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Equality, square roots and factors

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Equality, square roots and factors

Suppose A = B and both are complex.

How many real facts do we get? Two.

Let us see why that is so useful.

1 · When are two complex numbers equal?

a + bi = c + di if and only if a = c and b = d.

So one complex equation gives two real equations.

Book Example 4 ends with x = 3 and y = 2.

2 · Square root of a complex number

We want p + iq with (p + iq)2 = x + iy.

Comparing parts gives x = p2 − q2 and y = 2pq.

The book's formula (v) uses the modulus |z| = √(x2 + y2).

Book Example 5: 5 + 12i. Here |z| = 13.

p = √((13 + 5)/2) = 3 and q = √((13 − 5)/2) = 2.

The square roots are 3 + 2i and −3 − 2i.

3 · Polynomials and linear factors

A complex polynomial has complex coefficients. Example: (1 − i)z + 3i.

Fundamental theorem of algebra (book): a polynomial of degree n ≥ 1 has exactly n roots in ℂ.

So P(z) = a(z − z1)(z − z2)…(z − zn).

Example: z2 + 4 = (z + 2i)(z − 2i). No real factors exist, but complex ones do.

Book Example 6: z2 + (i − 3)z − 3i = (z + i)(z − 3).

Book Example 9: z3 − 3z2 + z + 5 = (z + 1)(z − 2 − i)(z − 2 + i).

Book Example 10 (completing the square): 2z2 − 12z + 50 = 0 gives z = 3 ± 4i.

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Notes, short questions and MCQs

Read the full notes: key terms, model answers and MCQs with answers

Equality, square roots and factors

Suppose A = B and both are complex.

How many real facts do we get? Two.

Let us see why that is so useful.

1 · When are two complex numbers equal?

a + bi = c + di if and only if a = c and b = d.

So one complex equation gives two real equations.

Book Example 4 ends with x = 3 and y = 2.

2 · Square root of a complex number

We want p + iq with (p + iq)2 = x + iy.

Comparing parts gives x = p2 − q2 and y = 2pq.

The book's formula (v) uses the modulus |z| = √(x2 + y2).

Book Example 5: 5 + 12i. Here |z| = 13.

p = √((13 + 5)/2) = 3 and q = √((13 − 5)/2) = 2.

The square roots are 3 + 2i and −3 − 2i.

3 · Polynomials and linear factors

A complex polynomial has complex coefficients. Example: (1 − i)z + 3i.

Fundamental theorem of algebra (book): a polynomial of degree n ≥ 1 has exactly n roots in ℂ.

So P(z) = a(z − z1)(z − z2)…(z − zn).

Example: z2 + 4 = (z + 2i)(z − 2i). No real factors exist, but complex ones do.

Book Example 6: z2 + (i − 3)z − 3i = (z + i)(z − 3).

Book Example 9: z3 − 3z2 + z + 5 = (z + 1)(z − 2 − i)(z − 2 + i).

Book Example 10 (completing the square): 2z2 − 12z + 50 = 0 gives z = 3 ± 4i.

Key terms

Equal complex numbers
Same real part and same imaginary part.
Square root of z
A number w with w2 = z. Every non-zero complex number has two, one the negative of the other.
Root of a polynomial
A value of z that makes P(z) = 0. Also called a zero of P.
Linear factor
A factor of the form z − z1.

Short questions with model answers

  1. Q1. Find the square root of 5 + 12i.

      Here x = 5, y = 12, so |z| = 13. √((13 + 5)/2) = 3 and √((13 − 5)/2) = 2. The roots are 3 + 2i and −3 − 2i. Check: (3 + 2i)2 = 5 + 12i.

    • Q2. Factorise z2 + 4 over ℂ.

        z2 + 4 = 0 gives z2 = −4, so z = ±2i. Then z2 + 4 = (z + 2i)(z − 2i). Check: (z + 2i)(z − 2i) = z2 − 4i2 = z2 + 4.

      Common mistakes

      • ✗ “The square root of 5 + 12i is only 3 + 2i.”

        ✓ There are two: 3 + 2i and −3 − 2i.

      • ✗ “From (3x − 2y) + (2x + 3y)i = 5 + 12i we get 3x − 2y + 2x + 3y = 17.”

        ✓ Do not mix real and imaginary parts. Write 3x − 2y = 5 and 2x + 3y = 12 separately.

      MCQs

      1. 1. What is the square root of 5 + 12i that has a positive real part?

        1. (a) 3 + 2i
        2. (b) 2 + 3i
        3. (c) 6 + 6i
        4. (d) 5 + 12i
        Show answer

        (a) (3 + 2i)2 = 9 + 12i − 4 = 5 + 12i.

      2. 2. What is |5 + 12i|?

        1. (a) 17
        2. (b) 13
        3. (c) 7
        4. (d) 60
        Show answer

        (b) √(52 + 122) = √169 = 13.

      3. 3. If a + bi = c + di, then:

        1. (a) a = c and b = d
        2. (b) a = d and b = c
        3. (c) a + b = c + d only
        4. (d) a = b
        Show answer

        (a) Book section 1.2: equal when real parts and imaginary parts are equal.

      4. 4. How many roots does a polynomial of degree 5 have in ℂ?

        1. (a) 1
        2. (b) 4
        3. (c) 5
        4. (d) 10
        Show answer

        (c) The book: degree n ≥ 1 means exactly n roots in ℂ.

      5. 5. Factorise z2 + 4 over ℂ.

        1. (a) (z + 2)(z − 2)
        2. (b) (z + 2i)(z − 2i)
        3. (c) (z + 4i)(z − 4i)
        4. (d) It cannot be factorised.
        Show answer

        (b) The book: z2 + 4 = (z + 2i)(z − 2i).

      Quick revision

      • Equal complex numbers have equal real parts and equal imaginary parts.
      • Every non-zero complex number has two square roots.
      • Over ℂ, a polynomial of degree n splits into n linear factors.

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