It is generally accepted nowadays that Hamilton’s greatest achievment is his general theory of dynamics. By comparison, quaternions have had less impact. At the same time, people continue to use and develop the theory and techniques of quaternionic algebra and analysis, and they continue to find new applications, so one cannot say what the verdict may be on the relative importance of the two inventions, in the long run. It is in the nature of mathematics that its abstract concepts find use far from their origins. Quaternions were invented because Hamilton wanted an algebra that would facilitate geometric work in three dimensions. Number theory (the theory of whole numbers) is quite a different area of mathematics, so I particularly like the fact that quaternions may be used to make an important step in the proof of a theorem in number theory.