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Scalar multiplication, addition and subtraction

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Like terms, but in two lanes

You already add 3x + 5x but leave 3x + 5y alone. Complex numbers work the same way. Real parts join real parts, and i parts join i parts.

A real number k scales both parts. On the Argand plane, 2z is twice as long, ½z is half as long, and −z points the opposite way (Fig. 1.3–1.5).

The three rules

Scalar: kz = (kx) + (ky)i, for real k.

Addition: z1 + z2 = (x1 + x2) + i(y1 + y2).

Subtraction: z1 − z2 = z1 + (−z2) = (x1 − x2) + i(y1 − y2).

The picture (Fig. 1.6–1.7)

z1 + z2 is the diagonal of the parallelogram made by z1 and z2.

Your book adds that Rafael Bombelli (1526–1572) first set out these rules.

Add or subtract the real parts, then the imaginary parts. Subtracting z2 means adding −z2.

Step 1 / 7

Notes, short questions and MCQs

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

Like terms, but in two lanes

You already add 3x + 5x but leave 3x + 5y alone. Complex numbers work the same way. Real parts join real parts, and i parts join i parts.

A real number k scales both parts. On the Argand plane, 2z is twice as long, ½z is half as long, and −z points the opposite way (Fig. 1.3–1.5).

The three rules

Scalar: kz = (kx) + (ky)i, for real k.

Addition: z1 + z2 = (x1 + x2) + i(y1 + y2).

Subtraction: z1 − z2 = z1 + (−z2) = (x1 − x2) + i(y1 − y2).

The picture (Fig. 1.6–1.7)

z1 + z2 is the diagonal of the parallelogram made by z1 and z2.

Your book adds that Rafael Bombelli (1526–1572) first set out these rules.

Add or subtract the real parts, then the imaginary parts. Subtracting z2 means adding −z2.

Key terms

Scalar multiple (kz)
A real number k stretches, shrinks or flips z.
Sum (z1 + z2)
Add matching parts.
Difference (z1 − z2)
Change both signs of z2, then add.
Parallelogram law
The sum is the diagonal from the origin.

Short questions with model answers

  1. Q1. Example 3 (p. 5): add (3 + 4i) and (5 − 2i).

    • = (3 + 5) + (4 − 2)i

    8 + 2i

  2. Q2. Example 4(i) (p. 5): simplify (8 + 2i) − (5 − 6i).

    • = 8 + 2i − 5 + 6i
    • = (8 − 5) + (2 + 6)i

    3 + 8i

Common mistakes

  • ✗ (8 + 2i) − (5 − 6i) = 3 − 4i.

    ✓ The minus also flips −6i to +6i, giving 3 + 8i.

  • ✗ 3 + 4i + 5 − 2i = 10i.

    ✓ Real and imaginary parts never combine: 8 + 2i.

  • ✗ 2(3 − 4i) = 6 − 4i.

    ✓ k multiplies both parts: 6 − 8i.

MCQs

  1. 1. (16 − 3i) + (9 + 2i) = (Exercise 1.2 Q1(ii))

    1. (a) 25 + 5i
    2. (b) 7 − 5i
    3. (c) 25 − i
    4. (d) 24i
    Show answer

    (c) 16 + 9 = 25 and −3 + 2 = −1.

  2. 2. (9 − 2i) − (7 − 3i) = (Exercise 1.2 Q1(iii))

    1. (a) 2 + i
    2. (b) 2 − 5i
    3. (c) 16 − 5i
    4. (d) 2 − i
    Show answer

    (a) 9 − 7 = 2 and −2 + 3 = 1.

  3. 3. For z = 3 − 4i, 2z is:

    1. (a) 6 − 4i
    2. (b) 5 − 2i
    3. (c) 3 − 8i
    4. (d) 6 − 8i
    Show answer

    (d) kz = (kx) + (ky)i: both parts double.

  4. 4. On the Argand plane, −z is:

    1. (a) z turned to point the opposite way
    2. (b) z reflected in the real axis
    3. (c) Half as long as z
    4. (d) Always on the imaginary axis
    Show answer

    (a) k = −1 in Fig. 1.5 (p. 4).

  5. 5. (4 − 3i) − (2 − 5i) = (Example 4(ii))

    1. (a) 2 − 8i
    2. (b) 2 + 2i
    3. (c) 6 − 8i
    4. (d) 2 − 2i
    Show answer

    (b) 4 − 2 = 2 and −3 + 5 = 2 (p. 6).

Quick revision

  • kz multiplies both parts by the real number k.
  • Add real with real and imaginary with imaginary; the sum is a parallelogram diagonal.
  • z1 − z2 = z1 + (−z2): flip both signs of z2 first.

Before you start

What is Class 10 Maths Scalar multiplication, addition and subtraction?
Scalar multiplication, addition and subtraction is the open Class 10 Maths lesson for Matric Part 2, Punjab board. It is a step-by-step lesson, not a PDF of the book.
Which boards is this Class 10 Maths lesson for?
The same Punjab textbook: Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Sargodha, Bahawalpur, Dera Ghazi Khan, Sahiwal.
Can I study this in Urdu?
Yes. English, Urdu or Roman Urdu. Science terms stay as the exam writes them.
Is this a notes PDF?
No. The book stays with you. This page is the lesson and the check.

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