Numbers
1
Natural numbers
What are the basic properties of the natural numbers N? First, we need the
number 1. Second, given a number n ∈ N, we can always find a next number
which we will write as s(n) and think of as the successor of n. Note that 1 is
not the successor of anything. Finally, we exhaust all the natural numbers
by taking 1 and all of its successors. We may summarize these properties by
saying the following: There is a function s : N → N (the successor function)
with the following properties:
1. The function s is injective. In other words, if s(n) = s(m), then n = m.
2. There is a unique element 1 ∈ N such that 1 is not in the image of s,
in other words there is no n ∈ N with s(n) = 1.
3. Let X be a subset of N with the following properties: 1 ∈ X, and, if
n ∈ X, then s(n) ∈ N. Then X = N.
Here (3) is referred to as the principle of mathematical induction.
Addition. Define n + 1 (naturally enough) to be s(n). Now we can define
addition inductively: suppose that n + m has already been defined. Then
define n + s(m) to be s(n + m). In other words, we define n + (m + 1)
to be (n + m) + 1, so that a partial form of associativity is built into the
definition. Somewhat tedious inductive arguments then show that addition
is associative and commutative, and the the cancellation law holds: n1 +m =
n2 + m =⇒ n1 = n2 . However, subtraction is only defined in some cases:
n − m is defined only when there exists a p ∈ N such that n = m + p, i.e.
when n > m (see below for a discussion of order), and in this case n−m = p.
Multiplication. The natural inductive definition is the following: n · 1 = n
for all n ∈ N, and n·(m+1) = (n·m)+1. With this definition multiplication
1
is commutative and associative and distributes over addition. There is also
divisibility, which we shall describe when we discuss integers.
Exponents. Define a1 = a and an+1 = an · a. With this definition we have
the usual rules of exponents:
an · am = an+m
(an )m = anm .
The proof is again by induction.
Factorials. Factorials can also be defined inductively: set 1! = 1 and
(n + 1)! = n!(n + 1).
Order. We define ≤ as follows: n ≤ m if either n = m or there exists some
p ∈ N such that m = n + p. One can then show:
Proposition 1.1 (Trichotomy property). If n, m ∈ N, then either n ≤ m
or m ≤ n. Moreover if n ≤ m and m ≤ n then m = n.
One importance of mathematical induction is that it is related to the
following ordering property of N:
Proposition 1.2 (Well-Ordering Principle). Let A be a nonempty subset of
N. Then A has a least element, i.e. an element x ∈ A such that for every
a ∈ A, x ≤ a.
We say that N is well-ordered. Note that Q, R, Z are not well-ordered.
We also have the strong principle of induction (or complete induction):
Proposition 1.3 (Principle of Complete Induction). Let A ⊆ N, and suppose that:
1. 1 ∈ A;
2. For all n ∈ N, if k ∈ A for all k ≤ n, then n + 1 ∈ A.
Then A = N.
2
2
Integers, rational numbers, real numbers
We shall just make some brief comments about each of these.
Integers: The integers Z are obtained by adding 0 and −n, n ∈ N. We
extend inverses to all of Z via: −(−n) = n. Two negative integers are added
via the rule (−n) + (−m) = −(n + m), and 0 + n = n + 0 = n for all n ∈ Z.
We add two integers, one positive and one negative, as follows:


if n > m;
n − m,
n + (−m) = (−m) + n = 0,
if n = m;


−(m − n), if n < m.
We multiply integers by the rules (−n)m = n(−m) = −nm and (−n)(−m) =
nm. In particular, we have the rule (−1)(−1) = 1. With these rules, addition and multiplication are associative and commutative, multiplication
distributes over addition, there is an additive identity 0 and additive inverses, and there is a multiplicative identity 1. However, the only integers
with multiplicative inverses are ±1. Thus divisibility is interesting for integers: given k, n ∈ Z, we say that k divides n, written k|n, if there exists an
integer d such that kd = n, i.e. n is a multiple of k. Thus, for all n ∈ Z, 1|n
and n|0. However 0|n ⇐⇒ n = 0, and n|1 ⇐⇒ n = ±1.
Rational numbers: The rational numbers Q can be thought of as the set
of “fractions” a/b, where a, b ∈ Z and b 6= 0. Here two fractions a/b and
c/d are equal ⇐⇒ ad = bc. We identify the integers with a subset of Q by
identifying n ∈ Z with n/1 ∈ Q. Note that n/1 = m/1 ⇐⇒ n = m, and
that n/m = 0/1 ⇐⇒ n = 0. Addition and multiplication are defined for
rational numbers by the formulas
a c
ad + bc
+ =
;
b d
bd
a c
ac
· = .
b d
bd
They satisfy the usual properties: addition and multiplication are associative and commutative, multiplication distributes over addition, there is an
additive identity 0 and additive inverses, and there is a multiplicative identity 1 and a multiplicative inverse b/a for every nonzero rational number a/b.
Note that, (n/1) + (m/1) = (n + m)/1 and that (n/1) · (m/1) = (nm)/1, so
that adding or multiplying integers either viewed as integers or as rational
numbers gives the same result. Since every nonzero rational number divides
every rational number, divisibility is not an interesting notion for rational
numbers.
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Real numbers: The construction of the real numbers R is not algebraic in
nature, so we shall say nothing about it here, except to note that addition
and multiplication are defined for real numbers and satisfy the usual properties. Explicitly, addition and multiplication are associative and commutative, multiplication distributes over addition, there is an additive identity
0 and additive inverses, and there is a multiplicative identity 1 and a multiplicative inverse for every nonzero real number. Additionally there is the
order relation ≤, and R is complete with respect to ≤. One way to say this
is that every nonempty set of real numbers which has an upper bound has
a least upper bound.
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3.1
Complex numbers
Algebra
The set C of complex numbers is formed by adding a square root i of −1
to the set of real numbers: i2 = −1. Every complex number can be written
uniquely as a + bi, where a and b are real numbers. We usually use a single
letter such as z to denote the complex number a + bi. In this case a is the
real part of z, written a = Re z, and b is the imaginary part of z, written
b = Im z. The complex number z is real if z = Re z, or equivalently Im z = 0,
and it is pure imaginary if z = (Im z)i, or equivalently Re z = 0. In general
a complex number is the sum of its real part and its imaginary part times i,
and two complex numbers are equal if and only if they have the same real
and imaginary parts.
We add and multiply complex numbers in the obvious way:
(a1 + b1 i) + (a2 + b2 i) = (a1 + a2 ) + (b1 + b2 )i;
(a1 + b1 i) · (a2 + b2 i) = (a1 a2 − b1 b2 ) + (a1 b2 + a2 b1 )i.
For example,
(2 + 3i)(−5 + 4i) = −22 − 7i.
In this way, addition and multiplication are associative and commutative,
multiplication distributes over addition, there is an additive identity 0 and
additive inverses (−(a + bi) = (−a) + (−b)i), and there is a multiplicative
identity 1. Note also that Re(z1 + z2 ) = Re z1 + Re z2 , and similarly for
the imaginary parts, but a corresponding statement does not hold for multiplication. Before we discuss multiplicative inverses, let us recall complex
conjugation: the complex conjugate z̄ of a complex number z = a + bi is by
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definition z̄ = a − bi. It is easy to see that:
1
Re z = (z + z̄);
2
1
Im z = (z − z̄).
2i
Thus z is real if and only if z̄ = z and pure imaginary if and only if z̄ = −z.
More importantly, we have the following formulas which can be checked by
direct calculation:
z1 + z2 = z̄1 + z̄2 ;
z1 · z2 = z̄1 · z̄2 ;
z n = (z̄)n ;
z̄ = z;
z · z̄ = a2 + b2 ,
where in the last line z = √
a + bi. √
Thus, z · z̄ ≥ 0, and z · z̄ = 0 if and
only if z = 0. We set |z| = z · z̄ = √a2 + b2 , the absolute value, length or
modulus of z.For example, |2 + i| = 5. Note that, for all z1 , z2 ∈ C,
|z1 |2 |z2 |2 = z1 z̄1 z2 z̄2 = z1 z2 z̄1 z̄2 = (z1 z2 )(z1 z2 ) = |z1 z2 |2 ,
and hence |z1 ||z2 | = |z1 z2 |. The link between the absolute value and addition
is somewhat weaker; there is only the triangle inequality
|z1 + z2 | ≤ |z1 | + |z2 |.
If z 6= 0, then z has a multiplicative inverse:
z̄
z −1 = 2 .
|z|
In terms of real and imaginary parts, this is the familiar procedure of dividing
one complex number into another by “rationalizing the denominator:” if at
least one of c, d is nonzero, then
a + bi
a + bi
c − di
(a + bi)(c − di)
=
=
.
c + di
c + di
c − di
c2 + d2
Thus it is possible to divide by any nonzero complex number. For example,
to express (2 + i)/(3 − 2i) in the form a + bi, we write
2+i
3 + 2i
(2 + i)(3 + 2i)
4 + 7i
4
7
2+i
=
=
=
=
+ i.
2
2
3 − 2i
3 − 2i
3 + 2i
3 +2
13
13 13
If z 6= 0, then z n is defined for every integer n, including the case n < 0,
and the formula z n = (z̄)n still holds. In particular, for z 6= 0, |z −1 | = |z|−1 .
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Definition 3.1. Let U (1) be the set of complex numbers of absolute value
1, i.e.
U (1) = {z ∈ C : |z| = 1}.
By the above discussion, if z1 , z2 ∈ U (1), then z1 z2 ∈ U (1) (U (1) is closed
under multiplication). Clearly 1 ∈ U (1), and, if z ∈ U (1), then z 6= 0 and
z −1 ∈ U (1) (U (1) is closed under taking inverses).
3.2
Geometry
Instead of thinking of a complex number z as a + bi, we can identify it
with the point (a, b) ∈ R2 . From this point of view, there is no difference
between a complex number and a 2-vector, and we sometimes refer to C as
the complex plane. The absolute value |z| is then the same as k(a, b)k, the
distance from the point (a, b) to the origin. Addition of complex numbers
then corresponds to vector addition. However, multiplication of complex
numbers is more complicated. One way to understand it is to use polar
coordinates: if z = a + bi, where (a, b) corresponds to the polar coordinates
(r, θ), then r = |z| and a = r cos θ, b = r sin θ. Thus we may write z =
r cos θ + (r sin θ)i = r(cos θ + i sin θ). This is sometimes called the polar
form of z; r = |z| is, as we have seen, called the modulus of z and θ is
called the argument, sometimes written θ = arg z. Note that the argument
is only well-defined up to an integer multiple of 2π. (More naturally, the
argument is an equivalence class, in other words an element of R/2πZ.) If
z = r(cos θ+i sin θ), then clearly r is real and nonnegative and cos θ+i sin θ is
a complex number of absolute value one; thus every complex number z is the
product of a nonnegative real number times a complex number of absolute
value 1. If z 6= 0, then this product expression is√unique. (What happens if
z = 0?) For example, the polar form of 1 + i is 2(cos(π/4) + i sin(π/4)).
Given two complex numbers z1 and z2 , with z1 = r1 (cos θ1 + i sin θ1 ) and
z2 = r2 (cos θ2 + i sin θ2 ), we can ask for the polar form of z1 z2 :
z1 z2 = r1 (cos θ1 + i sin θ1 ) · r2 (cos θ2 + i sin θ2 )
= r1 r2 ((cos θ1 cos θ2 − sin θ1 sin θ2 ) + i(cos θ1 sin θ2 + cos θ2 sin θ1 ))
= r1 r2 (cos(θ1 + θ2 ) + i sin(θ1 + θ2 )),
where we have used the standard addition formulas for sine and cosine. (We
will see in a minute where these addition formulas come from.) Thus the
modulus of the product is the product of the moduli (this is just the formula
|z1 z2 | = |z1 ||z2 | which we have already seen), but the really interesting
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formula is that the arguments add:
arg(z1 z2 ) = arg z1 + arg z2 .
Of course, this has to be understood as up to possibly adding an integer
multiple of 2π.
One often abbreviates the expression cos θ + i sin θ as cis θ. But it is
perhaps more natural to use Euler’s formula
eiθ = cos θ + i sin θ.
In this way, we see that the somewhat strange looking addition formulas for
cosine and sine are equivalent to the usual property of exponentials:
eiθ1 eiθ2 = eiθ1 +iθ2 = ei(θ1 +θ2 ) .
We can interpret geometrically the effect of multiplying by a complex
number z. If z is real, multiplying by z is just ordinary scalar multiplication
and has the usual geometric interpretation. If z = cos θ + i sin θ has absolute
value one, then multiplying a complex number x + iy by z is the same
as rotating the point (x, y) by the angle θ. For a general z, multiplying
the complex number x + iy by z is a combination of these two operations:
rotation by the angle θ followed by scalar multiplication by the nonnegative
real number |z|.
Using the formula for multiplication, it is easy to see that if z has polar
form r(cos θ + i sin θ), then
z n = rn (cos nθ + i sin nθ);
z −1 = r−1 (cos(−θ) + i sin(−θ)) = r−1 (cos θ − sin θ).
Here the first formula, which is easily proved by mathematical induction,
holds for all z and positive integers n, and the second holds for z 6= 0. From
this it is easy to check that, for z 6= 0, the first formula holds for all integers
n. This formula is called De Moivre’s Theorem.
We can use De Moivre’s Theorem to find powers
√ and roots of complex
numbers. For example, we have seen that 1 + i = 2(cos(π/4) + i sin(π/4)).
Thus
√
(1 + i)20 = ( 2)20 (cos(20π/4) + i sin(20π/4))
= 210 (cos(5π) + i sin(5π)) = 210 (−1) = −1024.
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De Moivre’s Theorem can be used to generate identities for sin nθ and
cos nθ via the binomial theorem. For example,
cos 3θ + i sin 3θ = (cos θ + i sin θ)3
= cos3 θ + 3i cos2 θ sin θ − 3 cos θ sin2 θ − i sin3 θ
= cos3 θ − 3 cos θ sin2 θ + i(3 cos2 θ sin θ − sin3 θ).
Equating real and imaginary parts, we see that cos 3θ = cos3 θ−3 cos θ sin2 θ,
and likewise sin 3θ = 3 cos2 θ sin θ − sin3 θ.
It is more interesting to find roots. Let z = r(cos θ +i sin θ) be a complex
number, which we assume to be nonzero (since the only nth root of 0 is zero–
why?), and let w = s(cos ϕ + i sin ϕ). Then wn = z if and only if sn = r and
nϕ = θ + 2kπ for some integer k. Thus s = r1/n , and ϕ = θ/n + 2kπ/n for
some integer k. But sometimes these numbers will be the same for different
values of k: if ϕ1 = θ/n + 2k1 π/n and ϕ2 = θ/n + 2k2 π/n, then
r1/n (cos ϕ1 + i sin ϕ1 ) = r1/n (cos ϕ2 + i sin ϕ2 )
if and only if ϕ1 and ϕ2 differ by an integer multiple of 2π, if and only
if 2k1 π/n and 2k2 π/n differ by an integer multiple of 2π, if and only if n
divides k1 − k2 , or equivalently ⇐⇒ k1 ≡ k2 (mod n). Moreover, we can
find a complete set of choices by taking the arguments
θ/n, θ/n + 2π/n, . . . , θ/n + 2(n − 1)π/n.
Thus we see: If n is a positive integer, then a nonzero complex number has
exactly n distinct nth roots given by the formula above.
Examples: the two square roots of i = cos(π/2) + i sin(π/2) are
π π √2 √2
cos
+ i sin
=
+
i;
4
4
2√
2√
π
π
2
2
+ π + i sin
+π =−
−
i.
cos
4
4
2
2
√
For another example, to find all of the fifth roots of 3 + i, first write
!
√
√
3 1
π
π
3+i=2
+ i = 2 cos + i sin
.
2
2
6
6
Thus the fifth roots are given by
2kπ
2kπ
π
π
1/5
2
cos
+
+ i sin
+
,
30
5
30
5
8
k = 0, 1, 2, 3, 4.
We can apply the above to the complex number 1 = cos 0 + i sin 0. Thus
there are exactly n complex numbers z such that z n = 1, called the nth roots
of unity: namely
2kπ
2kπ
cos
+ i sin
,
k = 0, 1, . . . , n − 1.
n
n
It is easy to see that, once we have found one nth root w of a nonzero
complex number z, then all of the nth roots of z are of the form
2kπ
2kπ
cos
+ i sin
w,
n
n
for k = 0, 1, . . . , n − 1, i.e. any two nth roots of a given nonzero complex
number differ by multiplying by an nth root of unity.
Warning: the usual rules for fractional exponents that hold for positive
real numbers do not usually hold for complex roots; this is connected with
the fact that there is not in general one preferred nth root of a complex
number. For example,
p
√
√ √
−1 = i2 = −1 −1 6= (−1)(−1) = 1 = 1.
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