Tag Archives: number

The square root of minus one asked me “Do I exist?”

Complex number.
“complex” as opposed to “simple” ?
“number” for what ?
Not for counting !
Not for measuring ! We’ll see about that !
“Square root of -1”, maybe, if that means anything at all !

Who needs the “i” ? It’s not essential.
Here goes…..

They say that (a+ib)(p+iq) = ap – bq + (bp + aq)i
But only if i is the square root of -1.

Getting rid of the i
Let us put the a and the b in a+ib together in brackets, as (a,b), and call this “thing” a “pair”.
This gets rid of the (magic) i straightaway.

Let us define an operation * to combine pairs:
(a,b)*(p,q) = (ap-bq, bp+aq)
This is the “pair” version of the “multiplication of complex numbers”.

It’s more interesting to read this as “(a,b) is applied to (p,q)”, and even better if we think of (p,q) as a “variable” and “apply (a,b)” as a function.
Ok, so we will write (x,y) instead of (p,q), and then
(a,b)*(x,y) = (ax-by, bx+ay)
Let us call the output of the “apply (a,b)” function the pair (X,Y)
Then
X = ax-by
Y = bx+ay
Now we can see this as a transformation of points in the plane:
The function “apply (a,b)” sends the point (x,y) to the point (X,Y)

Looking at some simple points we see that
(1,0)*(x,y) = (x,y)….no change at all
(-1,0)*(x,y) = (-x,-y)…the “opposite” of (x,y),
so doing (-1,0)* again gets us back to no change at all.
(0,1)*(x,y) = (-y,x)….which you may recognize as a rotation through 90 deg.
and doing (0,1)* again we get
(0,1)*(0,1)*(x,y) = (0,1)*(-y,x) = (-x,-y)….a rotation through 180 deg.

So with a bit of faith we can see that (0,1)*(0,1) is the same as (-1,0), and also that (-1,0)*(-1,0) = (1,0)…check it!
Consequently we have a system in which there are three interesting operations:
(1,0)* has no effect, it is like multiplying by 1
(-1,0)* makes every thing negative, it is like multiplying by -1, and
(0,1)*(0,1)* has the same effect as (-1,0)*

So we have found something which behaves like the square root of -1, and it is expressed as a pair of ordinary numbers.
It is then quite reasonable to give the name “i” to this “thing”, and use “i squared = -1”.

And generally, a complex number can be seen as a pair of normal (real) numbers, and bye-bye the magic !

When you think about it a fraction also needs two numbers to describe it.

Next post : matrix representation of “apply (a,b) to (x,y)”.

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Filed under complex numbers, meaning, ordered pairs

Complex Numbers via Rigid Motions

https://howardat58.files.wordpress.com/2015/02/complex-numbers-by-rotations.doc

Complex numbers via rigid motions
Just a bit mathematical !

I wrote this in response to a post by Michael Pershan:
http://rationalexpressions.blogspot.com/2015/02/could-this-introduce-kids-to-complex.html?

The way I have presented it is showing how mathematicians think. Get an idea, try it out, if it appears to work then attempt to produce a logical and mathematically sound derivation.
(This last part I have not included)
The idea is that wherever you have operations on things, and one operation can be followed by another of the same type, then you can consider the combinations of the operations separately from the things being operated on. The result is a new type of algebra, in this case the algebra of rotations.
Read on . . .

Rotations around the origin.
angle 180 deg or pi
Y = -y, and X = -x —> coordinate transformation
so (1,0) goes to (-1,0) and (-1,0) goes to (1,0)
Let us call this transformation H (for a half turn)

angle 90 deg or pi/2
Y = x, and X = -y
so (1,0) goes to (0,1) and (-1,0) goes to (0,-1)
and (0,1) goes to (-1,0) and (0,-1) goes to (1,0)
Let us call this transformation Q (for a quarter turn)

Then H(x,y) = (-x,-y)
and Q(x,y) = (-y,x)

Applying H twice we have H(H(x,y)) = (x,y) and if we are bold we can write HH(x,y) = (x,y)
and then HH = I, where I is the identity or do nothing transformation.
In the same way we find QQ = H

Now I is like multiplying the coodinates by 1
and H is like multiplying the coordinates by -1
This is not too outrageous, as a dilation can be seen as a multiplication of the coordinates by a number <> 1

So, continuing into uncharted territory,
we have H squared = 1 (fits with (-1)*(-1) = 1
and Q squared = -1 (fits with QQ = H, at least)

So what is Q ?
Let us suppose that it is some sort of a number, definitely not a normal one,
and let its value be called k.
What we can be fairly sure of is that k does not multiply each of the coordinates.
This appears to be meaningful only for the normal numbers.

Now the “number” k describes a rotation of 90, so we would expect that the square root of k to describe a rotation of 45

At this point it helps if the reader is familiar with extending the rational numbers by the introduction of the square root of 2 (a surd, although this jargon seems to have disappeared).

Let us assume that sqrt(k) is a simple combination of a normal number and a multiple of k:
sqrt(k) = a + bk
Then k = sqr(a) + sqr(b)*sqr(k) + 2abk, and sqr(k) = -1
which gives k = sqr(a)-sqr(b) + 2abk and then (2ab-1)k = sqr(a) – sqr(b)

From this, since k is not a normal number, 2ab = 1 and sqr(a) = sqr(b)
which gives a = b and then a = b = 1/root(2)

Now we have a “number” representing a 45 degree rotation. namely
(1/root(2)*(1 + k)

If we plot this and the other rotation numbers as points on a coordinate axis grid with ordinary numbers horizontally and k numbers vertically we see that all the points are on the unit circle, at positions corresponding to the rotation angles they describe.

OMG there must be something in this ! ! !

The continuation is left to the reader (as in some Victorian novels)

ps. root() and sqrt() are square root functions, and sqr() is the squaring function .

pps. Diagrams may be drawn at your leisure !

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Filed under abstract, algebra, education, geometry, operations, teaching

Fractions, at home and away.

So one day recently I was bored, and then the following rushed onto the page:

Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.

one quarter

Is this 3 hours or 1/4 of a pizza?

How many hours equals 1/4 of a pizza?

Apologies to those for whom a clock face is a historical artifact.

How do I know it’s a pizza and not a cake?

Does half a day include half the night?

There are four feet in our yard, mine and my sister’s.

Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?

Are ratios numbers or fractions (or neither) ?

Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)

You cut, I choose !

This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.

2/3 and 4/6 are equivalent fractions. Equivalent to what ?

The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.

The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.

Back to the heavy stuff next time !

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Filed under fractions, humor

“Number sentence”, what is this?

Being in complete agreement with  Dan Meyer on the term “Write an expression” I take exception to the vague instruction “Write a number sentence”.

Multiple choice question – Which of the following is a number sentence?

a) 3 + 2 = 5
b) three + two = five
c) three and two makes five
d) 2 + don’t know = 7
e) seven is 5 more than 2
f) they gave him 20 years
g) Mary gave three of her sweets to Jane and was left with 5
h) none of these, although they all have a verb

Answers on a postcard please, addressed to Santa Claus, North Pole

and next time I have much to say about “equations”

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Filed under algebra, arithmetic, humor, language in math

Fraction subtraction construction

So, you want to do a fraction subtraction. Here’s how, as a geometrical construction. You will need a piece of paper and a ruler.

Draw three number lines through a common point, which is the zero. Pick a nice point on the middle line to be the 1, say 6 inches away from the zero. Label the other two number lines 1,2,3,4,5,6,7 at equally spaced points, scale completely immaterial.

Now do what is shown in the picture below. (the pairs of lines are parallel)

Image

Now measure the distance with the ruler, and divide by 6 (if you put the 1 at the 6 inch point).

Bingo!

A simpler version of this (2 number lines) can be used to locate the point on the number line corresponding to any (relatively simple) fraction.

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