Brain Teasers ? or 50 Years On........... ?

My version of a "Tree" diagram.

Hope this helps.

The next Chapter in my school maths book is Complex Numbers ! Remember them ?

I can grasp the REAL part, but the IMAGINARY part is proving a bit more elusive..............

ISTR that anybody who succeeded with electrical power generation, distribution and its use in industry had a weird imagination and no problem with complex numbers, primarily because they substituted a "j" instead of an "i" and that made things easier.

After 10 or 12 years at school, I had rather formed the view that when any number is squared, say x for example, the result is either a positive number or zero ie x² ≥ 0. I had heard that these numbers (such as x above) are called real  numbers, but not given it much thought. It seemed obvious.

Then, one day our maths teacher pointed out that we had occasionally seen equations such as x² = -1 whose roots are clearly not real (and we had therefore discarded associated answers). Well, he now wanted us to be able to work with theses types of equations so introduced another category of numbers, namely the set of numbers whose squares are negative real numbers. he called members of this set of numbersimaginary numbers , and gave examples such as √-1; √-7; and √-16.

At this point, I had started day-dreaming, trying to imagine, imaginary numbers...................and failing !!

Does this sound familiar to anybody ??

Time to brush the cobwebs of those old A-Level maths books and let the imagination roll again.

Real + Imaginary gets rather Complex

A general member of the set of imaginary numbers is √-n² where n is a real number

But a bit of manipulation shows that √-n² = √n² x √-1 = ni where i = √-1

So, using the imaginary numbers from the post above, √-1 = i

And √-7 = i √7

And √-16 = 4i

We can add, subtract, multiply and divide imaginary numbers.........

So, 50 Years on........(just for starters !)

2i + 5i = ??

(√7i) - i = ??

2i x 5i = ??

6i ÷ 3i = ??

7i

i(√7 - 1)

-10

2

Someone with "real imagination", Well done sj.

You might recall your "powers of imagination" with the next bit !

Powers of i can often be simplified.

i³ = ??

i⁴ = ??

i⁵ = ??

i^-1 = ??

-i

1

i

-i (multiplying numerator and denominator by i produces (i / -1)

 I think I'll follow Roy's example and have a lie down now.

And for the engineers amongst us imaginary numbers [ sqrt(-1) ] are of course represented by j and not i.   

Simon-in-Suffolk posted:

And for the engineers amongst us imaginary numbers [ sqrt(-1) ] are of course represented by j and not i.   

ISTR that anybody who succeeded with electrical power generation, distribution and its use in industry had a weird imagination and no problem with complex numbers, primarily because they substituted a "j" instead of an "i" and that made things easier. (copied from a few posts above)

I think we are on the same wave-length here Simon................

Physicists, particularly those dealing with Quantum Theory, would say that √(-1) = ±i . Without the negative roots we would not have antiparticles and the theories just would not work. For an enlightening read on complex numbers, read Roger Penrose's 'The Road to Reality". Without complex numbers we could not solve many real life problems.

It also enjoyable to listen to music at the same time!

Filipe posted:

Physicists, particularly those dealing with Quantum Theory, would say that √(-1) = ±i . Without the negative roots we would not have antiparticles and the theories just would not work. For an enlightening read on complex numbers, read Roger Penrose's 'The Road to Reality". Without complex numbers we could not solve many real life problems.

It also enjoyable to listen to music at the same time!

You (well, in this case I did) learn something new every day. Thanks Filipe. And thanks for the reading reference.

And thanks to Simon for reminding us about "j" in engineering - easier to spot than "i"

Anybody got any "every-day" real-life examples of where Complex Numbers are used ?

After a certain point, you can't visualize what's going on in mathematics, you just have to accept. 

That was the point at which I switched to chemistry.

I know the feeling. I did engineering at university and the first year included structures, electrical, mechanical, and the maths to go with it. I eventual got some sort of concept about vibrations, power factors and gyros but none of them felt "tangible" at the time.

OTOH, organic chemistry had just seemed like a random jungle of H, C and O with a few suffix _{2 4 6 12} etc and the odd N thrown in for good measure.

Don Atkinson posted:

Anybody got any "every-day" real-life examples of where Complex Numbers are used ?

It is a remarkable fact that exp(iθ) = cos(θ) + isin(θ). It you think of polar coordinates and replace the x axis with the real axis and y with the imaginary axis then x = cosθ and y = sinθ helps to explain the above formula. This might help make complex numbers more real. The above and properties of powers such exp(θ)*exp(φ) = exp(θ+φ) allows cos(3θ) and other trig functions to be easily expanded in terms of cosθ and sinθ.

The unit circle (radius 1) is represented as exp(iθ), theta (θ) being measured anti-clockwise from the real axis. The nth roots of 1 ( you might have thought that 1 is a unique nth root of 1) are obtained by dividing 2π by n and multiplying by 1,2, ... n-1. If you measure these angle round the circle and draw lines between them you construct a regular n-polygon. Strictly speaking there are an infinite number of roots by adding any multiple of 2π.

Complex numbers are just a mathematical tool. All the mathematics developed from the concept is very useful for tackling real world problems in optics, linear amplifier circuits, etc.

Read Roger Penrose's book, which also takes you into modern physics to whatever ever depth you choose.

The following few teasers are a far cry from Roger Penrose................they come from my practise tests leading up to the "11 Plus" exam back in 1958 (I think !).

The "Rules" are................."No Calculators" - there weren't any back in 1958

In fact. all these were "Mental Arithmetic"

If the following fractions are re-arranged in ascending order of magnitude, which one is in the middle ?

A 1/2; B 2/3; C 3/5; D 4/7; E 5/9

This sentence contains the letter e __________ times.

seven   eight   nine   ten   eleven

How many of the five words shown above in italics can be placed in the blank space to make the sentence in red, true ?

A 0; B 1; C 2; D 3: E 4; F 5

Which of the following can be divided (without any remainder) by all the whole numbers from 1 to 10 inclusive ?

A 23 x 34;  B 34 x 45;  C 45 x 56;  D 56 x 67;  E 67 x 78

Don Atkinson posted:

..my practise tests...

Harrumph!  ;-)

C.

Grandma is four times as old as me now, but I recall that five years ago she was five times as old as I was then. How old are we both now ?

Oh ! I should have mentioned that we only got one minute to figure out each answer before the next one was read out.................

Christopher_M posted:

Don Atkinson posted:

..my practise tests...

Harrumph!  ;-)

C.

Which question Chris ?