# 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:

_{2}

What decimal value does 10110_{2} 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:

N^{o} | 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?