We should begin by defining a *numeral*. A *numeral* is a symbol that describes a *number*. Expressions such as: *4, IV, 78, 101*
are all *numerals* describing *numbers*. A *numeral* is not a *number*, it is an expression describing a *number*. A *number* is a concept of quantity. One
must know the difference because the same *number* may be described by many different *numerals*. An example would be *4 (in radix-10, or Base-10, the way we think), IV(in
Roman numerals), or (100, in Radix-2, or Base-2, otherwise known as Binary).* The fact is, for small numbers, it is actually easier to think in base-2, or binary, than
it is to think in base-10, or *decimal*. Multiplication and division is way easier in binary than the decimal way we were taught in school.

Thinking in different bases could be compared to a farmer looking into egg cartons, counting by 12s; more on this later. What this means is 3 dozen (base-12) is 3*12 =
36, in decimal (base 10, not base 12) that would be 3*10 + 6 = 36. Just keep reading, you'll get it, really! *Radix n* is a numeration system with a base of n. Later
we will learn about octal numbers (a numeration system with a base of 8, or radix 8 where the *n* in *Radix n* is 8).

A clock uses a Radix 60, numeration system, at least the seconds and minutes. These are just good examples to get you thinking about the concept, and difference between *numbers* and *numerals*.
A quick example of a an octal *number* would be 26_{8} The eight in subscript tells us we are looking at a *number* described in Radix-8 notation. We will get
into octals later, you do not have to understand this right now, this is just to introduce you to the concept. This *number* represents 2*8+6 in base-10 or 22_{10}.
This will make sense later.

When we write a *number* we assume it is in decimal format (radix-10), unless the subscript tells us otherwise, or it is obviously implied. This means 14_{10} is
unnecessary because we already will assume that the text means 14 in decimal.

Base-2, or the binary system is the simplest system one can count in because there are only two symbols to indicate a *number* 1 & 0 that's it. The decimal system
needs 10 different symbols to describe the same numbers 0-9. and each place from the right to the left is a multiple of 10.

10 to the Power of: | 10^{6} |
10^{5} |
10^{4} |
10^{3} |
10^{2} |
10^{1} |
10^{0} |

1000000 | 100000 | 10000 | 1000 | 100 | 10 | 1 |

This means that the place from right to left, determines the power of the numeral. The same is true for the binary system.

2 to the Power of: | 2^{6} |
2^{5} |
2^{4} |
2^{3} |
2^{2} |
2^{1} |
2^{0} |

64 | 32 | 16 | 8 | 4 | 2 | 1 |

That's all there is to it. Now lets try converting some numbers from decimal to binary.

First lets try 48

Decimal: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

32 + 16 = 48, this is 110000 in binary, read from right to left. This is six characters long (the most significant bit is six from the right) so this would be
at it's smallest, a six *bit* number. Usually, computers reserve a set amount of memory for numbers in sectors with an equal size, such as 8, 16, 32, or even
64 bits. So this number in an 8-bit register would be 00110000 (in other words the extra zeros to the left are put in as *padding*, this will be important later
when we talk about bit shifting, but for now don't worry about it.)

Let's do another one, how about 19.

Decimal: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |

This gives us the 8-bit binary number 00010011. This is because the largest power of 2 (16 in base-10) is the most significant bit, the remainder is 3

Back to Index

So what about a clock? Are minutes and seconds counted in Base-60?

By the common definition, *Radix* is the synonymous with *Base*. However it was explained to me as the *glyphs* or characters being the base, but the decimal being the Radix. So, and this is wrong, A clock's minutes and seconds would be BASE=10 RADIX=60