{"id":1790,"date":"2026-08-29T11:00:00","date_gmt":"2026-08-29T05:30:00","guid":{"rendered":"https:\/\/www.techandtrain.com\/blog\/?p=1790"},"modified":"2026-08-29T11:00:04","modified_gmt":"2026-08-29T05:30:04","slug":"sunday-mathematics-3-complex-numbers-when-real-numbers-are-not-enough","status":"publish","type":"post","link":"https:\/\/www.techandtrain.com\/blog\/2026\/08\/sunday-mathematics-3-complex-numbers-when-real-numbers-are-not-enough\/","title":{"rendered":"Sunday Mathematics #3 \u2014 Complex Numbers: When Real Numbers Are Not Enough"},"content":{"rendered":"\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/media.licdn.com\/dms\/image\/v2\/D4D12AQFIuV1NajSO_Q\/article-inline_image-shrink_1500_2232\/B4DaBL.HxKI8AQ-\/0\/1787980976505?e=1789603200&amp;v=beta&amp;t=MUQFnh-yvkS2rsCpy3H6FZoMsAhi4SuYhYYEwysyOVg\" alt=\"Article content\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Most of us first encounter complex numbers through a slightly uncomfortable equation:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>x\u00b2 + 1 = 0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>x\u00b2 = \u22121<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But no real number squared gives \u22121.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Mathematics solved this by extending the number system and defining:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>i = \u221a\u22121<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A complex number can therefore be written as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = a + bi<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\"><strong>a<\/strong> is the real part<\/li>\n\n\n\n<li class=\"\"><strong>b<\/strong> is the imaginary part<\/li>\n\n\n\n<li class=\"\"><strong>i\u00b2 = \u22121<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">At first glance, this can look like a mathematical trick.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It isn&#8217;t.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers are one of the most useful mathematical abstractions in science, engineering and computing.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">1. From a Number Line to a Number Plane<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Real numbers live on a one-dimensional number line.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers give us a two-dimensional plane:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = a + bi \u2194 (a, b)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The horizontal axis represents the real component and the vertical axis represents the imaginary component.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = 3 + 4i<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">can be represented by the point <strong>(3,4)<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its magnitude is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>|z| = \u221a(3\u00b2 + 4\u00b2) = 5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its angle or phase is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u03b8 = tan\u207b\u00b9(4\/3)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So a complex number can represent both:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Magnitude + Direction<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is where complex numbers become extraordinarily useful.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">2. Cartesian and Polar Forms<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The same complex number can be represented in different ways.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cartesian form<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = a + bi<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Polar form<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = r(cos \u03b8 + i sin \u03b8)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>r = \u221a(a\u00b2 + b\u00b2)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Euler&#8217;s formula:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03b8) = cos \u03b8 + i sin \u03b8<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">we get:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = re^(i\u03b8)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is a remarkably powerful representation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Instead of thinking only about two numbers, we can think in terms of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Amplitude + Phase<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And amplitude and phase appear everywhere in physical and computational systems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">3. Euler&#8217;s Formula \u2014 A Beautiful Mathematical Bridge<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">One of the most famous equations in mathematics is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03c0) + 1 = 0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It connects five fundamental mathematical constants:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>0, 1, e, i and \u03c0<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But Euler&#8217;s formula is much more than mathematical beauty.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03b8) = cos \u03b8 + i sin \u03b8<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">provides a bridge between:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>exponentials \u2194 trigonometry \u2194 rotation \u2194 oscillation<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That bridge is extremely useful when studying waves, signals, electrical systems, communications and control systems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">4. Complex Numbers and Rotation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = re^(i\u03b8)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplying it by:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03c6)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">gives:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z&#8217; = re^(i(\u03b8+\u03c6))<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In simple terms, multiplication by a complex exponential can rotate a point.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This gives us a very elegant mathematical mechanism for representing rotations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Instead of repeatedly manipulating sine and cosine equations, many rotation and oscillation problems become multiplication problems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This idea appears in graphics, robotics, signal processing, physics and engineering.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">5. Electrical Engineering and AC Circuits<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">One of the classic applications of complex numbers is alternating-current circuit analysis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Electrical quantities such as voltage and current oscillate.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Instead of repeatedly working with expressions such as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>V(t) = V\u2080 cos(\u03c9t + \u03c6)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">engineers can represent oscillating quantities using complex numbers and <strong>phasors<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Circuit impedance can be represented as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Z = R + jX<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\"><strong>R<\/strong> = resistance<\/li>\n\n\n\n<li class=\"\"><strong>X<\/strong> = reactance<\/li>\n\n\n\n<li class=\"\"><strong>j<\/strong> represents \u221a\u22121 in electrical engineering<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The magnitude tells us the overall opposition to current, while the phase captures the relationship between voltage and current.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A difficult time-domain problem can often become a much simpler algebraic problem.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">6. Signal Processing and Fourier Analysis<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose we have audio, vibration, radar, network or sensor data.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A signal that looks complicated in the time domain may actually contain combinations of simpler frequencies.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Fourier analysis decomposes signals into these frequency components.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Complex exponentials provide an elegant representation:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03c9t) = cos(\u03c9t) + i sin(\u03c9t)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This idea forms part of the mathematical foundation behind tools such as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Fourier Transform<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Discrete Fourier Transform (DFT)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Fast Fourier Transform (FFT)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These are used across:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Audio processing<\/li>\n\n\n\n<li class=\"\">Image processing<\/li>\n\n\n\n<li class=\"\">Telecommunications<\/li>\n\n\n\n<li class=\"\">Radar<\/li>\n\n\n\n<li class=\"\">Medical imaging<\/li>\n\n\n\n<li class=\"\">Vibration analysis<\/li>\n\n\n\n<li class=\"\">Sensor analytics<\/li>\n\n\n\n<li class=\"\">Spectral analysis<\/li>\n\n\n\n<li class=\"\">Scientific computing<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers therefore help us move between:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Time Domain \u2194 Frequency Domain<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">7. Communication Systems<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Modern communication systems depend heavily on amplitude and phase.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Wireless systems can encode information by changing these properties of a carrier signal.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, in <strong>Quadrature Amplitude Modulation (QAM)<\/strong>, symbols can naturally be represented as points on a complex plane.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Think of a transmitted symbol as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = I + jQ<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\"><strong>I<\/strong> = In-phase component<\/li>\n\n\n\n<li class=\"\"><strong>Q<\/strong> = Quadrature component<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The constellation of these complex-valued points represents digital information.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So when your phone communicates using sophisticated wireless networks, complex-number mathematics is operating underneath many layers of abstraction.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">8. Control Systems<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers also appear naturally when studying the stability and behaviour of dynamic systems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Engineers examine poles and zeros in the complex plane.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A pole might look like:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>s = \u03c3 + j\u03c9<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The real component can tell us about growth or decay.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The imaginary component relates to oscillation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This makes the complex plane extremely useful for reasoning about:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Stability + Oscillation + Damping + System Response<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Applications range from industrial automation to aerospace, robotics and power systems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">9. Quantum Mechanics<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers are fundamental to quantum mechanics.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Quantum states are represented using complex-valued wave functions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A simplified representation might be:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u03c8 = a + bi<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The directly observable probability is not simply \u03c8 itself.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Instead, quantities involving its magnitude, such as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>|\u03c8|\u00b2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">play a central role.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here complex numbers are not merely a convenient calculation technique\u2014they are embedded deeply in the mathematical framework used to describe quantum systems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">10. Computer Graphics and Robotics<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers can represent rotations elegantly in two dimensions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If a point is represented by:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>z = x + iy<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">multiplication by:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>e^(i\u03b8)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">rotates the point through an angle \u03b8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This provides a compact way of understanding transformations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 3D rotations, related mathematical ideas extend into structures such as <strong>quaternions<\/strong>, widely used in robotics, aerospace systems, simulations and computer graphics.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">11. Complex Numbers in Data Science and AI<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Most introductory machine-learning models operate on real-valued data.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But complex-valued representations become useful when the underlying information naturally contains <strong>phase, frequency, waves or spectral characteristics<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Examples can arise in:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Signal classification<\/li>\n\n\n\n<li class=\"\">Radar analytics<\/li>\n\n\n\n<li class=\"\">Wireless communications<\/li>\n\n\n\n<li class=\"\">Medical imaging<\/li>\n\n\n\n<li class=\"\">MRI reconstruction<\/li>\n\n\n\n<li class=\"\">Audio processing<\/li>\n\n\n\n<li class=\"\">Computer vision<\/li>\n\n\n\n<li class=\"\">Spectral methods<\/li>\n\n\n\n<li class=\"\">Scientific machine learning<\/li>\n\n\n\n<li class=\"\">Complex-valued neural networks<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This highlights an important lesson for data science:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The mathematical representation should follow the structure of the problem.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the phenomenon contains magnitude and phase, forcing everything prematurely into purely real-valued representations can sometimes hide useful structure.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">12. A Small Python Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Python supports complex numbers directly.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>z = 3 + 4j\n\nprint(z.real)\nprint(z.imag)\nprint(abs(z))<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">The result is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Real part = 3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Imaginary part = 4<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Magnitude = 5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Scientific Python libraries such as NumPy can also perform complex-valued numerical computations, Fourier transforms and linear algebra.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So the journey from:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>i = \u221a\u22121<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">to computational engineering is surprisingly short.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">The Bigger Lesson<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Complex numbers demonstrate something important about mathematics.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sometimes mathematics advances not by solving a problem inside the existing system, but by <strong>expanding the system itself<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Natural numbers were not enough.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We introduced integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Integers were not enough.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We introduced rational numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rational numbers were not enough.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We introduced real numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And real numbers were not enough.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We introduced complex numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">What initially looks &#8220;imaginary&#8221; can eventually become indispensable for describing reality.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">WHY \u2192 WHAT \u2192 WHERE \u2192 WHEN \u2192 HOW<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For learning complex numbers, I would approach the topic in this order:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>WHY?<\/strong> Real numbers alone cannot conveniently represent every mathematical and physical phenomenon.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>WHAT?<\/strong> A complex number combines real and imaginary components: <strong>a + bi<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>WHERE?<\/strong> Signals, circuits, communications, control systems, physics, graphics, robotics and scientific computing.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>WHEN?<\/strong> Especially when the problem involves oscillation, rotation, frequency, magnitude and phase.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>HOW?<\/strong> Complex algebra, Euler&#8217;s formula, polar representation, Fourier analysis\u2014and computational tools such as Python, NumPy, MATLAB and scientific libraries.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">AI can increasingly help us with the <strong>HOW<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But understanding the <strong>WHY and WHAT<\/strong> remains essential if we want to know whether the answer actually makes sense.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Sunday Mathematics<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The objective of this series is not mathematics for examinations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is mathematics for <strong>computer science, data science, AI, engineering, technology and decision-making<\/strong>\u2014connecting equations with the systems around us.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Sunday Mathematics #3: Complex Numbers<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From <strong>\u221a\u22121<\/strong> to signals, circuits, wireless communication, quantum mechanics, robotics and AI.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sometimes the numbers we call <em>imaginary<\/em> help us understand the real world.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>HSOPC \u2014 Harwani Systems<\/strong> <a href=\"https:\/\/www.harwanisystems.in\/\">https:\/\/www.harwanisystems.in\/<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>TechAndTrain<\/strong> <a href=\"https:\/\/www.techandtrain.com\/\">https:\/\/www.techandtrain.com\/<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Neil Harwani \u2014 LinkedIn<\/strong> <a href=\"https:\/\/www.linkedin.com\/in\/neil27\/\">https:\/\/www.linkedin.com\/in\/neil27\/<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Email:<\/strong> <a href=\"mailto:Neil@HarwaniSystems.in\">Neil@HarwaniSystems.in<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Narrative and concept:<\/strong> Neil Harwani<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Creation help:<\/strong> ChatGPT<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">#SundayMathematics #Mathematics #ComplexNumbers #DataScience #ArtificialIntelligence #Engineering #ComputerScience #SignalProcessing #FourierTransform #ElectricalEngineering #MachineLearning #QuantumComputing #Robotics #STEM #Education<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Most of us first encounter complex numbers through a slightly uncomfortable equation: x\u00b2 + 1 = 0 Therefore: x\u00b2 = \u22121 But no real number squared gives \u22121. Mathematics solved this by extending the number system and defining: i = \u221a\u22121 A complex number can therefore be written as: z = a + bi where: &hellip; <a href=\"https:\/\/www.techandtrain.com\/blog\/2026\/08\/sunday-mathematics-3-complex-numbers-when-real-numbers-are-not-enough\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Sunday Mathematics #3 \u2014 Complex Numbers: When Real Numbers Are Not Enough<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":1791,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[189,70],"tags":[450,552,499,447,108,556,494,554,451,278,555,558,553,557,551],"class_list":["post-1790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering","category-mathematics","tag-artificialintelligence","tag-complexnumbers","tag-computerscience","tag-datascience","tag-education","tag-electricalengineering","tag-engineering","tag-fouriertransform","tag-machinelearning","tag-mathematics","tag-quantumcomputing","tag-robotics","tag-signalprocessing","tag-stem","tag-sundaymathematics"],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7do02-sS","jetpack-related-posts":[{"id":73,"url":"https:\/\/www.techandtrain.com\/blog\/2016\/01\/balanced-software-architecture\/","url_meta":{"origin":1790,"position":0},"title":"Balanced software architecture","author":"Neil Harwani","date":"January 27, 2016","format":false,"excerpt":"Humanity is facing unprecedented challenges of climate change, violence, wars, diseases, etc. One of the reasons is development not in unison with universal rules and facts. In physics there are fundamental forces around gravity, atom, electricity, fusion\/fission, energy, thermodynamics, quantum, etc. All physical things are quantified using mathematics. If our\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2016\/01\/Balanced.jpg?fit=1200%2C675&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2016\/01\/Balanced.jpg?fit=1200%2C675&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2016\/01\/Balanced.jpg?fit=1200%2C675&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2016\/01\/Balanced.jpg?fit=1200%2C675&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2016\/01\/Balanced.jpg?fit=1200%2C675&ssl=1&resize=1050%2C600 3x"},"classes":[]},{"id":1426,"url":"https:\/\/www.techandtrain.com\/blog\/2025\/02\/top-100-mathematics-keywords-for-data-science-part-1\/","url_meta":{"origin":1790,"position":1},"title":"Top 100 mathematics keywords for Data Science &#8211; Part 1","author":"Neil Harwani","date":"February 22, 2025","format":false,"excerpt":"Whoever is teaching you data science without teaching you Mathematics especially optimization is not teaching it right to you. That's my biggest learning from Master of Data Science at IIT Gandhinagar - it will take you good 2 years to learn the related mathematics in all four major areas below.\u2026","rel":"","context":"In &quot;Data Science&quot;","block_context":{"text":"Data Science","link":"https:\/\/www.techandtrain.com\/blog\/category\/data-science\/"},"img":{"alt_text":"Image credit: www.Pixabay.com","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2025\/02\/Data.jpg?fit=1200%2C801&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2025\/02\/Data.jpg?fit=1200%2C801&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2025\/02\/Data.jpg?fit=1200%2C801&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2025\/02\/Data.jpg?fit=1200%2C801&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2025\/02\/Data.jpg?fit=1200%2C801&ssl=1&resize=1050%2C600 3x"},"classes":[]},{"id":1582,"url":"https:\/\/www.techandtrain.com\/blog\/2026\/06\/mathematics-for-computer-science-and-data-science-sunday-mathematics-2\/","url_meta":{"origin":1790,"position":2},"title":"Mathematics for Computer Science and Data Science &#8211; Sunday Mathematics: #2","author":"Neil Harwani","date":"June 20, 2026","format":false,"excerpt":"Mathematics for Computer Science and Data Science Why, What, Where, When and How These Concepts Matter Post #1 on Sunday Mathematics here. Data Science is much more than learning Python, SQL, or Machine Learning libraries. Mathematics provides the foundation that helps us understand why algorithms work, when to use them,\u2026","rel":"","context":"In &quot;Mathematics&quot;","block_context":{"text":"Mathematics","link":"https:\/\/www.techandtrain.com\/blog\/category\/mathematics\/"},"img":{"alt_text":"Credit: www.Pixabay.com","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/06\/Maths.jpg?fit=853%2C568&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/06\/Maths.jpg?fit=853%2C568&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/06\/Maths.jpg?fit=853%2C568&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/06\/Maths.jpg?fit=853%2C568&ssl=1&resize=700%2C400 2x"},"classes":[]},{"id":1606,"url":"https:\/\/www.techandtrain.com\/blog\/2026\/07\/understanding-synthetic-aperture-radar-sar-19-key-concepts-every-engineer-and-data-scientist-should-know-part-1\/","url_meta":{"origin":1790,"position":3},"title":"Understanding Synthetic Aperture Radar (SAR): 19 Key Concepts Every Engineer and Data Scientist Should Know &#8211; Part 1","author":"Neil Harwani","date":"July 8, 2026","format":false,"excerpt":"Synthetic Aperture Radar (SAR) has become one of the most important remote sensing technologies for Earth observation. Unlike optical cameras, SAR is an active microwave sensing system that transmits its own radio waves and measures the reflected signals, allowing it to produce high-quality images day or night and in most\u2026","rel":"","context":"In &quot;Synthetic Aperture Radar&quot;","block_context":{"text":"Synthetic Aperture Radar","link":"https:\/\/www.techandtrain.com\/blog\/category\/synthetic-aperture-radar\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/07\/SatelliteStation.jpg?fit=1200%2C800&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/07\/SatelliteStation.jpg?fit=1200%2C800&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/07\/SatelliteStation.jpg?fit=1200%2C800&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/07\/SatelliteStation.jpg?fit=1200%2C800&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/07\/SatelliteStation.jpg?fit=1200%2C800&ssl=1&resize=1050%2C600 3x"},"classes":[]},{"id":1074,"url":"https:\/\/www.techandtrain.com\/blog\/2023\/05\/short-list-of-good-courses-links-books-on-mathematics-operating-systems-and-aiml-chatgpt-part-1\/","url_meta":{"origin":1790,"position":4},"title":"Short list of good courses \/ links \/ books on Mathematics, Operating Systems and AIML \/ ChatGPT &#8211; Part 1","author":"Neil Harwani","date":"May 21, 2023","format":false,"excerpt":"Here is a short list of good courses \/ links \/ books on Mathematics, Operating Systems and AIML \/ ChatGPT - Part 1: Machine Learning Foundations: Linear Algebra Online Class | LinkedIn Learning, formerly Lynda.com Programming Foundations: Numerical Mathematics and Calculations Online Class | LinkedIn Learning, formerly Lynda.com Machine Learning\u2026","rel":"","context":"In &quot;AIML&quot;","block_context":{"text":"AIML","link":"https:\/\/www.techandtrain.com\/blog\/category\/aiml\/"},"img":{"alt_text":"Credits: www.Pixabay.com","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2023\/05\/geometry-g69584ea1e_1920.jpg?fit=1200%2C790&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2023\/05\/geometry-g69584ea1e_1920.jpg?fit=1200%2C790&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2023\/05\/geometry-g69584ea1e_1920.jpg?fit=1200%2C790&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2023\/05\/geometry-g69584ea1e_1920.jpg?fit=1200%2C790&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2023\/05\/geometry-g69584ea1e_1920.jpg?fit=1200%2C790&ssl=1&resize=1050%2C600 3x"},"classes":[]},{"id":322,"url":"https:\/\/www.techandtrain.com\/blog\/2018\/08\/mystery-of-perfect-numbers-resolved-perfect-number-is-always-even-and-predictable\/","url_meta":{"origin":1790,"position":5},"title":"Mystery of \u2018Perfect Numbers\u2019 Resolved \u2013 Perfect Number is Always Even and Predictable","author":"Dr. Vrajlal Sapovadia","date":"August 6, 2018","format":false,"excerpt":"Mystery of \u2018Perfect Numbers\u2019 Resolved \u2013 Perfect Number is Always Even and Predictable Vrajlal Sapovadia (Ph.D.) Sweta Patel (Ph.D.) Introduction In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. In other words, a perfect\u2026","rel":"","context":"In &quot;Mathematics&quot;","block_context":{"text":"Mathematics","link":"https:\/\/www.techandtrain.com\/blog\/category\/mathematics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2018\/08\/Numbers.jpg?fit=1200%2C790&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2018\/08\/Numbers.jpg?fit=1200%2C790&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2018\/08\/Numbers.jpg?fit=1200%2C790&ssl=1&resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2018\/08\/Numbers.jpg?fit=1200%2C790&ssl=1&resize=700%2C400 2x, https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2018\/08\/Numbers.jpg?fit=1200%2C790&ssl=1&resize=1050%2C600 3x"},"classes":[]}],"jetpack_likes_enabled":true,"jetpack_featured_media_url":"https:\/\/i0.wp.com\/www.techandtrain.com\/blog\/wp-content\/uploads\/2026\/08\/Waves.png?fit=1280%2C941&ssl=1","_links":{"self":[{"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/posts\/1790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/comments?post=1790"}],"version-history":[{"count":1,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/posts\/1790\/revisions"}],"predecessor-version":[{"id":1792,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/posts\/1790\/revisions\/1792"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/media\/1791"}],"wp:attachment":[{"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/media?parent=1790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/categories?post=1790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.techandtrain.com\/blog\/wp-json\/wp\/v2\/tags?post=1790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}