This primer presents Dirac’s and Penrose’s "square-root" methods in one language: real geometric(Clifford) algebra. Dirac’s square root of the wave operator forces the Clifford relation and leads to rotors, spinors, and the Dirac equation; Penrose’s square root of a null vector leads to 2-spinors, flags, twistors, and the spin-network idea that geometry may be recovered from quantum angularmomentum. The exposition is written for graduate students who know special relativity and elementary quantum mechanics. It keeps the historical claims modest, separates convention from invariant content, and works every major idea through a concrete computation: the belt trick, theHestenes form of the Dirac equation, the null flagpole, the Dirac—Penrose conformal bridge, thespin-geometry theorem, and a complete 2+1 dimensional miniature in which bivectors, Lorentzorbits, unitary representations of SU(1, 1), and Clifford-signature superselection all fit into onediagram. Sixteen exercises are included so that the reader can verify the algebra rather than merely admire it.