**Properties of Modulus of Complex Numbers :**

Following are the properties of modulus of a complex number z.

**Question 1 :**

Find the modulus of the following complex numbers

(i) 2/(3 + 4i)

**Solution :**

We have to take modulus of both numerator and denominator separately.

|(2/(3+4i))| = |2|/|(3 + 4i)|

= 2 / √(3^{2} + 4^{2})

= 2 / √(9 + 16)

= 2 / √25

= 2/5

(ii) (2 - i)/(1 + i) + (1 - 2i)/(1 - i)

**Solution :**

**(2 - i)/(1 + i) + (1 - 2i)/(1 - i)**

** = |(2 - i)|/|(1 + i)| + |****(1 - 2i)|/|(1 - i)|**

**|(2 - i)|**** = **√(2

**|1 + i|**** = **√(1

**|1 - 2i|**** = **√(1

**|1 - i|**** = **√(1

= (√5/√2) + (√5/√2)

= 2√5/√2

= √2√5

= √10

(iii) (1 - i)^{10}

**Solution :**

To solve this problem, we may use the property

|z^{n}| = |z|^{n}

(1 - i)^{10 }= {(1 - i)^{2}}^{5}

= (1^{2} + i^{2} - 2i)^{5}

= (1 - 1 - 2i)^{5}

= (- 2i)^{5}

= -32i^{5}

= |-32i|

= √(-32)^{2}

= 32

(iv) 2i(3− 4i)(4 − 3i)

**Solution :**

|2i(3− 4i)(4 − 3i)| = |2i| |3 - 4i||4 - 3i|

= √2^{2 } √3^{2 }+ (-4)^{2 }√4^{2 }+ (-3)^{2 }

= √4 √25 √25

= 2 (5)(5)

= 50

**Question 2 :**

For any two complex numbers z_{1} and z_{2} , such that |z_{1}| = |z_{2}| = 1 and z_{1} z_{2} ≠ -1, then show that z_{1} + z_{2}/(1 + z_{1} z_{2}) is a real number.

**Solution :**

Let z_{1} = 1 and z_{2 } = i

|z_{1}| = √1^{2 }+ 0^{2 } = 1

|z_{2}| = √0^{ }+ 1^{2 } = 1

z_{1 }+ z_{2 }= 1 + i

z_{1 } z_{1 }= i

By applying the values of z_{1 }+ z_{2 }and z_{1 } z_{2 }in the given statement, we get

z_{1} + z_{2}/(1 + z_{1} z_{2}) = (1 + i)/(1 + i) = 1

1 is real. Hence it is proved.

**Question 3 :**

Which one of the points 10 − 8i , 11 + 6i is closest to 1 + i

**Solution :**

Let the given points as A(10 - 8i), B (11 + 6i) and C (1 + i).

To find which point is more closer, we have to find the distance between the points AC and BC.

AC = √(1-10) = √9 = √(81 + 81) AC = √162 |
BC = √(1-11) = √10 = √(100 + 25) BC = √125 |

√162 > √125

Hence the point B is closer to C.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**