急 ! F4 Number Systems

23.
(a)
(9 - 15i)(9 + 15i)
= 9² - (15i)²
= 81 + 225
= 306
It is real.

(b)
(14 - 5i) + 11i - (9 + 6i)
= 14 - 5i + 11i - 9 - 6i
= 5
It is real.

(c)
6 ÷ (1 + √2i)
6(1 - √2i) / [(1 + √2i)(1 - √2i)]
= 6(1 - √2i) / (1 + 2)
= 2(1 - √2i)
It is non-real.

(d)
(2 - 3i)(4 - 2i)(8 - i)
= (8 - 4i - 12i - 6)(8 - i)
= (2 - 16i)(8 - i)
= 16 - 32i - 98i - 16
= -100i
It is non-real.


26.
(a)
(2a + 7i) - (12 - 4i)
= 2a + 7i - 12 + 4i
= (2a - 12) + 11i

Since it is a pure imaginary number, then
2a - 12 = 0
a = 6

(b)
(a + bi)(-10 + 25i)
= -10a + 25ai - 10bi - 25b
= -(10a + 25b) + (25a - 10b)i

Since it is a real number, then
25a - 10b = 0
25a = 10b
a/b = 10/25
a : b = 2 : 5


16.
(a)
√{[2a - (5/3)] [2a + (5/3)] + (25/9)}
= √[(2a)² - (5/3)²] +(25/9)]
= √[(2a)² - (25/9)+ (25/9)]
= √(2a)²
= 2a
It is rational.

(b)
√(3a + 1) x √[(a + 2)(a + 1) - (a² + 1)]
= √(3a + 1) x √(a² + 3a + 2 - a² - 1)
= √(3a + 1) x √(3a + 1)
= 3a + 1
It is rational.

(c)
√[2a - 1 + (a - 1)²] x √6
= √(2a - 1 + a² -2a + 1) x √6
= √(a²) x√6
= a√6
It is irrational.


17.
P = [2(n + 2) - 2] + 2(n + 2) + [2(n + 2) + 2]
= 6(n + 2)

Q = (n + 1) + (n + 2) + (n + 3)
= 3n + 6
= 3(n + 2)

(a)
√(P/Q)
= √[6(n + 2)]/[3(n + 2)]
= √2
It is an irrational number.

(b)
√(PQ/18)
= √{[6(n + 2)] x [2(n + 2)] / 18}
= √[18(n + 2)² /18]
= n + 2
It is a rational number.