Count On

Explorer

The Power of Two

The old English system of measuring volumes such as beer or seeds involved only halving or doubling:

  • 4 gills make one pint,
  • 2 pints are one quart (8 gills),
  • 4 quarts made one gallon (32 gills),
  • 2 gallons were one peck (64 gills),
  • 4 pecks one bushel,
  • 4 bushels made one barrel,
  • 2 barrels a hogshead,
  • 2 hogsheads are one pipe,
  • 2 pipes one tun.

No doubt there were other words for the intermediate measures, such as 2 quarts, 2 pecks and 2 bushels. This is an example of a binary system with the smallest unit the gill which is about half a cup of beer or a handful of seed. Any quantity (of beer or seed) could be measured in these units. For example, 25 gills is 16+4+1 gills which is 2 quarts, one pint and a gill.

The Binary number system is a way of writing down such combinations when all the powers of 2 are available. In our ordinary decimal number system, the columns represent powers of 10, so that 528 means 5 hundreds, 2 tens and 8 units. Each column has one of ten values, from 0 to 9.

In a binary system the columns are (in reverse order) units, twos, fours, eights, sixteens, etc and the "digits" are either 0 or 1 only. 12, for instance, is 1 eight and 1 four with 0 twos and 0 units, so we would write it as 1100 in binary. To make sure we do not confuse this with one thousand one hundred (i.e. a decimal value), we will put a small 2 as a suffix to indicate the value isin the binary system, so we would write:

12 = 11002

What decimal value does 101102 represent?

From the right, the columns are units, twos, fours, eights and sixteens. So we would have 1 sixteen, 0 eights, 1 four, 1 two and 0 units or 16+4+2=22.

Here is a table to complete to show all the binary values from 1 to 20. Some are filled in already:

No 16 8 4 2 1
0 0 0 0 0 0
1 0 0 0 0 1
2 0 0 0 1 0
3 0 0 0 1 1
4 0 0 1 0 0
5
6
7
8
9
10
11
12 0 1 1 0 0
13
15
15
16
17
18
19
20

A Puzzle about Ancestors

Each one of us has 2 biological parents. Each of our parents have two parents, so that makes 4 grandparents for each of us; and again they each have two parents, so that makes 8 great-grandparents for every one of us. So we have 16 great-great-grandparents, and so on.

When Johnny hears this at school, he says:

"So the further back in time we go, the more people there must have been and the population of the world must be getting smaller, not bigger!"

Is Johnny correct?! If not, why not?


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