A Square and A Cube
NoteWhat This Chapter Covers
This chapter explores two of the oldest and most useful operations in all of mathematics: squaring a number and cubing it, together with the return journeys back through square roots and cube roots. We start with a museum puzzle about a row of glass cabinets, and let it carry us into the world of square numbers. From there you will hunt for patterns in the last digits of numbers, learn a one-second trick for squaring numbers that end in 5, prove a striking fact about odd numbers using nothing but a drawing, meet the most celebrated taxicab in mathematical history, and follow these ideas back through Aryabhata’s India and the clay tablets of ancient Mesopotamia.
Learning Outcomes
By the end of this chapter, you will be able to:
- recognise square numbers and cube numbers, and write them with the notation n^2 and n^3;
- use the clues hidden in a number’s last digit and its prime factorisation to decide whether it is a perfect square or a perfect cube;
- find square roots and cube roots by prime factorisation, by subtracting odd numbers, and by clever estimation;
- explain why every square is a running total of consecutive odd numbers, using a picture as the proof;
- apply quick mental shortcuts and appreciate the long history behind these ideas.