1 Numbers and Arithmetic Flashcards

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Q

Definition 1.1:
a|b

A

Let a, b ∈ Z. a divides b, written a | b, if there exists n∈Z such that an=b.

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Definition 1.1:
Prime numbers

A

For p∈N, p is prime if:
- p≥2
- ∀a∈N, a|p ⇒ a=1or a=p.

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Lemma 1.3:
1. if a|b and b|a
1. if m|a and m|b

A

Let a, b, m ∈ Z.
1. If a|b and b|a, then a=±b.
1. If m|a and m|b, then m|λa+μb for all λ,μ∈Z.

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Proposition 1.4:
Product of primes

A

Every integer >1 is a product of primes,
i.e., each m∈Z satisfying m≥2 can be written as
m = Πᵏᵢ₌₁ (pᵢ) = p₁ p₂ … pₖ
for some k ≥ 1, where p₁,…,pₖ are primes that need not be distinct.

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Q

Definition 1.6:
lcm

A

Let m,n∈N.
lcm(m,n) is the smallest a∈N such that m|a and n|a.

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Q

Definition 1.6:
gcd

A

Let m,n∈N.
gcd(m,n) is the largest a∈N such that a|m and a|n.

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Q

Definition 1.6
coprime

A

If gcd(m,n) = 1.
m,n∈N

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Q

Remark 1.7:
gcd of m/g and n/g

A

Let g := gcd(m,n). We claim that m/g and n/g are coprime.

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9
Q

Theorem 1.9
Euclid’s theorem

A

There are infinitely many primes.

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10
Q

Proposition 1.10:
Bézout identity

A

For a, b ∈ N, let g be the smallest positive integer
in the set S = { λa + μb | λ, μ ∈ Z}.
Then g = gcd(a, b).

i.e. gcd(a, b) = λa + μb for some λ, μ ∈ Z.

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Lemma 1.14:
Computing all Bézout identities

A

For a, b ∈ N, write g = gcd(a, b).
Suppose that λ, μ ∈ Z give a particular Bézout identity g = λa + μb.
Then every Bézout identity is of the form g = (λ + α)a + (μ − β)b for some α, β ∈ Z, where αa = βb.

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Q

Lemma 1.16:
Euclid’s Lemma

A

Let a, b ∈ Z and p be a prime. Then p|ab ⇒ (p|a or p|b).
More generally p|(a₁… aₖ) ⇒ p|aᵢ for some 1 ≤ i ≤ k.

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Q

Theorem 1.17:
Fundamental Theorem of Arithmetic

A

Every integer >1 can be written uniquely (upto the choice of order) as a product of primes.
i.e., each m∈Z satisfying m≥2 can be written as
m = Πᵏᵢ₌₁ (pᵢ) = p₁ p₂ … pₖ
for some k ≥ 1, where the primes p₁,…,pₖ are uniquely determined by m.

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Q

Lemma 1.19:
gcd and lcm

A

Consider d, m, n ∈ N. Then:
1. gcd(m,n) · lcm(m,n) = m·n
1. m|d and n|d iff lcm(m,n)|d.

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Q

Definition 1.21:
Congruence modulo n

A

For n∈N and a,b∈Z,
a and b are congruent modulo n iff n|(a−b).
So a≡b (mod n).

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Q

Lemma 1.22:
Modulo

A

For a ∈ Z, there exists r satisfying 0 ≤ r < n such that a ≡ r (mod n).

i.e. every int is congruent modulo n to an integer r in the range 0≤r<n.

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Q

Definition 1.23:
Congruence classes

A

Let 𝑛∈ℕ. The congruence classes modulo 𝑛
are the 𝑛 distinct sets [0], [1], …, [n-1] into which ℤ is ‘partitioned’ by the ‘congruence modulo n’ relation.
We write ℤ𝑛={ [𝑎] ∣ 𝑎∈ℤ } = { [0],[1],…,[𝑛−1] } for the set of all congruences classes modulo n.

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Q

Lemma 1.24:
Arithmetic in Zn

A

There are well-defined sum and product operations on Zn given by
* [a]+[b]=[a+b]
* [a]·[b]=[a·b].

Sum and product are both associative and commutative in Zn.

[0] is the additive identity, [1] is the multiplicative identity.

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Q

Proposition 1.26:
Multiplicative inverse

A

Let n ∈ N and m ∈ Z. Then [m] ∈ Zn has a multiplicative inverse if and only if m and n are coprime.

20
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Corollary 1.27:
Property of Zp if p is prime

A

If p is prime, then each of [1], [2], . . . , [p−1] have a multiplicative inverse in Zp.

Every nonzero congruence class in Zp has a multiplicative inverse.

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Corollary 1.28:
Multiplication by a coprime element is invertible

A

If m, n ∈ Z are coprime, then the map Mₘ : Zₙ → Zₙ Mₘ([a]) = [a · m] has an inverse map.

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Q

Theorem 1.29:
Fermat’s Little Theorem

A

If p is prime, then aᵖ ≡ a (mod p) for
all a ∈ Z.
Furthermore if p does not divide a (a and p are coprime), then aᵖ⁻¹ ≡ 1 (mod p).

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Definition 1.31:
Congruence equation

A

Let 𝑛∈ℕ and 𝑎,𝑏∈ℤ.
Any statement of the form ‘𝑎𝑥≡𝑏 (mod 𝑛)’ is a congruence equation.
Solving any such equation means finding all 𝑥∈ℤ that satisfy the relation, or equivalently, solving [𝑎][𝑥]=[𝑏] in ℤ𝑛.

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Q

Proposition 1.33:
Solving congruences

A

For n ∈ N and a ∈ Z, let g = gcd(a,n). Then
ax≡b (mod n) has a solution iff g|b,
in which case, the congruence reduces to a/g x ≡ b/g (mod n/g), where a/g and n/g are coprime.

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Theorem 1.36: **Chinese Remainder Theorem**
Let m, n ∈ N be coprime. For any a, b ∈ Z, the congruence equations * x ≡ a (mod m), * x ≡ b (mod n) have a unique solution (mod mn).
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Definition 1.38: **Group of units**
Let 𝑛∈ℕ. Group of units mod n: ℤ𝑛* = { [𝑎]∈ℤ𝑛 ∣ [𝑎] is invertible}. Equivalently: ℤ𝑛={[𝑎] ∣ 1 ≤ 𝑎 ≤ 𝑛−1 and gcd(𝑎,𝑛)=1}.
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Definition 1.38: **Euler’s 𝜙-function**
Let 𝑛∈ℕ. It is the function 𝜙:ℕ→ℕ, 𝜙(𝑛) = * number of integers 𝑎∈{1,…,𝑛−1} where gcd(𝑎,𝑛)=1. * number of elements in ℤ𝑛*. | Counts the number of elements in the group of units mod n. ## Footnote 𝜙(1)=1.
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Lemma 1.41: **The group of units is a group**
If [a], [b] ∈ Zₙ* , then [ab] ∈ Zₙ* .
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Theorem 1.42: **Euler's** generalisation of Fermat's Little Theorem
If n ∈ N and a ∈ Z is coprime to n, then a^φ(n) ≡ 1 (mod n). | **a** to the power of **φ(n)**
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Definition 1.43: **Sum of complex numbers**
(𝑎+𝑖𝑏) + (𝑥+𝑖𝑦) = (𝑎+𝑥) + 𝑖(𝑏+𝑦)
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Definition 1.43: **Product of complex numbers**
(𝑎+𝑖𝑏) (𝑥+𝑖𝑦) = 𝑎𝑥 + 𝑖(𝑏𝑥+𝑎𝑦) + 𝑖2𝑏𝑦 = (𝑎𝑥−𝑏𝑦) + 𝑖(𝑏𝑥+𝑎𝑦)
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Definition 1.45: **Complex conjugate, real and imaginary parts, modulus**
Let 𝑧=𝑥+𝑖𝑦∈ℂ. * **z̄** =𝑥−𝑖𝑦 is the **complex conjugate** (reflection in the real axis). * **ℜ(𝑧)** := 𝑥 is the **real part** of 𝑧 * **ℑ(𝑧)** :=𝑦 is the **imaginary part** of 𝑧 * **|𝑧|**= sqrt(𝑥²+𝑦²) is the **modulus** (or magnitude) of 𝑧
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Lemma 1.47: Arithmetic with complex conjugate (4)
1. Re(z) = (z+z̄)/2, Im(z) = (z−z̄)/2i 2. complexconj(z₁+z₂) = z̄₁+z̄₂, complexconj(z₁·z₂) = z̄₁·z̄₂ 3. |z|² = z·z̄ 4. |z₁z₂| = |z₁|·|z₂|
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Definition 1.48: **Polar form**
The polar form of any nonzero 𝑧∈ℂ is **𝑧=𝑟𝑒ⁱᶿ=𝑟(cos𝜃+𝑖sin𝜃)**, where 𝑟=|𝑧| is the modulus of 𝑧 and 𝜃=arg(𝑧).
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Proposition 1.51: **The unit circle is a group**
The unit circle S¹ = { w∈C | |w|=1 } is **closed under complex multiplication**; more explicitly, eⁱᶿ¹ · eⁱᶿ² = eⁱ⁽ᶿ¹⁺ᶿ²⁾. ## Footnote Closed under complex multiplication: The product of two complex numbers that lie on the unit circle is a complex number on the unit circle.
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Comment 1.51.5 effect of multiplying complex numbers
For any nonzero w = reⁱᶿ, z = seⁱᵝ ∈ C, multiplication by w is **z ↦ (w·z) = reⁱᶿ·seⁱᵝ = (rs)eⁱ⁽ᶿ⁺ᵝ⁾**, i.e. a rotation by arg(w) = θ and scaling by |w|= r.
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Corollary 1.52: **de Moivre's Theorem**
For n ∈ Z, we have (eⁱᶿ)ⁿ = eⁱⁿᶿ, i.e. (cosθ+isinθ)ⁿ = cos(nθ)+isin(nθ).
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Definition 1.53: **Roots of unity**
Let 𝑛∈ℕ. An 𝑛th root of unity is any complex number 𝑧∈ℂ that satisfies 𝑧^𝑛=1.
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Proposition 1.54: Roots of unity: zⁿ-1
The nth roots of unity are precisely z = e^(2πik/n) for 0 ≤ k ≤ n − 1. In particular, zⁿ − 1 = Πₖ₌₀ⁿ⁻¹ ( z − e^(2πik/n) ).
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Theorem 1.55 / Corollary 1.56: **Fundamental Theorem of Algebra**
Every polynomial f(z) in one variable with coefficients in C has a root in C, that is, a solution to f(z) = 0. Basically: Every polynomial of degree n with coefficients in C has precisely n roots in C if we count roots ‘with multiplicity’. i.e. f(z) is a product of n linear factors: * if f(z) = aₙzⁿ +...+ a₁z + a₀ with aⱼ ∈C and aₙ≠0, then * f(z)=aₙ Πⁿᵢ₌₁ (z−αᵢ), for some αᵢ ∈ C.
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Definition 1.57: **Order of a root of unity**
If 𝜔∈ℂ is a root of unity (i.e. 𝜔ⁿ=1 for some 𝑛∈ℕ) then we say 𝜔 has order 𝑚, denoted ord(𝜔)=𝑚, if **𝑚=min{ 𝑛∈ℕ ∣ 𝑤ⁿ=1 }**. In this case, we call 𝜔 a primitive 𝑚th root of unity.
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Lemma 1.58: Order of a root of unity and its divisibility
If ωⁿ = 1 for some ω ∈ C and n ∈ N, then ord(ω) | n.