Professor Stewart's Hoard of Mathematical Treasures Read Free Page B

parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers. 2
    In it, he wrote: ‘To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe 3 lines of one lengthe: ========, bicause noe .2. thynges, can be moare equalle.’

    Robert Recorde and his equals sign.

Stars and Snips
    Betsy Ross, who was born in 1752, is generally credited with having sewn the first American flag, with 13 stars representing the 13 founding colonies. (On the present-day Stars and Stripes,
they are represented by the 13 stripes.) Historians continue to debate the truth of this story, since it is mainly based on word of mouth, and I don’t want to get tangled up in the historical arguments: see www.ushistory.org/betsy/
    The important thing for this puzzle is that the stars on the American flag are five-pointed. Apparently George Washington’s original design used six-pointed stars, whereas Betsy favoured the five-pointed kind. The committee objected that this type of star was too hard to make. Betsy picked up a piece of paper, folded it, and cut off a perfect five-pointed star with one straight snip of her scissors. The committee, impressed beyond words, caved in.
    How did she do that?
    Can a similar method make a six-pointed star?
     
    Answers on page 278

    Fold and cut this . . .

    . . . to make this.

By the Numbers of Babylon
    Ancient cultures wrote numbers in many different ways. The ancient Romans, for instance, used letters: I for 1, V for 5, X for 10, C for 100, and so on. In this kind of system, the bigger the numbers become, the more letters you need. And arithmetic can be tricky: try multiplying MCCXIV by CCCIX, using only pencil and paper.
    Our familiar decimal notation is more versatile and better suited to calculation. Instead of inventing new symbols for ever-bigger
numbers, it uses a fixed set of symbols, which in Western cultures are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Larger numbers are taken care of by using the same symbols in different positions. For instance, 525 means
    5×100 + 2×10 + 5×1
    The symbol ‘5’ at the right-hand end stands for ‘five’; the same symbol at the left-hand end stands for ‘five hundred’. A positional number system like this needs a symbol for zero, otherwise it can’t distinguish between numbers like 12, 102, and 1,020.
    Our number system is said to be base 10 or decimal, because the value of a digit is multiplied by 10 every time it moves one place to the left. There’s no particular mathematical reason for using 10: base 7 or base 42 will work just as well. In fact, any whole number (greater than 1) can be used as a base, though bases greater than 10 require new symbols for the extra digits.
    The Mayan civilisation, which goes back to 2000 BC, flourished in Central America from about AD 250 to 900, and then declined, used base 20. So to them, the symbols 5-2-14 meant
    5×20 2 + 2×20 + 14×1
which is 2,054 in our notation. They wrote a dot for 1, a horizontal line for 5, and combined these to get all numbers from 1 to 19. From 36 BC onwards they used a strange oval shape for 0. Then they stacked these 20 ‘digits’ vertically to show successive base-20 digits.

    Left: the numbers 0-29 in Mayan; right: Mayan for 5 × 20 2 + 2 × 20 + 14×1

    It is often suggested that the Mayans employed base 20 because they counted on their toes as well as their fingers. An alternative explanation occurred to me while I was writing this item. Maybe they counted on fingers and thumbs, with a thumb representing 5. Then each dot is a finger, each bar a thumb, and it can all be done on two hands. Admittedly, we don’t have three thumbs, but there are easy ways round this with hands and it’s not an issue for symbols. As for the oval shape for zero: don’t you agree that it looks a bit like a clenched fist? Meaning no fingers and no thumbs.
    This is