Algebra of Complex Numbers: Definition, Rules, Identities of Complex Numbers

Algebra of Complex Numbers: Complex numbers have wide applications in various fields of science, such as AC circuit analysis. Learning about the algebra of complex numbers serves the basic purpose of handling complex numbers well. Since the complex plane is very similar to the two-dimensional Cartesian plane, the rules that are associated with the complex number system has a lot in common with that of a real number system that we are already familiar with.

The algebraic identities that in the real number system studied in lower grades have their counterparts in complex numbers. This article explains all these and more.

Complex Numbers: Definition

Every complex number is a combination of real and imaginary numbers. The standard symbol used for the set of complex numbers is \(C\). It is customary to use \(z\) to denote a complex variable. The real and imaginary parts are often represented using lowercase letters such as \(a,\,b,\,c,\,d,\,x,\,y\) etc.

The standard form of a complex number has a real number part, an imaginary part, and an imaginary unit. So, a complex number looks like,

\(z = a + bi\)

where,

\(a \to \) real part
\(b \to \) imaginary part
\(i \to \) imaginary unit and \(i = \sqrt { – 1} \)

Two complex numbers \({z_1} = a + bi\) and \({z_2} = c + di\) are said to be equal if these two conditions are satisfied.

(i) Real parts are equal \(a = c\)

(ii) Imaginary parts are equal \(b = d\)

Let us now develop and learn the algebra of complex numbers.

Addition of Two Complex Numbers

Similar to the addition of polynomials where the like terms are added, the addition of complex numbers can also be performed. That is, add the real parts together and then the imaginary parts together.

So, if \({z_1} = a + bi\) and \({z_2} = c + di\) then:

\({z_1} + {z_2} = \left( {a + bi} \right) + \left( {c + di} \right) = \left( {a + c} \right) + \left( {b + d} \right)i\)

Listed below are the basic properties related to the addition of two complex numbers:

1. Closure Law: The sum of two complex numbers is also a complex number, and therefore, the set of complex numbers is closed under addition.
2. Commutative Law: Since adding two complex numbers involves adding their respective real and imaginary parts irrespective of the order of the numbers, it is commutative.
   That is, if \({z_1} = a + bi\) and \({z_2} = c + di\) then:
   \({z_1} + {z_2} = (a + c) + (b + d)i\)
   \( = \left( {c + a} \right) + \left( {d + b} \right)i\)
   \( = {z_2} + {z_1}\)
3. Associative Law: When three complex numbers are added, the order in which they are grouped does not matter. That is, for three complex numbers \({z_1},\,{z_2},\,{z_3}\) associativity can be defined as,
   \({z_1} + \left( {{z_2} + {z_3}} \right) = \left( {{z_1} + {z_2}} \right) + {z_3}\)
4. Additive Identity: A complex number with zero real and imaginary parts is known as the additive identity. That is, for any complex number \(z\), there exists \(0 + 0i\) (denoted as \(0\)), such that \(z + 0 = 0 + z = z\)
5. Additive Inverse: Any complex number \(z\), has an additive inverse \(-z\), with real and imaginary parts that are the additive inverses of the real and imaginary parts of \(z\). That is, for any complex number \(z = a + bi\), there exists \( – z = – a + ( – b)i\) such that \(z + ( – z) = – z + z = 0\).

Subtraction of Two Complex Numbers

Subtraction of a complex number from another is nothing but the addition of a negative number. That is, for the complex numbers \({z_1}\) and \({z_2},\,{z_1} – {z_2} = {z_1} + \left( { – {z_2}} \right)\).

Multiplication of Two Complex Numbers

While multiplying two complex numbers, consider them as binomials, and apply the FOIL rule of multiplication.

That is First terms – Outer terms – Inner Terms – Last terms.

Lastly, the fact \({i^2} = – 1\) is applied to simplify the expressions further.

Let \({z_1} = a + bi\) and \({z_2} = c + di\). On multiplying, we get,
\({z_1} \cdot {z_2} = (a + bi) \cdot (c + di)\)
\( = (a \cdot c) + (a \cdot di) + (bi \cdot c) + (bi \cdot di)\)
\( = ac + adi + bci + bd{i^2}\)
\( = (ac – bd) + (ad + bc)i\)
The basic properties related to the multiplication of complex numbers are:

1. Closure Law: The product of two complex numbers is also a complex number. Hence, the set of complex numbers is closed under multiplication.
2. Commutative Law: Since the multiplication and addition of real numbers are commutative, the multiplication of the real and imaginary parts is also commutative. Hence, the multiplication of complex numbers is commutative
   That is if \({z_1} = a + bi\) and \({z_2} = c + di\) then:
   \({z_1} \cdot {z_2} = (ac – bd) + (ad + bc)i\)
   \( = (ca + db) + (cb + da)i\)
   \( = {z_2} \cdot {z_1}\)
3. Associative Law: When three complex numbers are multiplied to get the product, the order in which they are grouped does not affect the product. That is, for three complex numbers \({z_1},\,{z_2},\,{z_3}\)
   \({z_1} \cdot \left( {{z_2} \cdot {z_3}} \right) = \left( {{z_1} \cdot {z_2}} \right) \cdot {z_3}\)
4. Multiplicative Identity: A complex number with \(1\) as the real and \(0\) as the imaginary parts zero is known as the additive identity. That is, for any complex number \(z\), there exists \(1 = 1 + 0i\) such that \(z \cdot 1 = 1 \cdot z = z\)
5. Multiplicative Inverse: For any non-zero complex number \(z\) there exists a multiplicative inverse \(\frac{1}{z}\) such that \(z\cdot \frac{1}{z} = \frac{1}{z} \cdot z = 1\)
   For a complex number \(z = a + bi\), the multiplicative inverse \(\frac{1}{{a + bi}}\)  can be simplified by multiplying the numerator and the denominator by \(a – bi\).
   \(\frac{1}{{a + bi}} = \frac{{(a – bi)}}{{(a + bi)(a – bi)}}\)
   \( = \frac{{a – bi}}{{{a^2} + {b^2}}}\)
   \( = \frac{a}{{{a^2} + {b^2}}} – \frac{b}{{{a^2} + {b^2}}}i\)
6. Distributive Law: For complex numbers, just as in real numbers, addition is distributive over multiplication.
   That is, for three complex numbers \({z_1},\,{z_2},\,{z_3}\)
   \({z_1} \cdot \left( {{z_2} + {z_3}} \right) = \left( {{z_1} \cdot {z_2}} \right) + \left( {{z_1} \cdot {z_3}} \right)\)
   or
   \(\left( {{z_1} + {z_2}} \right) \cdot {z_3} = \left( {{z_1} \cdot {z_3}} \right) + \left( {{z_2} \cdot {z_3}} \right)\)

Division of Two Complex Numbers

While dividing a complex number by another non-zero complex number, that is, \({z_1} \div {z_2} = \frac{{{z_1}}}{{{z_2}}} = {z_1} \times \frac{1}{{{z_2}}},\,{z_2} \ne {0_r}\), follow the steps:

Step 1: Set up the division problem as a fraction.
Step 2: Use the concept of the identity \(\left( {{z_1} + {z_2}} \right)\left( {{z_1} – {z_2}} \right) = z_1^2 – z_2^2\) to rationalize the denominator.
Step 3: Apply the fact \({i^2} = – 1\) to simplify the expressions.

Integral Power of i

By definition, \(i = \sqrt { – 1} \). Any integral power of \(i\) can be derived using the laws of indices.

\({i^2} = – 1\)
\({i^3} = {i^2} \cdot i = – 1 \cdot i = – i\)
\({i^4} = {i^2} \cdot {i^2} = ( – 1) \cdot ( – 1) = 1\)

In general, for any integer \(k,\,{i^{4k}} = 1,{i^{4k + 1}} = i,\,{i^{4k + 2}} = – 1\), and \({i^{4k + 3}} = – i\)

Complex Notation of the Square Root of a Negative Real Number

By using the definition of the imaginary unit \(i = \sqrt { – 1} \), if \(a\) is a positive real number then, \(\sqrt { – a} \) can be written as:
\(\sqrt { – a} = \sqrt a \cdot \sqrt { – 1} \)
\( = i\sqrt a \)

Example: \(\sqrt { – 4} = \sqrt 4 i = 2i\).

Complex Number Identities and Formulas

The algebraic properties of complex numbers, mainly commutative and distributive properties, can be used to derive algebraic identities similar to real numbers.

The important complex number identities are: 

1. \({\left( {{z_1} \pm {z_2}} \right)^2} = z_1^2 \pm 2{z_1}{z_2} + z_2^2\)
2. \({\left( {{z_1} \pm {z_2}} \right)^3} = \left( {z_1^3 \pm 3z_1^2{z_2} + 3{z_1}z_2^2 \pm z_2^3} \right)\)
3. \(\left( {{z_1} + {z_2}} \right)\left( {{z_1} – {z_2}} \right) = z_1^2 – z_2^2\)

Solved Examples – Algebra of Complex Numbers

Q.1. Find the value of \(\sqrt { – 36} \sqrt { – 25} \).
Ans: Use the fact that \(i = \sqrt { – 1} \).
\(\sqrt { – 36} \sqrt { – 25} = (\sqrt { – 1} \sqrt {36} )(\sqrt { – 1} \sqrt {25} )\)
\( = (6i)(5i)\)
\( = 30{i^2}\)
\( = -30\)

Q.2. Find the real value of \(a\) if \(4a{i^3} – 6{i^2} + ai – 7\) is a real number.
Ans: Simplify the given expression and write it in the form \(a + bi\).
\(4a{i^3} – 6{i^2} + ai – 7\)
\( = (4ai){i^2} – 6{i^2} + ai – 7\)
\( = – 4ai + 6 + ai – 7\)
\( = – 1 – 3ai\)
Since this is a real number, the imaginary part is zero.
That is, \( – 3a = 0 \Rightarrow a = 0\).

Q.3. Find the value of \({( – \sqrt { – 1} )^{4n + 3}}\).
Ans: Use the fact that \(i = \sqrt { – 1} \).
\({( – \sqrt { – 1} )^{4n + 3}} = {( – i)^{4n + 3}}\)
\( = {( – 1)^{4n + 3}}{(i)^{4n + 3}}\)
Here, \(4n + 3\) is an odd integer, and odd powers of \(-1\) are \(-1\).
In general, for any integer \(k,\,{i^{4k + 3}} = – i\)
\({( – 1)^{4n + 3}}{(i)^{4n + 3}} = ( – 1)( – i)\)
\(\therefore \,\,{( – \sqrt { – 1} )^{4n + 3}} = i\)

Q.4. Simplify \({(3 + 2i)^3}\).
Ans: Using the identity \({(3 + 2i)^3}\), 
\({\left( {{z_1} \pm {z_2}} \right)^3} = \left( {z_1^3 \pm 3z_1^2{z_2} + 3{z_1}z_2^2 \pm z_2^3} \right)\)
\({(3 + 2i)^3} = {3^3} + 3\left( {{3^2}} \right)(2i) + 3(3){(2i)^2} + {(2i)^3}\)
\( = 27 + 54i + 36{i^2} + 4{i^3}\)
\( = 27 + 54i – 36 – 4i\)
\(\therefore \,\,{(3 + 2i)^3} = – 9 + 50i\)

Q.5. Write \({\left( {\frac{{3 + i}}{{3 – i}}} \right)^2} – {\left( {\frac{{3 – i}}{{3 + i}}} \right)^2}\) in the form \(a + bi\).
Ans: Using the identity \(\left( {{z_1} + {z_2}} \right)\left( {{z_1} – {z_2}} \right) = z_1^2 – z_2^2\),
\({\left( {\frac{{3 + i}}{{3 – i}}} \right)^2} – {\left( {\frac{{3 – i}}{{3 + i}}} \right)^2} = \left[ {\left( {\frac{{3 + i}}{{3 – i}}} \right) + \left( {\frac{{3 – i}}{{3 + i}}} \right)} \right]\left[ {\left( {\frac{{3 + i}}{{3 – i}}} \right) – \left( {\frac{{3 – i}}{{3 + i}}} \right)} \right]\)
\( = \left[ {\frac{{{{(3 + i)}^2} + {{(3 – i)}^2}}}{{(3 – i)(3 + i)}}} \right]\left[ {\frac{{{{(3 + i)}^2} – {{(3 – i)}^2}}}{{(3 – i)(3 + i)}}} \right]\)
\( = \left[ {\frac{{18 + 2{i^2}}}{{9 – {i^2}}}} \right]\left[ {\frac{{4(3)(i)}}{{9 – {i^2}}}} \right]\)
\( = \left[ {\frac{{18 – 2}}{{10}}} \right]\left[ {\frac{{12i}}{{10}}} \right]\)
\( = \frac{{192}}{{100}}i\)
Thus, the simplified expression is a purely imaginary number whose real part is zero.

Summary

The article helps you handle complex numbers by explaining their algebraic operations. It outlines the basic properties of addition, subtraction, multiplication, and division of complex numbers. Further, some of these properties are applied to derive algebraic identities of complex numbers, similar to real numbers.

Another useful algebraic operation discussed in the article is the integral powers of the imaginary unit \(i\). Along with this, the complex notation of the square root of negative real numbers helps you simplify complex algebraic expressions into a standard form of a complex number.

Modulus and Conjugate of a Complex Number

Frequently Asked Questions (FAQs): Algebra of Complex Numbers

Q.1. What is a complex number in algebra?
Ans: A complex number is a combination of real and imaginary numbers.
The standard form of a complex number has a real number part, an imaginary part, and an imaginary unit. So, for a complex number
\(z = a + bi\)
\(a \to \) real part
\(b \to \) imaginary part
\(i \to \) imaginary unit and \(i = \sqrt { – 1} \)

Q.2. What is the symbol of complex numbers?
Ans: The standard symbol used for the set of complex numbers is \(C\). It is customary to use \(z\) to denote a complex variable. The real and imaginary parts are often represented using lowercase letters such as \(a,\,b,\,c,\,d,\,x,\,y\) etc.

Q.3. Is every real number a complex number?
Ans: The standard form of a complex number has a real number part, an imaginary part, and an imaginary unit. So, for a complex number
\(z = a + bi\)
\(a \to \) real part
\(b \to \) imaginary part
\(i \to \) imaginary unit
Real numbers are also complex numbers with a zero as the imaginary part. Therefore, \(5\) is also a complex number with a zero imaginary part.

Q.4. How do you divide complex numbers?
Ans: To divide a complex number by another, set up the division problem as a fraction and use the identity \(\left( {{z_1} + {z_2}} \right)\left( {{z_1} – {z_2}} \right) = z_1^2 – z_2^2\) to rationalize the denominator. The fact \({i^2} = – 1\) is then applied to simplify the expressions.
For example, to divide the complex number \( – 3 + 4i\) by \(1 + 2i\), multiply the numerator and denominator by \(1 – 2i\).
\(\frac{{ – 3 + 4i}}{{1 + 2i}} = \frac{{( – 3 + 4i)(1 – 2i)}}{{(1 + 2i)(1 – 2i)}}\)
\( = \frac{{( – 3 + 4i)(1 – 2i)}}{{{1^2} – {{(2i)}^2}}}\)
\( = \frac{{ – 3 + 6i + 4i – 8{i^2}}}{{1 – 4{i^2}}}\)
\( = \frac{{ – 3 + 10i + 8}}{{1 + 4}}\)
\( = \frac{{5 + 10i}}{5}\)
\( = 1 + 2i.\)

Q.5. How do you simplify complex numbers?
Ans: A complex number has a real part, an imaginary part and an imaginary unit.
To add two complex numbers, add the real parts together and the imaginary parts together. That is, if \({z_1} = a + bi\) and \({z_2} = c + di\) then: \({z_1} + {z_2} = \left( {a + bi} \right) + \left( {c + di} \right) = \left( {a + c} \right) + \left( {b + d} \right)i\)
Subtraction is nothing but adding a negative number, hence uses the same rule.
To multiply two complex numbers, consider the complex numbers similar to binomials. That is, if \({z_1} = a + bi\) and \({z_2} = c + di\) then:
\({z_1} \cdot {z_2} = \left( {a + bi} \right) \cdot \left( {c + di} \right) = \left( {ac – bd} \right) + \left( {ad + bc} \right)i\)
To divide a complex number by another, set up the division problem as a fraction and rationalize the denominator. The fact \({i^2} = \, – 1\) is then applied to simplify the expressions.
The identities used for simplifying complex numbers further are:
1. \({\left( {{z_1} \pm {z_2}} \right)^2} = z_1^2 \pm 2{z_1}{z_2} + z_2^2\)
2. \({\left( {{z_1} \pm {z_2}} \right)^3} = \left( {z_1^3 \pm 3z_1^2{z_2} + 3{z_1}z_2^2 \pm z_2^3} \right)\)
3. \(\left( {{z_1} + {z_2}} \right)\left( {{z_1} – {z_2}} \right) = z_1^2 – z_2^2\)