BORING EDUCATION — LIVE EXPERIMENTProduct is live. Experiments are running. Feedback is being collected. Improvements are underway.Welcome to Boring Education. We're glad you're here while we're figuring out how to make learning better.
BORINGEDUCATION

1.1.2-1.1.4Free

Operations on complex numbers

Your guide: Sir AhmedSays brackets have saved more marks than any calculator.

Operations on complex numbers

You can add 2 + 3i and 4 + i.

But can you divide them?

Yes. One trick makes it easy.

1 · Adding, subtracting and multiplying

Add the real parts. Add the imaginary parts. (a + ib) + (c + id) = (a + c) + i(b + d).

Subtract in the same way: (a + ib) − (c + id) = (a − c) + i(b − d).

For a real number k: k(a + ib) = ka + i kb.

Multiply every term by every term. Then use i2 = −1.

The book gives (a + ib)(c + id) = (ac − bd) + i(ad + bc).

2 · Complex numbers as ordered pairs

The book also writes a + ib as the pair (a, b).

Sum: (a, b) + (c, d) = (a + c, b + d).

Product: (a, b)(c, d) = (ac − bd, ad + bc).

Book Example 1 uses (8, 9) and (5, −6).

Sum (13, 3). Difference (3, 15). Product (94, −3).

3 · Special numbers and the inverse

The additive identity is (0, 0). The inverse of (a, b) is (−a, −b).

The multiplicative identity is (1, 0).

Every non-zero number has a multiplicative inverse: (a/(a2 + b2), −b/(a2 + b2)).

Complex numbers cannot be ordered. We cannot say one is greater than another.

4 · Dividing

To divide, multiply top and bottom by the conjugate of the bottom.

The bottom then becomes a real number.

Book Example 2 gives (4 + 2i) ÷ (3 − i) = 1 + i.

Step 1 / 7

Notes, short questions and MCQs

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

Operations on complex numbers

You can add 2 + 3i and 4 + i.

But can you divide them?

Yes. One trick makes it easy.

1 · Adding, subtracting and multiplying

Add the real parts. Add the imaginary parts. (a + ib) + (c + id) = (a + c) + i(b + d).

Subtract in the same way: (a + ib) − (c + id) = (a − c) + i(b − d).

For a real number k: k(a + ib) = ka + i kb.

Multiply every term by every term. Then use i2 = −1.

The book gives (a + ib)(c + id) = (ac − bd) + i(ad + bc).

2 · Complex numbers as ordered pairs

The book also writes a + ib as the pair (a, b).

Sum: (a, b) + (c, d) = (a + c, b + d).

Product: (a, b)(c, d) = (ac − bd, ad + bc).

Book Example 1 uses (8, 9) and (5, −6).

Sum (13, 3). Difference (3, 15). Product (94, −3).

3 · Special numbers and the inverse

The additive identity is (0, 0). The inverse of (a, b) is (−a, −b).

The multiplicative identity is (1, 0).

Every non-zero number has a multiplicative inverse: (a/(a2 + b2), −b/(a2 + b2)).

Complex numbers cannot be ordered. We cannot say one is greater than another.

4 · Dividing

To divide, multiply top and bottom by the conjugate of the bottom.

The bottom then becomes a real number.

Book Example 2 gives (4 + 2i) ÷ (3 − i) = 1 + i.

Key terms

Additive identity
The number you add and nothing changes. In pairs it is (0, 0).
Multiplicative identity
The number you multiply by and nothing changes. In pairs it is (1, 0).
Multiplicative inverse
The number that gives (1, 0) when multiplied. It exists for every non-zero number.
Ordered pair
Two numbers in a fixed order, like (a, b). (a, b) and (b, a) are different.

Short questions with model answers

  1. Q1. Find the sum, difference and product of (8, 9) and (5, −6).

      Sum: (8 + 5, 9 − 6) = (13, 3). Difference: (8 − 5, 9 + 6) = (3, 15). Product: (40 + 54, −48 + 45) = (94, −3).

    • Q2. Find the multiplicative inverse of (3, 4).

        Use (a/(a2 + b2), −b/(a2 + b2)). Here a2 + b2 = 25. The inverse is (3/25, −4/25). Check: (3 + 4i)(3 − 4i) = 25.

      Common mistakes

      • ✗ “(a, b)(c, d) = (ac, bd).”

        ✓ The book's rule is (ac − bd, ad + bc). For (8, 9)(5, −6) that is (94, −3), not (40, −54).

      • ✗ “(4 + 2i) ÷ (3 − i) is found by dividing the real parts and the imaginary parts.”

        ✓ Use the conjugate of the bottom. The answer is 1 + i.

      MCQs

      1. 1. What is (8, 9) + (5, −6)?

        1. (a) (13, 3)
        2. (b) (3, 15)
        3. (c) (94, −3)
        4. (d) (13, 15)
        Show answer

        (a) Add the parts: (8 + 5, 9 − 6) = (13, 3).

      2. 2. Which pair is the multiplicative identity?

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

        (c) The book says the multiplicative identity is (1, 0).

      3. 3. What is the additive inverse of (a, b)?

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

        (a) The book gives (−a, −b) because (a, b) + (−a, −b) = (0, 0).

      4. 4. What is (4 + 2i) ÷ (3 − i)?

        1. (a) 1 + i
        2. (b) 1 − i
        3. (c) 10 + 10i
        4. (d) 2 + i
        Show answer

        (a) Book Example 2: the answer is 1 + i.

      5. 5. Which statement is true for complex numbers?

        1. (a) One is always greater than another.
        2. (b) They do not satisfy the order axioms.
        3. (c) Only real numbers can be added.
        4. (d) They have no multiplicative identity.
        Show answer

        (b) The book's note: there is no sense in saying one complex number is greater or less than another.

      Quick revision

      • Add and subtract real with real, imaginary with imaginary.
      • Multiply term by term. Use i2 = −1.
      • Divide with the conjugate of the bottom.

      Free right now. No ads. Tell us what to fix.