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Simplify the expression 7/10i

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  • Last Updated : 11 Mar, 2022

The sum of a real number and an imaginary number is termed a Complex number. These are the numbers that can be written in the form of a+ib, where a and b both are real numbers. It is denoted by z. Here in complex number form the value ‘a’ is called the real part which is denoted by Re(z), and ‘b’ is called the imaginary part Im(z). It is also called an imaginary number. In complex numbers form a +bi, ‘i’ is an imaginary number called “iota”. The value of i is (√-1) or we can write as i2 = -1. For example,

  • 3 + 4i is a complex number, where 3 is a real number (Re) and 4i is an imaginary number (Im).
  • 2 + 5i is a complex number where 2 is a real number (Re) and  5i is an imaginary number (im)

The Combination of real number and imaginary number is called a Complex number.

Real and Imaginary numbers

The numbers represent a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are called real numbers. These are represented as Re(). Example: 19, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.

The numbers which are not real are termed Imaginary numbers. After squaring an imaginary number, it gives a result in negative. Imaginary numbers are represented as Im(). Example: √-8, √-9, √-11 are all imaginary numbers. here ‘i’ is an imaginary number called “iota”.

Simplify the expression 7/10i

Solution:

Given: 7/10i

Standard form of numerator, 7 = 7 +0i

Standard form of denominator, 10i = 0 +10i

Conjugate of denominator, 0 + 10i = 0 – 10i

Multiply the numerator and denominator with the conjugate,

Therefore, {(7 + 0i) / (0 + 10i)} × {(0 – 10i)/(0 – 10i)}

= {7(0 – 10i)} / {0 – (10i)2}

= {0 – 70i} / {0 – (100(-1))}

= {-70i} / 100           

= 0 – 70/100i

= -7/10i

Similar Problems

Question 1: Solve (1 – 5i) / (-2i)

Solution:  

Given: (1 – 5i) / (-2i)

Standard form of denominator, -2i = 0 – 2i

Conjugate of denominator, 0 – 2i = 0 + 2i

Multiply the numerator and denominator with the conjugate,

Therefore, {(1 – 5i) / (0 – 2i)} × {(0 + 2i)/(0 + 2i)}

= {(1 – 5i)(0 + 2i)} / {0 – (2i)2}

= {0 + 2i – 5i – 10i2} / {0 –  (4(-1))}

= {-3i – 10 (-1)} / 4

= (-3i + 10) / 4

= 10 /4 – 3i/ 4

= 5/2 – (3/4) i

Question 2: Simplify (8 + 4i) / (3 + 2i)

Solution: 

Given: (8 + 4i) / (3 + 2i)

Multiplying the numerator and denominator with the conjugate of denominators,

= {(8 + 4i) / (3 + 2i)} × {( 3 – 2i) / (3 – 2i)}

= {(8 + 4i) × (3 – 2i)} / {(3 + 2i) × (3 – 2i)}

= (24 – 16i +12i – 8i2) / {9  -(2i)2}

= (32 – 4i) / (13)

= (32 – 4i) / 13

= 32/13 – 4/13i

Question 3: Express (2 – i)/(1 + i) in standard form ?

Solution:

Given: (2 – i)/(1 + i)

Multiplying the numerator and denominator with the conjugate of denominators,

= {(2 – i)/(1 + i) × (1 – i)/(1 – i)}

= {(2 – i)(1 – i)} / {(1)2 – (i)2}

= {2 – 2i – i – i2} / (1 – i2)

= {2 – 3i – (-1)} / (1 + 1)

= (3 – 3i) / 2

= 3/2 – 3/ 2i

Question 4:  Express in form of a + ib, 2(5 + 3i) + i(7 + 7i)

Solution:  

Given: 2(5 + 3i) + i(7 + 7i)

= 10 + 6i + 7i + 7i2

= 10 + 13i  + 7(-1)

= 3 + 13i

Question 5: Perform the following operation and find the result in form of a + ib?

(2 – √-25) / (1 – √-16) 

 

Solution:  

 

Given: (2 – √-25 ) / (1 – √-16)                                                  

= {2 – (i)(5)} / {1 – (i)(4)}, {i = √-1}

= (2 – 5i) / (1 – 4i)

= {(2 – 5i) / (1 – 4i)} × {(1 + 4i) / (1 + 4i)}

= {(2 – 5i) (1 + 4i)} / {(1 – 4i) (1 + 4i)}

= {2 + 8i – 5i – (20i2)} /  {(1-16i2)}, {i2 = -1}

= {2 + 3i + 20} / {1 – 16(-1)}

= (22 + 3i ) / (1 + 16)

= (22 + 3i)/17

= {(22/17) + (3i/17)}

= 22/17 + 3i/17

 

Question 6: Perform the following operation and find the result in form of a + ib?

(3 – √-16) / (1 – √-9) 

 

Solution:  

 

Given: (3 – √-16) / (1 – √-9)                                                  

= {3 – (i)(4)} / {1 – (i)(3)}, {i = √-1}

= (3 – 4i ) / (1 – 3i) {Multiplying the numerator and denominator with the conjugate of denominators}

= {(3 – 4i) / (1 – 3i)} × {(1 + 3i) / (1 + 3i)}

= {(3 – 4i) (1 + 3i)} / {(1 – 3i) (1 + 3i)}

= {3 + 9i – 4i – (12i2)} / {(1 – 9i2)}, {i2 = -1}

= {3 + 5i + 12} / {1 – 9(-1)}

= (15 + 5i) / (1 + 9)

= (15 + 5i)/10     

= 15/10 + 5i/10

= 3/2 + 1/2i

 

Question 7: Simplify the given expression 4/10i

 

Solution: 

 

Given: 4/10i

Standard form of numerator, 4 = 4 + 0i

Standard form of denominator, 10i = 0 + 10i

Conjugate of denominator, 0 +10i = 0 – 10i

Multiply the numerator and denominator with the conjugate,

Therefore, {(4 + 0i) / (0 + 10i)} × {(0 – 10i)/(0 – 10i)}

= {4(0 – 10i)} / {0 – (10i)2}

= {0 – 40i} / {0 – (100(-1))}

= {-40i} / 100          

= 0 – 40/100i

= -2/5i

 

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