ชุดการเรียนการสอน รายวิชาคณิตศาสตร์เพิ่มเติม 4 รหัสวิชา ค30204 เรื่อง จำนวนเชิงซ้อนสำหรับนักเรียนชั้นมัธยมศึกษาปีที่ 5 ชุดที่ 1

1. 1
คคคคคคคคคคคคคคคคคคคคค
1. 4 30204 5 12
1.1 1 (a,b) a + bi
1.2 2
1.3 3
1.4 4
1.5 5
1.6 6 2
1.7 7
1.8 8
1.9 9

2. 2
2. 4 30204 1 (a,b) a + bi 5
3.
-
-
-
-
-
-
-
- , ,
-
-
-
4. 1
2
1.10 10
1.11 11
1.12 12 n

3. 3
1.
2. 1
3.
4.
5.
6.
7. 1
8.
9. 1 1
10.
11. ไไไ

4. 4
1.
2. 1
3.
4.
5.
6.
7. 1 1
8.
9. –
คคคคคคคคคคคคคคคค
คคคคคคคค

5. 5
(a,b) a + bi
(a,b) a + bi a, b a (real
part) b (imaginary part)
1.
2. (a,b) a + bi
a + bi (a,b)
3.
4.
5. i i
n
n

6. 6
คคคคคคคคคคคคคคคคคคคคค
คคคคคค 1 คคคคคค คคคคคคคคคคคคคคคคคคคคคคคค
คคค (a,b) คคคค a + bi
1.
2.
3.
-
-
-
4.
2
5.

7. 7
1.
2. 1
3. (a,b) a + bi
4.
5.
6. 1 1
7.
8.
9. –

8. 8
คคคคคคคคค
คคคคคคคคค 1. 1 (a,b) a + bi 10
2. × ไ
, ,
10
1. คคคคคคค
. 6i3(3,6) −=
. 2i1(1,2) +=
. 5i44,5)( −−=−
. 2i62)6,( −=−−
2. คคคคคคคคคค
. 1)2,(i2 −−=−
. (6,2)2i6 =+
. 4)3,(4i3 −−=−−
. 8,5)(5i8 −=+−
3. 3)(2,− bia+
. 3i2+
. 3i2+−
. 3i2−−
. 3i2−
4. 5)(1,Z −=
. 5−
. 0
. 1
. 2
คคคคคคคคคคคคคคคคค
คคคคคค 1 คคคคคค คคคคคคคค
คคคคคคคคคคคคคคคคคคค (a,b) คคคค a + bi

9. 9
5. i8+ b)(a,
. 1)(8,−
. (8,1)
. 1)8,( −−
. 8,1)(−
6. 3i4Z −=
. 4
. 3
. 4−
. 3−
7. 25i
. i
. 1−
. 1
. i−
8.
. 5
. 2i5+
. i3−
. i2
9. 24i20i16i8i +++
. 0
. 2
. 4
. 8
10. 8)1002(i2i +
. 9−
. 7−
. 5−
. 3−

10. 10
1
(a,b) a + bi
1
(a,b) a + bi
1
2
3
4
5
6
7
8
9
10
................................
...
............................... ................................
..................... .......

11. 11
1
(a,b) a + bi
1
(a,b) a + bi
1. ข
2. ข
3. ข
4. ข
5. ข
6. ข
7. ข
8. ข
9. ข
10. ข

12. 12
1. ขขขขขขขขขขขขขขขขขขขขขขขขขขข (a,b) ขขขข a +
bi
a + bi a (real part) b (imaginary part)
ขขขขขขขขขขขขขขขขขขข ข ขขขขขขขขขขขขขขขขข
ขข
ขขขขขข
ขขขขขข
ขข
ขขขขข
(a,b)
ขขขขขข
ขขขขขข
ขข
ขขขขข a
+ bi
ขขขข
ขขขข
ขขขข
ขขขขขข
ข
ขขขขขขขขข
ขขขขขขขขขข
ขขขข
1 (4,0) 4 + 0 i 4 0 (a,b) a +
bi
2 (–8,0) –8 + 0
i
–8 0 a ≠ 0 b
= 0
3 (9,0) 9 + 0 i 9 0
4 (2,7) 2 + 7 i 2 7 (a,b) a +
bi
5 (–4,8) –4 + 8
i
–4 8 a ≠ 0 b
≠ 0
6 (–3, –
5)
–3 – 5
i
–3 –5
7 (0,3) 0 + 3 i 0 3 (a,b) a +
bi
8 (0, –7) 0 – 7 i 0 –7 a = 0
ขขขขขขข (a,b)
a + bi a, b
i
2
= –1 i (0,1)

13. 13
1. (a,b) a + bi a ≠ 0 b = 0
ขขขขขขขขขข a ( 1 – 3)
2. (a,b) a + bi a ≠ 0 b ≠ 0
ขขขขขขขขขขขขข ( 4 – 6)
3. (a,b) a + bi a = 0 b ≠ 0
ขขขขขขขขขขขขขขขข ( 7 – 9)
ขขขขขขขขขขข 1 (a,b) a + bi
1. (2, –4)
2. (1,8)
3. ( 5,–1)
ขขขขขขข 1. (2,–4) = 2 – 4i 2 –4
2. (1,8) = 1 + 8i 1 8
3. ( 5,–1) = 5 – i 5 –1

14. 14
i ร i
ขขขขขขขขขขข 2 a + bi (a,b)
1. 4 + 4i
2. –2i
3. 11
ขขขขขขข 1. 4 + 4i= (4,4) 4 4
2. –2i = (0,–2) 0 –2
3. 11 = ( 11,0) 11 0
i 1−
ขขขขขขขขขขข 3 ขขขขขขขขขขขขขขขขขขขขขขขขขขขขข
ขขขขขขขขขขข
1. 3−
2. 9−
3. 8−
4. 1442 −+
5. 121 −−−
ขขขขขขข 1. 3− = 13 − = i3
2. 9− = 19 − = 3i
3. 8− = 18 − = i22
4. 1442 −+ = 11442 −+ = 12i2+
ขขขขขขข a ia1aa =−=−

15. 15
2. ขขขขขขขข i
n
ขขขขข n ขขขขขขขขขขขขขขขขข
ขขข
ขขข
i ขขขขขข
ขขขขข
ขขขขขข
ขขขขขข
i
1 i i
2 i
2 –1
3 i
3 –i
4 i
4 1
5 i
5 i
6 i
6 –1
7 i
7 –i
8 i
8 1
9 i
9 i
10 i
10 –1
11 i
11 –i
12 i
12 1
13 i
13 i
i
n
1 ,–1 ,i ,–i 4 i i
n







=
04n1
34ni-
24n1-
14ni
ni

16. 16
ขขขขขขขขขขข 1 i
136
i
2345
ขขขขขขข i
136
= 1 (4 136 0 )
i
2345
= i (4 2345 1 )
ขขขขขขขขขขข 2
1. 20i18i15i9i +−+
2. )49i28(i36i −
ขขขขขขข ni
1. 20i18i15i9i +−+ = 11)(i)(i +−−−+
= 11ii ++−
= 2
ขขข 20i18i15i9i +−+ 2
2. )49i28(i36i − = i)1(1−
= i1−
ขขข )49i28(i36i − i1−

17. 17
1.
ขขข
ข
ขข
ขข
ขขขขขข
ขขขขขข
ขข
ขขขขขขขข
ขขขข
ขขขขข a +
bi
ขขข
ขขข
ขขขขข
ขขขขข
ขขขข
ขขขขขขขข
ขขขข
ขขขขข
(a,b)
1.1 (6,2) 1.4 5 +
12i
1.2 (–3,4) 1.5 –3 +
6i
1.3 (7, –
12)
1.6 –4 –
5i
2.
ขขข
ขขข
ขขขขขขขข
ขขขขขขข
ขขขขขขขข
ขขขข
ขขขขขขข
ขขขขขขขขข
ขขขขขขขขข
ขขขขข
2.1 12 – 5i
2.2 (–7,2)
2.3 –12 + 0i
2.4 0 + 9i
3. i
n
3.1 i
28
= ..……………………………..
1.11.1
ขขขขขขขขข

18. 18
Ùß ÝÜĞćĊĚĒǰǰĔĀšîĆÖđøĊ÷îđêĉöÙĞćêĂï ĔîĒêŠúą×šĂêŠĂĕðîĊĚĔĀšëĎÖêšĂÜǰ
1.
ขขข
ข
ขข
ขข
ขขขขขข
ขขขขขข
ขข
ขขขขขขขข
ขขขข
ขขขขข a +
bi
ขขข
ขขข
ขขขขข
ขขขขข
ขขขข
ขขขขขขขข
ขขขข
ขขขขข
(a,b)
1.1 (6,2) 6 + 2i 1.4 5 +
12i
(5,12)
1.2 (–3,4) –3 + 4i 1.5 –3 +
6i
(–3,6)
1.3 (7, –
12)
7 – 12i 1.6 –4 –
5i
(–4, –5)
2.
ขขข
ขขข
ขขขขขขขข
ขขขขขขข
ขขขขขขขข
ขขขข
ขขขขขขข
ขขขขขขขขข
ขขขขขขขขขข
ขขขข
2.1 12 – 5i 12 –5 จจจจจจจ
จจจจจจจ
2.2 (–7,2) –7 2 จจจจจจจ
จจจจจจจ
2.3 –12 + 0i –12 0 จจจจจจจ
จจจ
2.4 0 + 9i 0 9 จจจจจจจ
จจจจจจจ
จจจ
1.11.1

19. 19
ขขขขขขขขข () ()
1.21.2
...........1. 20i –1
...........2. 19i –i
...........3. 9 3i
...........4. 4− –2
...........5. 25− 5i
...........6. 2− i2
...........7. 18i17i16i15i +++ 1
...........8. )3i5(i20i + 0
...........9. 40i30i20i10i ⋅⋅⋅ 1
...........10. 4)(3,− bia+ 3 – 4i
...........11. 1)(5,− bia+ 5 + i
...........12. ,2)2( bia+ 2i2+
...........13. 8i2− b)(a, 8)(2,−
...........14. i53−− b)(a, )5,3( −−
...........15. bi 0b≠

20. 20
ขขขขขขขขข () ()
1.21.2
..... ..... 1. 20i –1
..... ..... 2. 19i –i
.....  ..... 3. 9 3i
.....  ….. 4. 4− –2
.....  ..... 5. 25− 5i
.....  ..... 6. 2− i2
.....  ….. 7. 18i17i16i15i +++ 1
.....  ..... 8. )3i5(i20i + 0
.....  ….. 9. 40i30i20i10i ⋅⋅⋅ 1
..... ..... 10. 4)(3,− bia+ 3 - 4i
.....  ….. 11. 1)(5,− bia+ 5 + i
.....  ..... 12. ,2)2( bia+ 2i2+
.....  ..... 13. 8i2− b)(a, 8)(2,−
.....  ….. 14. i53−− b)(a,
)5,3( −−
.....  ..... 15. bi 0b≠

21. 21
ขขขขขขขขข
1 (a,b) a + bi
1. 2
1.1………………………………………………
……………………….
2. biaZ +=
2.1 a .................................................
2.2 b ..................................................

22. 22
3. a a−
..........................................................
........................................
4. 1−
..........................................................
..........................................
5. “ ni ” n
1……………………………………………
…….
2……………………………………………
…….

23. 23
ขขขขขขขขข 1. 1 (a,b) a + bi 10
2. ×
, ,
10
1. 24i20i16i8i +++
. 8
. 4
. 2
. 0
2. i8+ b)(a,
. 1)8,( −−
. 8,1)(−
. 1)(8,−
. (8,1)
3. 25i
. i
. i−
. 1−
. 1
4. 3i4Z −=
. 4−
. 3−
. 3
. 4
ขขขขขขขขขขขขขขขขข
ขขขขขข 1 ขขขขขข ขขขขขขขข
ขขขขขขขขขขขขขขขขขขข (a,b) ขขขข a + bi

24. 24
5. 8)1002(i2i +
. 3−
. 5−
. 7−
. 9−
6. 5)(1,Z −=
. 2
. 1
. 0
. 5−
7. 3)(2,− bia+
. 3i2−
. 3i2−−
. 2 + 3i
. 3i2+−
8. ขขขขขขขขขข
. 4)3,(4i3 −−=−−
. 8,5)(5i8 −=+−
. 1)2,(i2 −−=−
. (6,2)2i6 =+
9. ขขขขขขข
. 2i62)6,( −=−−
. 5i44,5)( −−=−
. 6i3(3,6) −=
. 2i1(1,2) +=
10.
. i2
. i3−
. 2i5+
. 5

25. 25
1
(a,b) a + bi
1
(a,b) a + bi
............................... ................................
..................... .......
1
2
3
4
5
6
7
8
9
10
...................................
...................................

26. 26
1
(a,b) a + bi
1
(a,b) a + bi
1. ข
2. ข
3. ข
4. ข
5. ข
6. ข
7. ข
8. ข
9. ข
10. ข
...
...

27. 27
.................................. ................................
......... ..........
..................... ............... ..........................
ขขขขขขขขข
1.
........................................................................
.................................................
........................................................................
.................................................
........................................................................
2.
........................................................................
.................................................
........................................................................
.................................................
........................................................................

28. 28
3.
.........................................................................
................................................
.........................................................................
................................................
4.
.........................................................................
................................................
.........................................................................
................................................
5.
.........................................................................
................................................
.........................................................................
................................................

29. 29
6.
...................................................................
..................................................................
...................................................................
..................................................................
...................................................................
7.
...................................................................
..................................................................
...................................................................
..................................................................
8.
...................................................................
..................................................................
...................................................................
..................................................................
...................................................................

30. 30
9.
....................................................................
.................................................................
....................................................................
.................................................................
....................................................................
10.
....................................................................
.................................................................
....................................................................
.................................................................
....................................................................
...

31. 31
ขขขขขขขขขข
. ขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขข
ข.5 ขขขข 4 ข 014, ข 044. : SIENCE CENTER,
...
. “ขขขขขขขขขขขขขข” ขขขขขขขขขข ข.ขขขข. 16(7) :
145–152
2539.
. “ขขขขขขขขขขขขขข” ขขขขขขขขขข ข.ขขขข. 8(18) :
26–31 ;
2541.
. ขขขขขขขขขขขขขขขขขข ข 412 ขขขขขขขขขข ขขขข
ขขขขขขขขขข
ขขขขขขข. : , 2520.
. ขขขขขขขขขขขขขขขขขขขขขขขขขขขข ข 014 ข.5. :
, ...
. “ขขขขขขขขขขขขขข ขขขขขข 1” ขขขขขขขขขข ข.ขขขข.
10(7) :
182–188 ; 2540.
. “ขขขขขขขขขขขขขข ขขขขขข 2” ขขขขขขขขขข ข.ขขขข.
10(8) :
176–181 ; 2540.
. Modern Acadamic Math
ขขขขขขขขขข ข.5 ขขขข 4. : , 2535.
, . ขขขขขขขขขขขขขขขขขขขขขขขขขขข

32. 32
ขขขขขขขขขขขขขขขขขขข ขขขข 2 ขขขขขขขขขขขข
ขขขขขขขขขขขขขขขขขข ขขขขขขขขขขขขขข ขขขขข 5. : ,
2547.
, . ขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขขข
ขขขขขขขขขขขขขขขขขขข ขขขข 2 ขขขขขขขขขขขข
ขขขขขขขขขขขขขขขขขข ขขขขขขขขขขขขขขขข
ขขข 5. : , 2548.
. “ขขขขขขขขขขขขขข (Complex
Number).” ขขขขขขขขขข ข.ขขขขขขขขข. 1(6) : 79-
87 ; 2531.
. ขขขขขขขขขข ข 014. : , 2535.