Learning the Binary System

This is a VERY easy concept to learn, it's amazing that the binary system is not common knowledge! In fact it's easier than the system you already know!

What is it?

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 268 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 2210. 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 1410 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: 106 105 104 103 102 101 100
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: 26 25 24 23 22 21 20
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 310, this means 20 will go into that once with a remainder of 110 which is 21.

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Hang On!

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

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