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PROOF 89

Prove.

The product of a complex number and its conjugate is real

On the complex numbers ℂ.

What you may use

A1a(b + c) = ab + ac
distributive law
A2(ab)c = a(bc)
associativity of multiplication
A3ab = ba
commutativity of multiplication
A4a × 1 = a
multiplicative identity
A5(a + b) + c = a + (b + c)
associativity of addition
A6a + b = b + a
commutativity of addition
A7a + 0 = a
additive identity
A8a + (−a) = 0
additive inverse
A9a × 0 = 0
lemma of Proof 01
A10a(−b) = −(ab)
lemma of Proof 14
A11(−a)(−b) = ab
product of negatives (Proof 01)
A12i × i = −1
definition of i (Proof 14)
A13a ≠ 0 ⟹ there exists b with ab = 1
multiplicative inverse of reals
A140 ≤ a × a
the square of a real is nonnegative
A15a ≠ 0 ⟹ 0 < a × a
lemma of Proof 10
A160<a, 0≤b ⟹ 0 < a + b
order and addition
A17exactly one of a<b, a=b, b<a
trichotomy of order
A18p, q ∈ ℝ ⟹ p+q, pq ∈ ℝ
closure of the reals
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For z = a + bi with a, b real, z̄ = a + (−b)i is called the complex conjugate of z.

Proof.

Related labs

Complex Numbers lab →