The Decimal Number Base Systems

The Decimal Number System uses base 10. It includes the digits from 0 through 9. The weighted values for each position is as follows:

 

 

 

You have been using the decimal (base 10) numbering system for so long that you often take it for granted. When you see a number like "123", you don't think about the value 123. Instead, you generate a mental image of how many items this value represents. In reality, however, the number 123 represents:

 

 

(1 * 102) + (2 * 101) + (3 * 100) = (1 * 100) + (2 * 10) + (3 * 1) = 100 + 20 + 3 = 123

 

 

Binary Number System

The binary number system works like the decimal number system with the following exceptions:

binary uses base 2 binary includes only the digits 0 and 1 (any other digit would make the number an invalid binary number)

Binary Number Formats

We typically write binary numbers as a sequence of bits (bits is short for binary digits). We have defined boundaries for these bits. These boundaries are:

 

In any number base, we may add as many leading zeroes as we wish without changing its value. However, we normally add leading zeroes to adjust the binary number to a desired size boundary. For example, we can represent the number five as:

 

 

We'll number each bit as follows:

 

The rightmost bit in a binary number is bit position zero. Each bit to the left is given the next successive bit number.

 

Bit position:

7

6

5

4

3

2

1

0

 

 

Bit zero is usually referred to as the LSB (least significant bit). The left-most bit is typically called the MSB (most significant bit). We will refer to the intermediate bits by their respective bit numbers.

The Bit

The smallest "unit" of data on a binary computer is a single bit. Since a single bit is capable of representing only two different values (typically zero or one) you may get the impression that there are a very small number of items you can represent with a single bit. Not true! There are an infinite number of items you can represent with a single bit.

 

With a single bit, you can represent any two distinct items. Examples include zero or one, true or false, on or off, male or female, and right or wrong. However, you are not limited to representing binary data types (that is, those objects which have only two distinct values).

 

To confuse things even more, different bits can represent different things. For example, one bit might be used to represent the values zero and one, while an adjacent bit might be used to represent the values true and false. How can you tell by looking at the bits? The answer, of course, is that you can't. But this illustrates the whole idea behind computer data structures: data is what you define it to be.

 

If you use a bit to represent a boolean (true/false) value then that bit (by your definition) represents true or false. For the bit to have any true meaning, you must be consistent. That is, if you're using a bit to represent true or false at one point in your program, you shouldn't use the true/false value stored in that bit to represent red or blue later.

 

Since most items you will be trying to model require more than two different values, single bit values aren't the most popular data type. However, since everything else consists of groups of bits, bits will play an important role in your programs. Of course, there are several data types that require two distinct values, so it would seem that bits are important by themselves. however, you will soon see that individual bits are difficult to manipulate, so we'll often use other data types to represent boolean values.

The Nibble

A nibble is a collection of bits on a 4-bit boundary. It wouldn't be a particularly interesting data structure except for two items: BCD (binary coded decimal) numbers and hexadecimal (base 16) numbers. It takes four bits to represent a single BCD or hexadecimal digit.

 

With a nibble, we can represent up to 16 distinct values. In the case of hexadecimal numbers, the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F are represented with four bits. BCD uses ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and requires four bits. In fact, any sixteen distinct values can be represented with a nibble, but hexadecimal and BCD digits are the primary items we can represent with a single nibble.

 

The Byte

Without question, the most important data structure used by the microcontroller is the byte. A byte consists of eight bits and is the smallest addressable datum (data item) in the microprocessor.

 

The bits in a byte are numbered from bit zero (0) through seven (7). The bit positions are shown with their weighted values in the following table:

 

 

Bit-0 is the low order bit or least significant bit, bit-7 is the high order bit or most significant bit of the byte. We'll refer to all other bits by their number.

A byte also contains exactly two nibbles. Bits 0 through 3 comprise the low order nibble, and bits 4 through 7 form the high order nibble. Since a byte contains exactly two nibbles, byte values require two hexadecimal digits.

 

Since a byte contains eight bits, it can represent 2⁸, or 256, different values. Generally, we'll use a byte to represent:

1. unsigned numeric values in the range 0 to 255
2. signed numbers in the range -128 to +127
3. ASCII character codes
4. other special data types requiring no more than 256 different values. Many data types have fewer than 256 items so eight bits is usually sufficient.

One of the most important uses for a byte is holding a character code. Characters displayed on an LCD, and sent via serial communication all have numeric values. To allow communication with the rest of the world, microcontrollers use the ASCII character set. There are 128 defined codes in the standard ASCII character set. IBM uses the remaining 128 possible values for extended character codes including European characters, graphic symbols, Greek letters, and math symbols.

The Word

The bits in a word are numbered from bit zero (0) through fifteen (15). The bit positions are shown with their weighted values in the following table:

 

 

 

Like the byte, bit-0 is the LSB and bit-15 is the MSB. When referencing the other bits in a word use their bit position number.

Notice that a word contains exactly two bytes. Bits 0 through 7 form the low order byte (byte-0), bits 8 through 15 form the high order byte byte-1.

With 16 bits, you can represent 2¹⁶ (65,536) different values.  The major uses for words are:

1. unsigned numeric values in the range 0 to 65,535
2. signed numbers in the range -32,768 to +32,767
3. 16-bit memory addresses
4. any number system requiring 16 bits or less

Number Base Conversion

Binary to Decimal

It is very easy to convert from a binary number to a decimal number. Just like the decimal system, we multiply each digit by its weighted position, and add each of the weighted values together. For example, the binary value 11001010 represents:

 

(1*2⁷) + (1*2⁶) + (0*2⁵) + (0*2⁴) + (1*2³) + (0*2²) + (1*2¹) + (0*2⁰) =

(1 * 128) + (1 * 64) + (0 * 32) + (0 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1) =

128 + 64 + 0 + 0 + 8 + 0 + 2 + 0 =

202

Decimal to Binary

To convert decimal to binary is slightly more difficult. There are two methods, that may be used to convert from decimal to binary, repeated division by 2, and repeated subtraction by the weighted position value.

Repeated Division By 2

For this method, divide the decimal number by 2, if the remainder is 0, on the side write down a 0. If the remainder is 1, write down a 1. This process is continued by dividing the quotient by 2 and dropping the previous remainder until the quotient is 0. When performing the division, the remainders which will represent the binary equivalent of the decimal number are written beginning at the least significant digit (right) and each new digit is written to more significant digit (the left) of the previous digit. Consider the number 2671.

 

 

  

The Subtraction Method

For this method, start with a weighted position value greater that the number.

• If the number is greater than the weighted position for the digit, write down a 1 and subtract the weighted position value.
• If the number is less than the weighted position for the digit, write down a 0 and subtract 0.

This process is continued until the result is 0. When performing the subtraction, the digits which will represent the binary equivalent of the decimal number are written beginning at the most significant digit (the left) and each new digit is written to the next lesser significant digit (on the right) of the previous digit. Consider the same number, 2671, using a different method.