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How to get a negative out of a square root?

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Complex numbers are those numbers that are composed of two parts Real part + Imaginary part. For the complex number x = a + ib, a is called the real part, and b is called the imaginary part. Imaginary numbers are those numbers whose square root is a negative number. Examples – 7 + 4i, 5 + 2i, 2 – 2i, 0 + 4i

In the first example, 7 is a real number and 4i is an imaginary number, here the alphabet i is known as iota, 4i is an imaginary part of the complex number 7+4i, here iota( i ) is used to represent the imaginary part of the complex number.

History of Complex Numbers

So this need for complex numbers was first realized by the Italian mathematician Girolamo Cardano. The mathematician realized that during the calculations of cubic equations he was encountered negative square roots many times, that was the time when he realized the need for complex numbers.

Algebra of Complex Numbers

Algebra of complex numbers explains the different operations that are done on complex numbers. The operations include addition, subtraction, multiplication, and division. Let’s take a look at these in detail,

  • Addition of complex numbers

Let x = a + ib and y = m+ iu are two complex numbers. So the sum of these two complex numbers will be,

x + y = (a + m) + i(b + u)

For example: 

5 + 6i and 7 + 9i

x + y = (5 + 7) + i(6 + 9) = 12 + i15

  • Subtraction of complex numbers

Let x = a + ib and y = m+ iu are two complex numbers. So the difference between these two complex numbers will be,

x – y = x + (-y)

For example: 

5 + 7i and 3 + 4i

x – y = 5 + 7i + {-( 3 + 4i)} 

= 5 + 7i + (-3 – 4i) 

= 5 + 7i  – 3 – 4i 

= 2 + 3i

  • Multiplication of complex numbers

Let x = a + ib and y = m+ iu are two complex numbers . So the product of these two complex numbers will be –

xy = (am – bu) + i(au + bm)

For example 

2 + 4i and 3 – 5i

xy = [2 × 3 – 4 × (-5)] + i[2 × (-5) + 4 × 3] 

= 26 + i2

  • Division of complex numbers

Let x = a + ib and y = c+ id are two complex numbers . So the pr of these two complex numbers will be –

x  ⁄  y = a + ib / c + id where c ≠ 0 and a ≠ 0 

Multiply the numerator and denominator by the complex conjugate of the denominator.

Complex Conjugate 

The complex conjugate of c + id is m – id. To get conjugate of any complex number, change the sign of the imaginary part and keep the sign of the real part the same.

(ac + bd) + i(bc -ad) / c2 + d =  (ac + bd) / c2 + d + (bc – ad)i / c2 + d2 

How to get a negative out of a square root?

i or iota is used to get a negative out of a square root by substituting iota in place of a negative number’s square root that is √-1, with the help of iota square root of negative number is calculated. The values of iota are listed below.

  1. i = √-1
  2. i2 = -1
  3. i3 =  i.i2 = i(-1) = -i
  4. i4 = (i2)2 = (-1)2= 1
  5. i4n = 1
  6. i4n + 1 =  i
  7. i4n + 2 =  -1
  8. i4n + 3 =  -i

The method is simple to break the problem statement into parts such that it contains √-1, then substitute it with i. 

Let’s say, there is a negative square root represented as √-y.

Now it can be written as √{y × (-1)}

= √y × √-1

= √y i.

Here are some sample problems describing the above discussed method step by step.

Sample Problems

Question 1: Find the square root of -4.

Solution: 

√-4 or √-1.√4

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-4 or √-1.√4 = 2i

Final answer = 2i    

Question 2: Find the square root of -16.

Solution: 

√-16 or √-1.√16

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-16 or √-1.√16 = 4i

Final answer = 4i  

Question 3: Find the square root of -27.

Solution:

√-27 or √-1.√27

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-27 or √-1.√27 = 3√3i

Final answer = 3√3i  

Question 4: Find the square root of -31.

Solution: 

√-31 or √-1.√31

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-31 or √-1.√31 = √31i

Final answer = √31i

Question 5: Find the square root of -78.

Solution: 

√-78 or √-1.√78

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-78 or √-1.√78 = √78i

Final answer = √78i

Question 6: Find the square root of -25.

Solution: 

√-25 or √-1.√25

From the above-listed table i = √-1, now substitute i in place of √-1 to get negative out of a square root.

√-25 or √-1.√25 = 5i

Final answer = 5i


Last Updated : 25 Dec, 2023
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