Complex Numbers
Begin with the real numbers, R Add “the” square root of −1; call it i.
You have just constructed the complex numbers, C, in the form
A complex number, represented as z = a + bi, consists of two real numbers (a, b) It can be visualized as a point in the complex plane with coordinates (a, b) or interpreted as a vector with components a and b.
History
Complex numbers emerged in the 16th century to solve cubic equations, particularly in cases where real solutions necessitated their use The formal acknowledgment of complex numbers as a significant mathematical concept is often credited to Rafael Bombelli, who contributed to this recognition in 1572.
first to formalize the rules of complex arithmetic (and also, at the same time, the first to write down the rules for manipulating negative numbers).
The termimaginary was introduced only later, by Ren´e Descartes in 1637.
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6 The Geometry of the Octonions
Algebra
Complex numbers are not only a vector space but also form an algebraic structure, allowing for multiplication To compute the product of complex numbers, simply multiply them out.
(a+bi)(c+di) = (a+bi)c+ (a+bi)di
The expression (ac−bd) + (bc+ad)i illustrates the fundamental properties of complex numbers, specifically the distribution of multiplication over addition Additionally, it highlights that i serves as the square root of -1, satisfying the equation i² = -1.
Third, we have used associativity, that is
(xy)z=x(yz) (2.5) for any complex numbersx,y,z Finally, we have used commutativity, i.e. xy=yx (2.6) to replacebicwithbci 1
The complex conjugate of a complex number \( z = a + bi \) is defined as \( z = a - bi \), effectively changing the sign of the imaginary part This operation can also be viewed as a real linear transformation that maps 1 to 1 and \( i \) to \( -i \) Additionally, the norm \( |z| \) of a complex number \( z \) is defined accordingly.
|z| 2 =zz=a 2 +b 2 (2.8) The only complex number with norm zero is zero Furthermore, any nonzero complex number has a unique inverse, namely z −1 = z
Since complex numbers are invertible, linear equations such as c=az+b (2.10) can always be solved forz, so long asa= 0.
The norms of complex numbers satisfy the following identity:
|yz|=|y||z| (2.11) Squaring both sides and expanding the result in terms of components yields
(ac−bd) 2 + (bc+ad) 2 = (a 2 +b 2 )(c 2 +d 2 ) (2.12) (where, say,z=a+biandy =c+di), which is called the2-squares rule.
Multiplication is considered linear over the real numbers, which guarantees both distributivity and commutativity among real numbers and the complex unit, i This property is sufficient to establish complete commutativity in this context.
Euler's formula, \( e^{i\theta} = \cos\theta + i\sin\theta \), allows complex numbers to be expressed in polar coordinates using their magnitude and phase angle \( \theta \) This means any complex number can be represented as \( z = re^{i\theta} \), where \( r = |z| \) and \( |e^{i\theta}| = 1 \) Consequently, each complex number has a specific direction in the complex plane, defined by the angle \( \theta \).
Euler’s formula provides an elegant derivation of the angle addition formulas for sine and cosine We have
The equation cos(α+β) + isin(α+β) can be analyzed through complex multiplication, which allows us to derive the standard formulas for trigonometric functions by comparing the real and imaginary components.
We can use (2.16) to provide a geometric interpretation of complex multiplication We have
The multiplication of two complex numbers, represented as \( (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)} \), results in the stretching of the first complex number \( z_1 \) by the magnitude \( r_2 \) of the second complex number \( z_2 \), while simultaneously rotating it counterclockwise by the phase angle \( \theta_2 \) of \( z_2 \) This operation can also be viewed by reversing the roles of \( z_1 \) and \( z_2 \) Notably, multiplying by \( i \) specifically rotates a complex number counterclockwise by \( \frac{\pi}{2} \) without altering its norm.
Euler's formula is often demonstrated through the comparison of power series expansions, but an alternative proof can be derived by observing that both sides satisfy the same differential equation, d²f/dθ² = -f, along with identical initial conditions A notable instance of Euler's formula is the renowned equation e^(iπ) + 1 = 0, which connects five fundamental mathematical constants.
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The Geometry of the Quaternions
What happens if we include another, independent, square root of−1? Call itj Then the big question is, what isij?
Hamilton eventually proposed thatk=ijshould be yet another square root of−1, and that the multiplication table should be cyclic, that is ij=k=−ji, (3.1) jk=i=−kj, (3.2) ki=j=−ik (3.3)
We refer toi, j, andk asimaginary quaternionic units Notice that these units anticommute!
The multiplication table for quaternionic units is illustrated in Figure 3.1 When two quaternionic units are multiplied in the direction indicated by the arrow, the result is the third unit; however, multiplying against the arrow introduces an additional negative sign.
Quaternions, denoted by H in honor of Hamilton, consist of the identity element 1 and three imaginary units A quaternion, represented as four real numbers (q₁, q₂, q₃, q₄), can be expressed in the form q = q₁ + q₂i + q₃j + q₄k This representation allows quaternions to be visualized as points or vectors in R⁴ Additionally, it can be reformulated as q = (q₁ + q₂i) + (q₃ + q₄i)j, highlighting their complex structure.
1 The symbol Q is used to denote the rational numbers, and is therefore not available for the quaternions.
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10 The Geometry of the Octonions k j i
Fig 3.1 The quaternionic multiplication table. we see that a quaternion can be viewed as a pair of complex numbers
(q 1C , q 2C ) = (q 1 +q 2 i, q 3 +q 4 i), or equivalently that we can write
H=C⊕Cj (3.6) in direct analogy to the construction ofCfromR.
The quaternions were discovered by Sir William Rowan Hamilton in 1843, after struggling unsuccessfully to construct an algebra in three dimensions.
On October 16, 1843, while walking along a canal in Dublin, mathematician William Rowan Hamilton discovered how to create an algebra in four dimensions In a historic moment of mathematical expression, he famously carved the multiplication table of quaternions—i² = j² = k² = ijk = -1—onto the base of the Brougham Bridge Although the original carving no longer exists, a plaque now commemorates this significant event in the history of mathematics.
(complete with graffiti!), as shown in Figure 3.2 The inscription on the plaque reads: Here as he walked by on the 16th of October 1843 Sir William
Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplicationi 2 =j 2 =k 2 =ijk= 1& cut it on a stone of this bridge.
Quaternions were the first natural language used to discuss electromagnetism, paving the way for later developments It wasn't until the mid-1880s that vector analysis, introduced by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz, became the modern framework for this field After considerable debate, this new language of vector analysis gained acceptance.
Fig 3.2 The Brougham Bridge in Dublin (left), and the plaque there commemorating
Hamilton’s discovery (right). vector analysis won out, and to this day electromagnetism is taught almost exclusively using vectorial methods 2
Quaternions have regained popularity in recent years due to their effectiveness in describing spatial rotations Today, they are widely utilized in various fields, including aeronautics, robotics, and video game development.
The quaternionic multiplication table closely resembles the vector cross product, with one key distinction: imaginary quaternions yield a negative square, while the cross product of a vector with itself results in zero.
The identification of vectors \( v \) and \( w \) with imaginary quaternions allows us to express them as \( v = v_x i + v_y j + v_z k \) and \( w \) similarly Consequently, the imaginary part of the quaternionic product \( vw \) corresponds to the cross product \( v \times w \), represented as \( v \times w \leftrightarrow \text{Im}(vw) \) In contrast, the real part of this product reflects the negative of the dot product \( v \cdot w \), indicating that \( v \cdot w \) is equivalent to the real component of \( vw \).