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complex numbers on brilliant maths for quantum states skip to main content quantumcomputing courses com courses all courses course platforms coursera edx udemy brilliant hardware providers google quantum ai ibm quantum ionq quantinuum amazon braket azure quantum quera rigetti d wave tutorials all tutorials hello world qiskit hello world cirq hello world pennylane hello world braket quantum gates grover s algorithm shor s algorithm reference all frameworks qiskit cirq pennylane amazon braket pyquil tket d wave ocean q explore learn learning paths prerequisites programming guide case studies glossary books quantum news podcasts tools bloch sphere quantum pinball guides algorithm guide hardware guide qubit types framework comparison migration guide language timeline cheat sheets career events 2026 jobs careers certifications salary guide universities interview prep faq troubleshooting about company editorial policy search browse courses home courses brilliant complex numbers brilliant online course complex numbers paid one time view on brilliant affiliate link we may earn a commission at no extra cost to you how this works interactive problem solving learn at your own pace builds intuition not just facts quantum states are described using complex numbers amplitudes phases and the complex plane this course builds the fluency you need before tackling quantum mechanics beginner self paced by brilliant org scores from the listing the provider s level plus maths and coding load from the topics named on the page this site has not sat the course difficulty easy maths medium coding low in this guide 7 sections 01 what you ll learn 02 course structure 03 who is this for 04 prerequisites 05 hands on practice 06 why take this course 07 topics covered quantum state amplitudes are complex numbers quantum interference relies on phases without comfort with the complex plane quantum mechanics stays opaque no matter how good your intuition for other parts of the subject is this course builds exactly the fluency needed as a prerequisite for quantum computing study not through dry calculation but through brilliant s visual interactive approach that makes complex numbers genuinely intuitive what you ll learn what complex numbers are why the equation x² 1 has no real solution and how introducing i solves it the complex plane representing complex numbers as points in two dimensions with the real part on one axis and imaginary part on the other arithmetic with complex numbers addition subtraction multiplication and division treated geometrically polar form expressing complex numbers as a magnitude modulus and angle argument instead of real and imaginary parts why polar form is useful multiplication becomes rotation and scaling which maps directly to quantum gate operations powers and roots using polar form to compute powers of complex numbers and to find all the roots of an equation including multiple roots complex conjugates and division what conjugates are and how they make division workable later in quantum mechanics squaring the modulus of a complex amplitude gives the measurement probability via the born rule course structure the course follows a build up approach across 50 lessons and 647 exercises organised into 11 levels you begin with the motivation some equations have no real solutions and complex numbers arise naturally as the answer with immediate geometric interpretation on the complex plane arithmetic is covered first in rectangular form a bi then revisited in polar form to show how multiplication becomes rotation and scaling a much cleaner picture conjugates and division follow and the later levels work up to powers and roots where polar form really pays off the quantum payoff is something you bring with you once amplitudes phases and the complex plane are second nature qubit states like ψ α 0 β 1 and the role of relative phase in interference stop being obstacles each concept in the course is explored through interactive diagrams before symbolic manipulation is introduced who is this for anyone preparing to study quantum computing who wants mathematical confidence students who covered complex numbers in school but never felt they truly understood them self learners who have tried reading quantum computing textbooks and got stuck on amplitudes and state vectors anyone taking the brilliant quantum computing or quantum objects courses who wants thorough preparation before starting physicists returning to study who want a refresher with modern interactive tools prerequisites brilliant lists comfort with the cartesian and polar coordinate planes quadratic equations and basic trigonometry sine and cosine as assumed background no calculus is needed no prior exposure to complex numbers is assumed hands on practice all practice is browser based and interactive you will drag points around the complex plane to see how operations move them work multiplication and division problems geometrically as rotations and scalings enter complex number calculations and receive immediate feedback solve powers and roots problems where polar form turns hard algebra into simple angle arithmetic the emphasis is on building geometric intuition before algebraic fluency you understand what an operation does before you practise calculating it why take this course complex numbers sit at the heart of every quantum computing formalism a qubit s state lives in a two dimensional complex vector space quantum gates are unitary matrices with complex entries quantum interference the phenomenon that gives quantum algorithms their power is entirely a consequence of complex amplitude arithmetic amplitudes for unwanted outcomes cancel while amplitudes for correct outcomes reinforce without fluency here every quantum computing textbook and course requires constant back referencing to definitions brilliant s interactive approach makes this fluency enjoyable to build the course is self paced with 50 lessons you can work through in short sessions and it will pay dividends across everything you study afterwards topics covered complex numbers quantum states amplitudes phases mathematics enrol in this course complex numbers is a paid course on brilliant at beginner level and runs self paced view on brilliant more brilliant courses affiliate link we may earn a commission at no extra cost to you on this page 01 what you ll learn 02 course structure 03 who is this for 04 prerequisites 05 hands on practice 06 why take this course 07 topics covered at a glance provider brilliant price paid certificate not offered level beginner difficulty easy 2 5 maths medium 3 5 coding low 1 5 related tutorials bra ket notation explained 25 min complex numbers in quantum computing 20 min quantum measurement explained 20 min similar courses group theory paid quantum mechanics with sabine paid computer science fundamentals paid paid complex numbers view on brilliant get one quantum email a week new tutorials courses worth taking and what changed in qiskit cirq pennylane this week no spam unsubscribe anytime email address subscribe 112 courses 220 tutorials 241 glossary terms 26 framework references 34 case studies quantumcomputing courses com free tutorials curated courses framework references and tools for anyone learning quantum computing an independent catalog of quantum computing courses and tutorials published by hadamard llc learn all courses free tutorials learning paths compare frameworks algorithm guide case studies quantum news reference glossary framework docs hardware guide qubit types history timeline cheatsheets bloch sphere quantum programming careers careers guide salary guide certifications interview questions jobs team training post a job talent pool about company editorial policy faq events 2026 podcasts books contact as an amazon associate quantumcomputingcourses com earns from qualifying purchases 2026 hadamard llc quantumcomputingcourses com affiliate disclosure privacy terms cookies we use 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