to the value of zero equal to 1

[Hardchina]

Hardchina wrote,..

Hello Hardchina,

My understanding,...and I may have misunderstood...;) is that unlike the integers that you have given in your example, infinity is not an numerical quantity, that is it is without bound, hence the reason the answer was given as indeterminite. I can see your logic though.

What about 1^i? What does it equal?

Ron.

what does 0 x infinity equal? :p

[Landy Smurf]

zero

[Ferret]

I call bull****. 1 ^ infinity equals 1 :)

Really, it hardly matters but --- Indeterminate form

[robzilla]

https://www.aulro.com/afvb/

..is the answer

[Redback]

mmmm

if 1^1 = 1
and 1^1000000 = 1
and 1 ^ kazillion = 1
and we plot a graph...... the line does not deviate from 1 at any time
all the way to infinity and beyond.

I call bull****. 1 ^ infinity equals 1 :)

Buzz Lightyear = 1

[Bushie]

http://www.camargo.eti.br/42.png

..is the answer

Got it in 1 :D:D:D:D



Martyn

[Hardchina]

42 is the answer, but whats the bloody question?

How many dirt roads must a man drive down in a series land rover? :)

[Tombie]

42 is the answer, but whats the bloody question?

Q: Why are there bloody maths questions on a Landrover Board :wasntme:

[Rosco]

The easiest way to give your calculator a headache ....

sq root(2-3) ..... It should smoke for a little while .. :eek::cool:

[Quarks]

... Not if I put it into "complex" mode beforehand ;) :p

[Hardchina]

Hardchina wrote,..

Hello Hardchina,

What about 1^i? What does it equal?

Ron.


Stupid imaginary numbers are doing my head in, trying to work it out but it just aint going to happen :(..... It can't be 1 though. Is it even a valid concept?

[Rosco]

...... Is it even a valid concept?


Yep ... unfortunately for us mere mortals. Some time back the mathematicians decided to give the square root of (-1) the value of "i". Then they could play with the numbers to their heart's content.

I spent some time with a serious cramp in the forehead trying to get my head around it when I did pure maths in my engineering degree.

It was of no value to me whatsoever (civil), however it is used quite a bit in electrickery ..... (edit) ... for AC if I recall.

[RoverP6B]

Hardchina wrote,..
It can't be 1 though. Is it even a valid concept?

Hello Harchina,

It does and it is...;)

The clue as to where to begin is in the title of the thread...."Elementary my dear Watson",...Elementary implies exponential and natural logarithmic functions as they are known in mathematics,..not simplicity.

The answer to the question,..what does 1^i equal?

1^i = e^-2PIk ; where k = 0, +/- 1, 2, 3 etc

I am quite happy to show the derivation on how this solution is obtained if anyone is interested.

1^i = 1 when k = 0 which is the principal value of the argument, along with k other solutions all of which do not equal one.

Complex numbers are used by Engineers in applications as diverse as heat flow, elasticity, hydrodynamics and aerodynamcis, fluid flow, electrostatics and electrical circuit calculations involving time varying fields to name but a few.

There are also many applications in pure mathematics.

Ron.

[Hardchina]

You lost me at "hello" :p

[UncleHo]

BUT will any of this assist me rebuild a Series 2a gearbox???????????

[robzilla]

BUT will any of this assist me rebuild a Series 2a gearbox???????????
Yeh, i'm struggling to find how this helps me remove a D1 gearbox as well.

maybe i need an extender that's 1^i long to reach the top nuts?

[Quarks]

BUT will any of this assist me rebuild a Series 2a gearbox???????????

Of course!

Just get someone competent to say "i will do it" !!! :p:p:p

:D

[The ho har's]

Well this certaintly helped with the restoration on my series 2a.....NOT:confused:


Mrs hh:angel:

[Landy Smurf]

lol

[awabbit6]

The answer to the question,..what does 1^i equal?

1^i = e^-2PIk ; where k = 0, +/- 1, 2, 3 etc

I am quite happy to show the derivation on how this solution is obtained if anyone is interested.



I'd like to see your derivation Ron.

[RoverP6B]

awabbit6 wrote,..
I'd like to see your derivation Ron.

Hello Paul,

Absolutely,...

1^i can be expressed using the exponential as e^lni just as any base and exponent can be.

now looking at lni express 'i' which is the imaginary operator in exponential form as a complex number...1e^i0
so lni = ln(1e^io)
= ln(1) + lne^i(0 + 2PIk)
= i(2PIk)

So 1^i = e^i(ln1 + lne^i(0 + 2PIk))
= e^i(ln1 + i(0 + 2PIk))
= e^iln1.e^-2PIk
= e^-2PIk,..........where k is an integer including zero.

How cool is that as they say,...the principal value of the argument is equal to one, but all other values of which there is essentially an infinite number are not. What is even more amazing is that the results are all real even though we raised one to an imaginary number. Advanced mathematics is just amazing...:)

Ron.

[Rosco]

Ron

As one of the blokes said earlier ................ you need to get out more .. ;):p:D