| กำหนดให้ |
z1 |
= |
a + bi |
| |
z2 |
= |
c + di |
| จะได้ |
z1z2 |
= |
(a + bi)(c + di) |
| |
|
= |
a(c + di) + (bi)(c +di) |
| |
|
= |
ac + adi + bic + bidi |
| |
|
= |
ac + adi + bci + bdi2 |
| |
|
= |
ac + adi + bci + bd(-1) |
| |
|
= |
ac - bd + adi + bci |
| |
|
= |
(ac - bd) + (adi + bci) |
| ดังนั้น |
z1z2 |
= |
(ac - bd) + (ad + bc)i |
|
|
| |
และ |
z2 |
= |
c + di |
| |
จะได้ว่า |
z1z2 |
= |
(ac - bd) + (ad + bc)i |
|
|
 |
| |
จงหาค่าของ (4 + 3i)(5 + 6i) |
|
| |
วิธีที่ 1 ใช้สูตร |
|
|
|
| |
|
(4 + 3i)(5 + 6i) |
= |
[(4)(5) - (3)(6)] + [(4)(6) + (3)(5)]i |
|
|
| |
|
|
= |
(20 - 18) + (24 + 15)i |
|
|
| |
|
|
= |
2 + 39i |
|
|
| |
วิธีที่ 2 ใช้วิธีคูณกระจาย |
|
|
|
| |
|
(4 + 3i)(5 + 6i) |
= |
(4)(5) + (4)(6i) + (3i)(5) + (3i)(6i) |
|
|
| |
|
|
= |
20 + 24i + 15i + 18i2 |
|
|
| |
|
|
= |
20 + 24i + 15i + 18(-1) |
|
|
| |
|
|
= |
20 + 39i - 18 |
|
|
| |
|
|
= |
2 + 39i |
|
|
|
|
| |
จงหาค่าของ (3 + 2i)(3 - 2i) |
|
| |
วิธีที่ 1 ใช้สูตร |
|
|
|
| |
|
(3 + 2i)(3 - 2i) |
= |
[(3)(3) - (2)(-2)] + [(3)(-2) + (2)(3)]i |
|
|
| |
|
|
= |
(9 + 4) + (-6 + 6)i |
|
|
| |
|
|
= |
13 + 0i |
|
|
| |
|
|
= |
13 |
|
|
| |
วิธีที่ 2 ใช้วิธีคูณกระจาย |
|
|
|
| |
|
(3 + 2i)(3 - 2i) |
= |
(3)(3) + (3)(-2i) + (2i)(3) + (2i)(-2i) |
|
|
| |
|
|
= |
9 - 6i + 6i + (-4)i2 |
|
|
| |
|
|
= |
9 + (-4)(-1) |
|
|
| |
|
|
= |
9 + 4 |
|
|
| |
|
|
= |
13 |
|
|
|
|
| |
กำหนดให้ z1 = 2 + 3i, z2 = 6 + i, z3 = -2i และ z4 = -1 - 4i จงหาค่าของ |
| |
| 1. 2(z1 + z2) |
2. 3z1 + 4z3 |
3. 3z1z4 - 4z2 |
4. |
 |
|
|
| |
1. |
2(z1 + z2) |
= |
2[(2 + 3i) + (6 + i)] |
|
|
| |
|
|
= |
2[(2 + 6) + (3 + 1)i] |
|
|
| |
|
|
= |
2(8 + 4i) |
|
|
| |
|
|
= |
16 + 8i |
|
|
| |
2. |
3z1 + 4z3 |
= |
3(2 + 3i) + 4(-2i) |
|
|
| |
|
|
= |
6 + 9i - 8i |
|
|
| |
|
|
= |
6 + i |
|
|
| |
3. |
3z1z4 - 4z2 |
= |
3[(2 + 3i)(-1 - 4i)] - 4(6 + i) |
|
|
| |
|
|
= |
3[(2)(-1) + 2(-4i) + (3i)(-1) + (3i)(-4i)] - 24 - 4i |
|
|
| |
|
|
= |
3[-2 - 8i - 3i - 12i2] - 24 - 4i |
|
|
| |
|
|
= |
3[-2 - 11i - 12(-1)] - 24 - 4i |
|
|
| |
|
|
= |
3[-2 - 11i + 12] - 24 - 4i |
|
|
| |
|
|
= |
3(10 - 11i) - 24 - 4i |
|
|
| |
|
|
= |
30 - 33i - 24 - 4i |
|
|
| |
|
|
= |
6 - 37i |
|
|
| |
4. |
|
= |
 |
|
|
| |
|
|
= |
i - 1 - 4i |
|
|
| |
|
|
= |
-1 - 3i |
|
|
|
|
|
