Download Why division as “repeated subtraction” works

Why division as “repeated subtraction” works
2
1
2
7
8
2
6
1
1
1
3
8
3
5
rem 5
Why division as “repeated subtraction” works
2
1
2
7
8
2
6
1
1
1
3
8
3
5
rem 5
278
dividend
=
2 · 10 · 13 + 18
=
2 · 10 · 13 + 1 · 1 · 13 + 5
=
(2 · 10 + 1 · 1) · 13 + 5
=
quotient · divisor + remainder
Why division as “repeated subtraction” works
2x
+1
2x
2
+7x
+8
2x
2
+6x
x
x
1·x
+3
+8
+3
5
rem 5
−3 | 2
7
8
|
−6 −3
2
1 5
Why division as “repeated subtraction” works
2x
+1
2x
2
+7x
+8
2x
2
+6x
x
x
1·x
+3
rem 5
−3 | 2
7
8
|
−6 −3
2
1 5
+8
+3
5
2x 2 + 7x + 8
dividend
=
2 · x · (1 · x + 3) + 18
=
2 · x · (1 · x + 3) + 1 · (1 · x + 3) + 5
=
(2 · x + 1) · (1 · x + 3) + 5
=
quotient · divisor + remainder
Back to Complex Numbers ...
Please work on the following problems:
1
2
3
(a)
(b)
(a)
(b)
(c)
√
Find a number that, when added to √7, yields an integer.
Find a number that, when multiplied 7,√yields an integer.
Find a number that, when added to 2 + √7, yields an integer.
Find a number that, when multiplied 2 + 7, yields an integer.
Find√as many different numbers as possible that, when added to
number. Do the same for numbers that
2 + 7, yield a rational
√
multiply with 2 + 7 to yield a rational number.
(If you have time) Find a number that, when multiplied with 2 +
yields a rational number.
√
3
7,
Today’s Class
Themes:
Representations: Graphical/Geometric, Algebraic/Symbolic
Generalizing from Key Properties
Topics:
Various representations of complex numbers
Complex roots of polynomials
Two for the Price of One: the Complex Conjugate Theorem!
Can we make the sale last a little longer? . . . Other Conjugates.
Various representations of complex numbers
geometric
algebraic
points in C
or
vectors in C
coordinates (a, b)
or
“Cartesian” form a + bi
. . . others?
Complex roots of polynomials
What does it mean to factor a polynomial “completely”?
What if we only allow factors to be ones that we can graph on the
xy -plane?
Complex roots of polynomials
What does it mean to factor a polynomial “completely”?
What if we only allow factors to be ones that we can graph on the
xy -plane?
Degree n Theorem. A nonconstant, degree n polynomial has exactly
n roots. Suppose the polynomial is
f (x) = an x n + an−1 x n−1 + · · · + a1 x1 + a0 .
Then f (x) can always be represented in the form
f (x) = a(x − α1 )(x − α2 ) . . . (x − αn ),
where α1 , α2 , . . . , αn are the roots of f , and they are complex – not
necessarily real – numbers.
How would we rephrase the Degree n Theorem if we only allowed
polynomials with real coefficients?
Complex conjugate: very special pairs
The arithmetic of conjugate pairs is what makes them special as pairs,
and what makes each complex number have a unique conjugate.
Notation: α, Re (α), Im (α)
Complex conjugate: very special pairs
The arithmetic of conjugate pairs is what makes them special as pairs,
and what makes each complex number have a unique conjugate.
Notation: α, Re (α), Im (α) . . .
Complex conjugate: very special pairs
The arithmetic of conjugate pairs is what makes them special as pairs,
and what makes each complex number have a unique conjugate.
Notation: α, Re (α), Im (α) . . . α??
Complex conjugates and polynomials
1
(a) What is the simplest polynomial that you can think of that has i as a
root? That √
has 1 − i as a root?
√
(b) How about 7 as a root? That has 2 + 7 as a root?
2
How would your solutions to these problems change if we required:
(a) . . . the polynomials have to have real coefficients?
(b) . . . rational coefficients?
Complex conjugates and polynomials
1
(a) What is the simplest polynomial that you can think of that has i as a
root? That √
has 1 − i as a root?
√
(b) How about 7 as a root? That has 2 + 7 as a root?
2
How would your solutions to these problems change if we required:
(a) . . . the polynomials have to have real coefficients?
(b) . . . rational coefficients?
Complex conjugates and polynomials
1
(a) What is the simplest polynomial that you can think of that has i as a
root? That √
has 1 − i as a root?
√
(b) How about 7 as a root? That has 2 + 7 as a root?
2
How would your solutions to these problems change if we required:
(a) . . . the polynomials have to have real coefficients?
(b) . . . rational coefficients?
2(a) changes 1(a) . . . and not 1(b)
2(b) changes 1(b) . . . and not 1(a)
Complex conjugates and polynomials
1
(a) What is the simplest polynomial that you can think of that has i as a
root? That √
has 1 − i as a root?
√
(b) How about 7 as a root? That has 2 + 7 as a root?
2
How would your solutions to these problems change if we required:
(a) . . . the polynomials have to have real coefficients?
(b) . . . rational coefficients?
2(a) changes 1(a) . . . and not 1(b)
2(b) changes 1(b)
√ . . . and not 1(a)
i is to R . . . as . . . 7 is to
√ Q.
C is to R . . . as . . . {a + b 7|a, b ∈ Q} is to Q
Complex conjugates and polynomials
i is to R . . .
C is to R . . .
as
as
√
. . . 7 is to
√ Q.
. . . {a + b 7|a, b ∈ Q} is to Q
Both are square roots outside the home number system that we got by
taking a square root of a number in the home system.
Both can be roots of polynomials whose coefficient is in the home number
system
Complex conjugates and polynomials
i is to R . . .
C is to R . . .
as
as
√
. . . 7 is to
√ Q.
. . . {a + b 7|a, b ∈ Q} is to Q
Both are square roots outside the home number system that we got by
taking a square root of a number in the home system.
Both can be roots of polynomials whose coefficient is in the home number
system . . . and come in pairs!
In the case of complex numbers, the pairing looks like:
The Conjugate Pair Theorem, a.k.a. Two for the Price of One!
If f (x) is a real polynomial and a + bi is a root of f (x), then a − bi must
also be a root of f (x).
[Let’s work this out on the board. What are the conditions that we want to
establish? What are the conditions that are given to use?]
Complex Roots and Polynomials
Why are roots of a polynomials called roots?
Factor x 3 − 1 into linear factors
Factor x 4 − 1 into linear factors
When you are finished factoring them, plot their zeros as vectors on the
complex plane.
Complex Roots and Polynomials
Why are roots of a polynomials called roots?
Factor x 3 − 1 into linear factors
Factor x 4 − 1 into linear factors
When you are finished factoring them, plot their zeros as vectors on the
complex plane.
What about x 5 − 1? x 6 − 1? x 51 − 1
????
Complex Roots and Polynomials
Why are roots of a polynomials called roots?
Factor x 3 − 1 into linear factors
Factor x 4 − 1 into linear factors
When you are finished factoring them, plot their zeros as vectors on the
complex plane.
What about x 5 − 1? x 6 − 1? x 51 − 1
????
Next time: DeMoivre’s Theorem – The Holy Grail of Complex Numbers!