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Complex numbers: real part, imaginary part, conjugate and modulus

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Read: a number whose square is −1

Try to solve x2 + 1 = 0.

No real number works. Its square is never negative.

So mathematicians made a new kind of number. Let us meet it.

1 · The imaginary number i

From x2 + 1 = 0 we get x2 = −1. So x = ±√−1.

√−1 is not a real number. The book calls it an imaginary number.

We write it as i. It is read as iota.

So i2 = −1.

2 · What is a complex number?

A number of the form z = a + ib is a complex number. Here a and b are real numbers.

Examples from the book: 3 + 4i, 2 − (5/7)i and −7 − 2i.

The set of all complex numbers is written ℂ.

3 · Real part and imaginary part

In z = a + ib, a is the real part. It is written Re z.

b is the imaginary part. It is written Im z.

For z = 3 + 4i: Re z = 3 and Im z = 4.

A non-zero real number times i is imaginary. Examples: 2i, −3i, √5 i, −(11/2)i.

Every real number is a complex number. Its imaginary part is 0.

4 · The conjugate

The conjugate of a + ib is a − ib. It is written z̄.

5 − 4i is the conjugate of 5 + 4i. −2 − 3i is the conjugate of −2 + 3i.

A real number is its own conjugate.

5 · The Argand diagram

Every complex number is one point on the plane. No two numbers share a point.

The x-axis is the real axis. The y-axis is the imaginary axis.

The number x + iy is the point (x, y). This plane is the complex plane.

Book points: A is 3 + 2i, B is −2 + 2i, C is −3 − 2i, D is 2 − 2i.

Argand diagram: the points A(3, 2), B(−2, 2), C(−3, −2) and D(2, −2) on the complex plane, with the real axis across and the imaginary axis up.
Four points from the book, each a complex number: A is 3 + 2i, B is −2 + 2i, C is −3 − 2i, D is 2 − 2i.

6 · The modulus

The modulus of x + iy is √(x2 + y2). It is written |x + iy|.

It is the distance from the origin (0, 0) to the point.

Book example: z = −2 + i gives |z| = √5.

Step 1 / 7

Notes, short questions and MCQs

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

Read: a number whose square is −1

Try to solve x2 + 1 = 0.

No real number works. Its square is never negative.

So mathematicians made a new kind of number. Let us meet it.

1 · The imaginary number i

From x2 + 1 = 0 we get x2 = −1. So x = ±√−1.

√−1 is not a real number. The book calls it an imaginary number.

We write it as i. It is read as iota.

So i2 = −1.

2 · What is a complex number?

A number of the form z = a + ib is a complex number. Here a and b are real numbers.

Examples from the book: 3 + 4i, 2 − (5/7)i and −7 − 2i.

The set of all complex numbers is written ℂ.

3 · Real part and imaginary part

In z = a + ib, a is the real part. It is written Re z.

b is the imaginary part. It is written Im z.

For z = 3 + 4i: Re z = 3 and Im z = 4.

A non-zero real number times i is imaginary. Examples: 2i, −3i, √5 i, −(11/2)i.

Every real number is a complex number. Its imaginary part is 0.

4 · The conjugate

The conjugate of a + ib is a − ib. It is written z̄.

5 − 4i is the conjugate of 5 + 4i. −2 − 3i is the conjugate of −2 + 3i.

A real number is its own conjugate.

5 · The Argand diagram

Every complex number is one point on the plane. No two numbers share a point.

The x-axis is the real axis. The y-axis is the imaginary axis.

The number x + iy is the point (x, y). This plane is the complex plane.

Book points: A is 3 + 2i, B is −2 + 2i, C is −3 − 2i, D is 2 − 2i.

Argand diagram: the points A(3, 2), B(−2, 2), C(−3, −2) and D(2, −2) on the complex plane, with the real axis across and the imaginary axis up.
Four points from the book, each a complex number: A is 3 + 2i, B is −2 + 2i, C is −3 − 2i, D is 2 − 2i.

6 · The modulus

The modulus of x + iy is √(x2 + y2). It is written |x + iy|.

It is the distance from the origin (0, 0) to the point.

Book example: z = −2 + i gives |z| = √5.

Key terms

Imaginary number
A number like 2i or −3i: a non-zero real number times i.
Real part and imaginary part
In a + ib, the real part is a and the imaginary part is b. Note: b is a real number, without the i.
Conjugate
Change the sign of the imaginary part: a + ib becomes a − ib.
Complex plane
The plane with a real axis and an imaginary axis. The diagram is the Argand diagram.
Modulus
The distance of the point from the origin: |x + iy| = √(x2 + y2).

Short questions with model answers

  1. Q1. Write the real part and imaginary part of 3 + 4i.

      It has the form a + ib with a = 3 and b = 4. So Re z = 3 and Im z = 4.

    • Q2. Find |z| if z = (1 + 2i)2 / (2 − i).

        First z = −2 + i (see the solver). Then |z|2 = (−2)2 + 12 = 5. So |z| = √5.

      Common mistakes

      • ✗ “In 3 + 4i the imaginary part is 4i.”

        ✓ The book says Im z = 4. The imaginary part is the real number 4.

      • ✗ “(1 + 2i)2 = 1 + 4i + 4.”

        ✓ 4i2 = −4, not +4. So (1 + 2i)2 = −3 + 4i.

      • ✗ “The conjugate of −2 + 3i is 2 − 3i.”

        ✓ Only the sign of the imaginary part changes. The conjugate is −2 − 3i.

      MCQs

      1. 1. What is i2?

        1. (a) 1
        2. (b) −1
        3. (c) i
        4. (d) 0
        Show answer

        (b) The book defines i as √−1, so i2 = −1.

      2. 2. For z = 3 + 4i, what is Im z?

        1. (a) 3
        2. (b) 4i
        3. (c) 4
        4. (d) 7
        Show answer

        (c) Im z is the number b in a + ib. Here b = 4.

      3. 3. What is the conjugate of 5 + 4i?

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

        (b) The book gives 5 − 4i as the conjugate of 5 + 4i.

      4. 4. Which point is the number −3 − 2i on the Argand diagram?

        1. (a) (3, 2)
        2. (b) (−2, 2)
        3. (c) (−3, −2)
        4. (d) (2, −2)
        Show answer

        (c) x + iy is the point (x, y). Here x = −3 and y = −2. The book calls this point C.

      5. 5. What is |3 + 4i|?

        1. (a) 7
        2. (b) 5
        3. (c) 25
        4. (d) 1
        Show answer

        (b) |3 + 4i| = √(32 + 42) = √25 = 5.

      6. 6. Which statement is true?

        1. (a) Every real number is a complex number.
        2. (b) No real number is a complex number.
        3. (c) i is a real number.
        4. (d) A real number has no conjugate.
        Show answer

        (a) The book's note: every real number is a complex number with 0 as its imaginary part.

      Quick revision

      • A complex number is a + ib with real a, b and i2 = −1.
      • The conjugate flips the sign of the imaginary part.
      • The modulus √(x2 + y2) is the distance from the origin.

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