For my Laidlaw Research project, I had the pleasure of working on the design of computer-generated holograms. I explored a speculative technique to generate holographic images in the near field, namely, the use of singular variable decomposition with a cost-function minimisation approach. Throughout this work, complex analysis, one of the classical branches of mathematics, has been of great importance. Along with their use in my project, complex numbers are integral to our understanding of physics, and yet they are often misunderstood. Some even seem to think they imply some “imaginary” component to our reality. I hope to provide some intuition on their significance and purpose in physics.

*What is a Complex Number?*

In mathematics, complex numbers extend the real numbers with the addition of the imaginary unit *i*. This unit *i *equates to the square root of a negative one. All complex numbers can be expressed in the form *a+bi** *where *a* and *b* are real numbers. Much of the confusion surrounds the term ‘imaginary’ which is used for the unit* i*. This misnomer was catalysed by the early mathematicians and philosophers who considered the 'imaginary' unit to be fictitious or useless much like negative numbers once were. However, *i *is ‘real’ or ‘imaginary’ as any other number despite its ill-adapted notation.

Complex numbers are referred to as 2-dimensional numbers. An interesting way to understand this is through an expansion of the number line. Thinking back to the number line introduced in school, numbers are assigned some scalar value and a direction, positive or negative. Complex numbers effectively expand the possible directions of this scalar value, where a number is not necessarily positive or negative but somewhere in between.

*A Classical Wave *

One of the most useful applications of complex numbers is their ability to describe periodic motion. Consider a 2-dimensional wave, such as a cross-sectional view of an ocean wave. Take a point on the wave moving downwards and another point of equal elevation moving upwards. Can these points be meaningfully differentiated? Complex numbers give us the ability to encode this information. More specifically waves can be represented by *Ae^{iθ}* where *A* is the amplitude (size) and *θ* is the phase (position within a cycle). This representation is called the complex amplitude whereby taking the absolute value gives the real amplitude and the angle gives the phase. With this complex-valued exponential form the multiplication of waves with the same scalar becomes trivial.

*Complex Numbers in Quantum Theory *

So, do complex numbers exist just for mathematical ease? Famously, complex numbers appear in Schrodinger’s equation which describes how particles in quantum physics behave. This prompts the question of whether complex numbers are a requirement or just one of the many ways to describe these systems. Erwin Schrodinger himself was dissatisfied with the use of complex numbers and called upon the development of a real quantum theory. Here, the single complex-valued vector is replaced with two real-valued vectors. But this real numbered approach has been squashed by recent experiments suggesting complex numbers are an indispensable tool if we accept the standard framework of quantum mechanics based on several postulates. So for now complex numbers may be considered a necessity to our understanding of reality.

Complex numbers may just be another mathematical abstraction, but their impact on our understanding of the world is profound. The elegance and utility of this expanded number system cannot be overstated, bridging the gap between abstract mathematics and tangible reality, and driving advancements in science and technology.

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