Complex Numbers: Introduction and Algebra

Introduction to Complex Numbers and the Imaginary Unit ($i$)

Expanding the Number System

Throughout our study of arithmetic and basic algebra, we have primarily worked within the set of real numbers, denoted by $\mathbb{R}$. This set includes rational numbers (integers, fractions like $1/2$) and irrational numbers (like $\sqrt{2}, \pi$). The real number system is sufficient for solving many equations, such as $x - 3 = 5$ (solution $x=8$) or $x^2 = 4$ (solutions $x=2, x=-2$).

However, if we consider the simple quadratic equation $x^2 + 1 = 0$, we find that it has no solution within the real number system. Rearranging the equation gives $x^2 = -1$. In the real number system, the square of any real number is always greater than or equal to zero ($y^2 \ge 0$ for any $y \in \mathbb{R}$). There is no real number whose square is a negative number like $-1$. This indicates a limitation of the real number system.

To overcome this limitation and provide solutions for such equations, mathematicians extended the number system by introducing new types of numbers called imaginary numbers and, more generally, complex numbers. This expanded number system, called the complex number system, provides solutions to all polynomial equations, including $x^2 = -1$.

The Imaginary Unit ($i$)

The foundation of the complex number system is the introduction of the imaginary unit, denoted by the symbol $i$. The imaginary unit is formally defined as the number whose square is $-1$.

From this definition, it follows that $i$ can be thought of as the square root of $-1$: $i = \sqrt{-1}$.

The imaginary unit $i$ allows us to express the square roots of any negative real number in terms of a real number multiplied by $i$. For any positive real number $k$, the square root of $-k$ is given by:

Examples:

• $\sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2 \times i = 2i$.
• $\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i$.
• $\sqrt{-25} = \sqrt{25} \times i = 5i$.
• $\sqrt{-2} = \sqrt{2} \times i = \sqrt{2}i$.
• $\sqrt{-7} = \sqrt{7} \times i = \sqrt{7}i$.

Imaginary Numbers

An imaginary number is a number that can be written in the form $bi$, where $b$ is any real number and $i$ is the imaginary unit ($i^2 = -1$).

If $b \neq 0$, the number is considered a purely imaginary number (e.g., $3i, -5i, \sqrt{2}i, -i/2$). If $b=0$, then $0 \times i = 0$, which is a real number. So, the set of imaginary numbers includes the real number 0.

The set of all imaginary numbers is sometimes denoted as $i\mathbb{R} = \{bi \mid b \in \mathbb{R}\}$.

Complex Numbers

A complex number is a number that combines a real number and an imaginary number through addition. Complex numbers are written in the standard form (also called the rectangular or binomial form):

where:

• $a$ is a real number.
• $b$ is a real number.
• $i$ is the imaginary unit ($i^2 = -1$).

In the complex number $z = a + bi$:

• $a$ is called the real part of $z$. We denote this as $\text{Re}(z) = a$.
• $b$ is called the imaginary part of $z$. It is the real coefficient of $i$. We denote this as $\text{Im}(z) = b$. It is important to note that the imaginary part is the real number $b$, not $bi$.

Examples of complex numbers and their parts:

The set of all complex numbers is denoted by $\mathbb{C}$. The real numbers ($\mathbb{R}$) are a subset of the complex numbers (when the imaginary part is 0, $a+0i = a$). Purely imaginary numbers are also a subset of complex numbers (when the real part is 0, $0+bi = bi$).

The complex number system is a complete algebraic system where equations like $x^2+1=0$ have solutions. The roots of $x^2+1=0$ are $x^2 = -1$, so $x = \pm \sqrt{-1} = \pm i$. These are the complex numbers $0+1i$ and $0-1i$. The Fundamental Theorem of Algebra states that any polynomial equation with complex coefficients has all its roots within the complex number system.

The Complex Conjugate

For a complex number $z = a + bi$, its complex conjugate is a related complex number where the real part is the same, but the sign of the imaginary part is reversed. The complex conjugate of $z$ is denoted by $\overline{z}$ (read as "z-bar") or sometimes $z^*$.

Examples:

• The conjugate of $3 + 4i$ is $\overline{3+4i} = 3 - 4i$.
• The conjugate of $2 - 5i$ is $\overline{2-5i} = 2 + 5i$.
• The conjugate of $-1 + i$ is $\overline{-1+i} = -1 - i$.
• The conjugate of $7$ (which is $7+0i$) is $\overline{7} = 7 - 0i = 7$. The conjugate of a real number is the number itself.
• The conjugate of $-6i$ (which is $0-6i$) is $\overline{-6i} = 0 - (-6i) = 6i$. The conjugate of a purely imaginary number $bi$ is $-bi$.

A significant property of complex conjugates is that their product is always a non-negative real number. Let $z = a + bi$ and $\overline{z} = a - bi$. Their product is:

Using the difference of squares identity $(A+B)(A-B) = A^2 - B^2$ where $A=a$ and $B=bi$:

Since $(bi)^2 = b^2 i^2 = b^2 (-1) = -b^2$:

Since $a$ and $b$ are real numbers, $a^2 \ge 0$ and $b^2 \ge 0$, so $a^2 + b^2 \ge 0$. This value $a^2 + b^2$ is the square of the modulus (or magnitude) of the complex number $z$, denoted as $|z|^2$. The concept of conjugates is fundamental in complex number arithmetic, particularly in division.

Algebra of Complex Numbers (Addition, Subtraction, Multiplication, Division)

Just as we perform arithmetic operations on real numbers, we can also perform addition, subtraction, multiplication, and division on complex numbers. These operations follow rules that are extensions of the rules for real numbers and algebraic expressions (like binomials), with the crucial understanding that $i^2 = -1$.

Let $z_1$ and $z_2$ be two complex numbers in standard form, $z_1 = a + bi$ and $z_2 = c + di$, where $a, b, c,$ and $d$ are real numbers.

1. Addition of Complex Numbers

To add two complex numbers, we add their corresponding real parts and their corresponding imaginary parts separately. Think of it like combining like terms in algebraic expressions, where the real numbers are one kind of term and the imaginary terms (those with $i$) are another kind.

Group the real parts and the imaginary parts:

The result is a new complex number whose real part is $(a+c)$ and whose imaginary part is $(b+d)$.

Answer:

Addition of complex numbers satisfies the commutative property ($z_1 + z_2 = z_2 + z_1$) and the associative property ($(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$), just like real number addition.

2. Subtraction of Complex Numbers

To subtract one complex number from another, we subtract their real parts and their imaginary parts separately.

Distribute the negative sign and group the real and imaginary parts:

The result is a new complex number whose real part is $(a-c)$ and whose imaginary part is $(b-d)$.

Answer:

3. Multiplication of Complex Numbers

To multiply two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, we use the distributive property, similar to multiplying two binomials (like using the FOIL method). The key difference is that whenever $i^2$ appears, we substitute $-1$.

Using the distributive property:

Substitute $i^2 = -1$ into the last term:

Now, group the terms that do not contain $i$ (the real part) and the terms that do contain $i$ (the imaginary part):

The result is a new complex number.

Answer:

Multiplication of complex numbers is commutative ($z_1 z_2 = z_2 z_1$), associative ($(z_1 z_2) z_3 = z_1 (z_2 z_3)$), and the distributive property holds ($z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3$).

4. Division of Complex Numbers

Dividing complex numbers is slightly more involved than the other operations because we generally want the result to be in the standard form $a + bi$, which means the denominator should be a real number. We achieve this by using the concept of the complex conjugate.

To divide a complex number $z_1 = a + bi$ by a non-zero complex number $z_2 = c + di$ (where $c+di \neq 0$), we multiply both the numerator and the denominator by the complex conjugate of the denominator, $\overline{z_2} = c - di$. Since multiplying a complex number by its conjugate results in a real number ($c^2 + d^2$), this process makes the denominator real.

Multiply numerator and denominator by $\overline{z_2} = c - di$:

Perform the multiplication in the numerator: $(a + bi)(c - di)$. Using the multiplication formula $(ac - bd) + (ad + bc)i$ with $d$ replaced by $-d$: $(a \times c - b \times (-d)) + (a \times (-d) + b \times c)i = (ac + bd) + (-ad + bc)i = (ac + bd) + (bc - ad)i$.

Perform the multiplication in the denominator: $(c + di)(c - di)$. This is a product of a complex number and its conjugate, which equals $c^2 + d^2$. Since $z_2 \neq 0$, $c$ and $d$ are not both zero, so $c^2+d^2 > 0$ and is a real number.

So, the fraction becomes:

To write the result in standard form $A + Bi$, separate the real and imaginary parts by dividing both terms in the numerator by the real denominator:

Answer:

Mastering the algebra of complex numbers is essential for working with polynomial roots, electrical engineering, quantum mechanics, and many other areas where complex numbers are applied.

Powers of $i$

Cyclical Nature of Powers of $i$

The imaginary unit $i$ is defined by the property $i^2 = -1$. Based on this fundamental definition, the higher integer powers of $i$ exhibit a repeating, cyclical pattern. Understanding this cycle allows us to easily determine the value of any integer power of $i$.

Calculating the First Few Powers of $i$

Let's calculate the values for the first few non-negative integer powers of $i$:

• $i^0$: By definition, any non-zero number raised to the power of $0$ is $1$. So, $i^0 = 1$.
• $i^1$: Any number raised to the power of $1$ is the number itself. So, $i^1 = i$.
• $i^2$: By the definition of the imaginary unit, $i^2 = -1$.
• $i^3$: We can write $i^3$ as $i^2 \times i^1$. Substituting the known values: $i^3 = (-1) \times i = -i$.
• $i^4$: We can write $i^4$ as $i^2 \times i^2$. Substituting the known value $i^2 = -1$: $i^4 = (-1) \times (-1) = 1$. Alternatively, $i^4 = (i^2)^2 = (-1)^2 = 1$.

Let's continue to the next few powers:

• $i^5$: We can write $i^5$ as $i^4 \times i^1$. Since $i^4 = 1$: $i^5 = 1 \times i = i$.
• $i^6$: We can write $i^6$ as $i^4 \times i^2$. Since $i^4 = 1$ and $i^2 = -1$: $i^6 = 1 \times (-1) = -1$.
• $i^7$: We can write $i^7$ as $i^4 \times i^3$. Since $i^4 = 1$ and $i^3 = -i$: $i^7 = 1 \times (-i) = -i$.
• $i^8$: We can write $i^8$ as $i^4 \times i^4$. Since $i^4 = 1$: $i^8 = 1 \times 1 = 1$.

The sequence of values for the powers of $i$ is $1, i, -1, -i, 1, i, -1, -i, \ldots$. This pattern repeats every 4 powers. The cycle is $\textbf{1, i, -1, -i}$.

Finding Any Integer Power of $i$

Because the powers of $i$ repeat every 4 values, the value of any integer power of $i$, say $i^n$ (where $n$ is an integer), depends only on the remainder when the exponent $n$ is divided by $4$.

According to the Division Algorithm for integers, for any integer $n$ and divisor $4$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $n = 4q + r$, where $0 \le r < 4$. The possible remainders are $0, 1, 2,$ or $3$.

Using the laws of exponents, we can write $i^n$ as:

We can rewrite $i^{4q}$ as $(i^4)^q$. Since $i^4 = 1$, $(i^4)^q = 1^q = 1$ (for any integer $q$).

So, the value of $i^n$ is determined by the value of $i$ raised to the power of the remainder $r$ when $n$ is divided by $4$.

• If the remainder $r = 0$, then $i^n = i^0 = 1$.
• If the remainder $r = 1$, then $i^n = i^1 = i$.
• If the remainder $r = 2$, then $i^n = i^2 = -1$.
• If the remainder $r = 3$, then $i^n = i^3 = -i$.

Steps to find $i^n$ for an integer $n$:

1. Divide the exponent $n$ by $4$.
2. Note the remainder $r$ (where $r$ is $0, 1, 2,$ or $3$).
3. The value of $i^n$ is $i^r$.

Examples

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Understanding the cyclical nature of powers of $i$ and how to use the remainder when dividing the exponent by 4 is essential for simplifying complex expressions and performing calculations involving $i^n$.

Identities of Complex Numbers

Algebraic Identities Extended to Complex Numbers

The fundamental algebraic identities that hold true for all real numbers are also valid for complex numbers. This is because the basic properties of arithmetic operations – commutativity, associativity, and distributivity – are valid for complex number addition and multiplication. Therefore, we can substitute complex numbers into identities like $(a+b)^2 = a^2 + 2ab + b^2$, $(a-b)^2 = a^2 - 2ab + b^2$, etc., and the equalities will hold true when the operations are performed using the rules of complex number algebra ($i^2 = -1$).

Let $z_1$ and $z_2$ be any two complex numbers. Then the following standard algebraic identities hold for $z_1$ and $z_2$:

• Square of a Sum: $(z_1 + z_2)^2 = z_1^2 + 2z_1 z_2 + z_2^2$
• Square of a Difference: $(z_1 - z_2)^2 = z_1^2 - 2z_1 z_2 + z_2^2$
• Difference of Squares: $(z_1 + z_2)(z_1 - z_2) = z_1^2 - z_2^2$
• Cube of a Sum: $(z_1 + z_2)^3 = z_1^3 + 3z_1^2 z_2 + 3z_1 z_2^2 + z_2^3$
• Cube of a Difference: $(z_1 - z_2)^3 = z_1^3 - 3z_1^2 z_2 + 3z_1 z_2^2 - z_2^3$
• Sum of Cubes: $z_1^3 + z_2^3 = (z_1 + z_2)(z_1^2 - z_1 z_2 + z_2^2)$
• Difference of Cubes: $z_1^3 - z_2^3 = (z_1 - z_2)(z_1^2 + z_1 z_2 + z_2^2)$

These identities can be verified by substituting $z_1 = a+bi$ and $z_2 = c+di$ and performing the complex arithmetic, or by appealing to the properties of complex arithmetic that mirror real arithmetic properties.

Additionally, complex numbers allow for factorisations not possible with real numbers. For instance, a sum of two squares, $a^2+b^2$ (where $a, b$ are real), cannot be factored into linear factors with real coefficients unless $a=0$ and $b=0$. However, over the complex numbers, it can be factored:

Proof: Expand the RHS: $(z_1 + iz_2)(z_1 - iz_2) = z_1(z_1 - iz_2) + iz_2(z_1 - iz_2) = z_1^2 - iz_1 z_2 + iz_1 z_2 - i^2 z_2^2 = z_1^2 - (-1)z_2^2 = z_1^2 + z_2^2$.

Identities Involving Complex Conjugates

The complex conjugate introduces specific identities that are particularly useful when working with complex numbers. Let $z = a + bi$ and $w = c + di$ be complex numbers.

• Conjugate of a Conjugate:

  Taking the conjugate twice returns the original complex number.

  $\overline{\overline{z}} = z $

  [Identity 1]

  Proof: $\overline{\overline{a+bi}} = \overline{a-bi} = a - (-b)i = a + bi = z$.

• Sum of a Complex Number and its Conjugate:

  The sum of a complex number and its conjugate is twice its real part (a real number).

  $z + \overline{z} = 2 \text{Re}(z) = 2a $

  [Identity 2]

  Proof: $z + \overline{z} = (a + bi) + (a - bi) = (a+a) + (b-b)i = 2a + 0i = 2a$.

• Difference of a Complex Number and its Conjugate:

  The difference between a complex number and its conjugate is twice its imaginary part multiplied by $i$ (a purely imaginary number).

  $z - \overline{z} = 2i \text{Im}(z) = 2bi $

  [Identity 3]

  Proof: $z - \overline{z} = (a + bi) - (a - bi) = (a-a) + (b-(-b))i = 0 + (b+b)i = 2bi$.

• Product of a Complex Number and its Conjugate:

  The product of a complex number and its conjugate is the square of its magnitude (a non-negative real number).

  $z \overline{z} = a^2 + b^2 $

  [Identity 4]

  Proof: As shown in the introduction, $z \overline{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2$.

• Conjugate of a Sum:

  The conjugate of the sum of two complex numbers is the sum of their conjugates.

  $\overline{z + w} = \overline{z} + \overline{w} $

  [Identity 5]

  Proof: Let $z = a+bi, w = c+di$. $\overline{z+w} = \overline{(a+c) + (b+d)i} = (a+c) - (b+d)i$. $\overline{z}+\overline{w} = (a-bi) + (c-di) = (a+c) + (-b-d)i = (a+c) - (b+d)i$. The results are equal.

• Conjugate of a Difference:

  The conjugate of the difference of two complex numbers is the difference of their conjugates.

  $\overline{z - w} = \overline{z} - \overline{w} $

  [Identity 6]

  Proof: Similar to the conjugate of a sum.

• Conjugate of a Product:

  The conjugate of the product of two complex numbers is the product of their conjugates.

  $\overline{z \times w} = \overline{z} \times \overline{w} $

  [Identity 7]

  Proof: Let $z = a+bi, w = c+di$. $z \times w = (ac - bd) + (ad + bc)i$. $\overline{z \times w} = (ac - bd) - (ad + bc)i$. $\overline{z} \times \overline{w} = (a-bi)(c-di) = (ac - (-b)(-d)) + (a(-d) + (-b)c)i = (ac - bd) + (-ad - bc)i = (ac - bd) - (ad + bc)i$. The results are equal.

• Conjugate of a Quotient:

  The conjugate of the quotient of two complex numbers is the quotient of their conjugates (provided the denominator is non-zero).

  $\overline{z / w} = \overline{z} / \overline{w} \quad (w \neq 0) $

  [Identity 8]

  Proof: This can be proven using the definition of division and Identity 7.

These identities are powerful tools for manipulating complex expressions and are fundamental in various areas of complex analysis.

Answer:

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Familiarity with these identities, particularly those involving conjugates, greatly simplifies operations and manipulations of complex numbers and is essential for solving problems in this area of mathematics.