Data lines, bits, nibbles, bytes, words, binary and HEX
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Information (often called data) inside a computer, as well as on the board used in this tutorial, is exchanged among the various components by means of metallic conductors called data lines. A group of data lines is called a data bus.
Each data line carries a unit of data called a bit. A bit can be on or off. On is usually considered to be 5 volts, and off is considered to be 0 volts, although modern systems often use lower on voltages to decrease power consumption.
Data can be represented on paper as a series of ones and zeros. A one means a bit is on, and a zero means it is off. A group of 8 bits is called a byte. A byte with a value of 0 would be represented as 00000000. Nonzero bytes can be any combination of 1s and 0s. 01100010 will be used as an example here. In the C language, a byte is called a character and is abbreviated char.
When data is represented as a series of ones and zeros, it is said to be a binary representation, or to have a base of 2 because it uses 2 digits. We humans use a base of 10, probably because we have 10 fingers.
The leftend bit of a number represented in binary is called the most significant bit, abbreviated msb, and the rightend bit is called the least significant bit, abbreviated lsb.
A little review might be helpful to those who are a little rusty on
raising a number to a power. No high math here  to raise a number to a
power just write it down the exponent number of times and multiply. The
exponent is the power to which a number is raised. One way to recognize
an exponent is by the fact that it is often ^{
raised} when written:
5^{2} = 5 * 5 = 25
2^{3} = 2 * 2 * 2 = 8
4^{4} = 4 * 4 * 4 * 4 = 256
Each bit position has a weight. For
all numbering systems I am aware of (the mathematicians probably know of
others), the right, leastsignificant position is known as the 1's
place. There, the weight is equal to the base raised to the power of 0.
Any number raised to the power of 0 is equal to 1.
The exponent is increased by 1 with each move to the left. Thus, the
second place from the right has a weight equal to the base raised to the
power of 1.
Any number raised to the power of 1 is equal to itself.
We were taught in grade school that the second place from the right is the
10's place. That's because we were using a base of 10 and we were
raising it to the power of 1. Since a base of 2 is used in binary, the
second place from the right has a weight of 2 because it is 2 raised to
the power of 1. The next weight is 2^{2} = 2 * 2 = 4, then 2^{3} = 2 * 2 * 2 = 8 and so on.
The exponents are often used to designate a bit in a binary number.
Bit 0 is on the right end
of the byte and bit 7 is on the left end. Bit 0 is the lsb and bit 7 is the
msb. Data bits are often abbreviated using the letter D  D0, D1, D2,
etc.
Bit  D7  D6  D5  D4  D3  D2  D1  D0 
Base^{exponent}  2^{7}  2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Weights  128  64  32  16  8  4  2  1 
The example binary number above was 01100010. To figure out what the decimal value is, simply add the weights for the bits that are turned on. In this case, bits 6, 5 and 1 are on. The total of their weights equals 64 + 32 + 2 = 98.
A more general description of the procedure is to multiply the
position weights by the values at the positions, then add them up. The
example 01100010 would be:
(0 * 128) + (1 * 64) + (1 * 32) + (0 * 16) + (0 * 8) + (0 * 4) + (1 * 2) + (0 * 1) = 98.
A common way of showing numbers in a C program is to use hexadecimal notation, or HEX. It uses a base of 16. Break a byte into two groups of 4 bits each: nnnn nnnn. Each group is called a nibble. A nibble with all low bits, 0000, is equal to 0. With all of its bits turned on, 1111, a nibble has a value of 15 (8 + 4 + 2 + 1). Thus, we are dealing with the 16 values from 0 through 15, and a base of 16.
Hexadecimal notation is simple. Just use digits for 0 through 9,
and A through F for 10 through 15. The following table shows all of
the combinations.
Binary  Decimal  Hexadecimal  Binary  Decimal  Hexadecimal 
0000  00  0  1000  08  8 
0001  01  1  1001  09  9 
0010  02  2  1010  10  A 
0011  03  3  1011  11  B 
0100  04  4  1100  12  C 
0101  05  5  1101  13  D 
0110  06  6  1110  14  E 
0111  07  7  1111  15  F 
The right nibble of a byte is the least significant nibble. It's the 1's place because it's 16^{0}. Next is the 16's place because it's 16^{1}, then 16^{2} = 256, and so on. To get the decimal value, take the value of the nibbles, multiply by the position weight values and add them up. Thus, the HEX value 9B = (9 * 16) + (11 * 1) = 155.
To show a number is hexadecimal in the C language, prefix it with 0x. The above would be represented as 0x9B or 0x9b. This particular notation is not casesensitive, although many things in C are.
The following shows the byte table again, but this time with the
weights also expressed in hexadecimal notation, as often seen in
C operations.
Bit  D7  D6  D5  D4  D3  D2  D1  D0 
Base^{exponent}  2^{7}  2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Weights  128  64  32  16  8  4  2  1 
HEX Weights  0x80  0x40  0x20  0x10  0x08  0x04  0x02  0x01 
A word is usually 16 bits, D0 through D15. A table with the bit names and their relationship to the binary base of 2 is below.
The following two tables show the bits with their HEX weights.
A word can be broken up into 4 nibbles. It can be represented by
showing its 4 nibbles as a 4place hexadecimal number. For example, the
decimal number 19070 can be represented as the hexadecimal number 0x4A7E.
0x4A7E = (4 * 16^{3}) + (10 * 16^{2})
+ (7 * 16^{1}) + (14 * 16^{0})
= ( 4 * 4096) + (10 *
256) + (7 * 16) + (14 * 1)
= 19070.
In the C language, a word is most often called an integer, abbreviated int. An integer can be used to represent numbers that range from negative to positive values, or numbers that have only positive values. In other words, an integer can be signed or unsigned. A signed integer can have either positive or negative values. An unsigned integer can only be positive. An unsigned 16bit integer can have values from 0 through 65535. It is often abbreviated simply as unsigned.
Bit 15 is used as a sign bit for signed integers. If it is on, the
number is negative. If it is off, it is positive. Positive values can
range from 0 to 32767. Negative values can range from 1 to 32768. Some
examples are shown below. Notice that the signed version is equal to 1
* (65536  unsigned version). For example, to get the signed number from
the unsigned value 49151,
signed = 1 * (65536  49151) = 16385.
A long word is generally considered to be 4 bytes or 32 bits. A long is used for very large numbers. Longs can also be signed or unsigned. Signed longs have a range from 2,147,483,648 to 2,147,483,647. The maximum unsigned value is 0xFFFFFFFF = 4,294,967,295. The minimum unsigned value is 0.
The following is a selftest over this section. It would be a very good idea to make sure you know the answers to all of the questions since the sections that follow will build on this one.
1) Data inside computers is exchanged among the different components by means of metal conductors called __1__. A group is called a __2__.
A) Data Bus, Weight
B) Nibble, HEX
C) Binary, Bit
D) Data Lines, Data Bus
2) If the voltage is 5 volts, the bit is on. If the bit is off, the voltage is 0 volts.
A) True
B) False
3) A group of 8 bits is a __1__ and is called a __2__ in the C programming language.
A) Data Bus, Nibble
B) Byte, Character (or char)
C) Weight, Bit
D) Unsigned, Signed
4) It is said to be a _____ when data is represented with a base of 2, because of the two digits used.
A) Integer (or int)
B) Most Significant Bit
C) Binary
D) HEX
5) The leftend bit can also be referred to as the __1__. The __2__ is the rightend bit.
A) Word, Integer (or int)
B) Character (or char), Data Bus
C) Data Lines, Long Word
D) Most Significant Bit (or msb), Least Significant Bit (or lsb)
6) In the 1's place, the _____ is equal to the base number raised to the power of 0.
A) Byte
B) Weight
C) Long Word
D) Data Bus
7) _____ is a common way of showing numbers in a C program. It uses a base of 16.
A) Data Lines
B) HEX
C) Word
D) Byte
8) If you break a byte into 2 groups of 4 bits each, then each group is a _____.
A) HEX
B) Binary
C) Nibble
D) Volt
9) A __1__ is usually 16 bits, or 2 bytes. Something with 32 bits, or 4 bytes is usually called a __2__.
A) Signed, Unsigned
B) Byte, Bit
C) Word, Long Word
D) HEX, Binary
10) The __1__ integer can be a positive or negative word, and the __2__ integer can only be a positive word.
A) Nibbles, HEX
B) Byte, Data lines
C) Bit, Integer
D) Signed, Unsigned
In the table below, fill in the blanks and correct errors where needed
Hints:MS Binary Nibble Means Most Significant Binary Nibble.  LS Binary Nibble Means Least Significant Binary Nibble.
MS HEX Nibble Means Most Significant HEX Nibble.  LS HEX Nibble Means Least Significant HEX Nibble.
Remember 8,4,2,1 for binary.
Remember that the base for HEX is 16.
The Weight for the Most Significant HEX Nibble = 16
The Weight for the Least Significant HEX Nibble = 1.
Remember, when you count in HEX you use 0 through 9, then A through F.
If you have binary but no HEX, use the binary to figure out the HEX.
If you have HEX but no binary, use the HEX to figure out the binary.
The first 16 numbers in the table simply count from 0 through 15.
Remember the binary and HEX for 0 to 15 and you have done over half of the table.
Notice that half of the numbers are on the left side of the table, and half are on the right side.
0 is on the left, 1 is on the right, 2 is on the left, 3 is on the right and so on.
That's true all the way through 15.
MS 
LS 
MS 
LS 
Decimal 

MS 
LS 
MS 
LS 
Decimal 
0000 
0000 
0 
0 
0 

0000 
0001 


1 
0000 
0010 
0 
2 




0 
3 
3 
0000 
0100 


4 

0000 
0101 
0 
5 



0 
6 
6 

0000 
00111 


7 
0000 
1000 
0 
8 




0 
9 
9 
0000 
1010 


10 

0000 
1011 
0 
B 

0000 
1100 
0 
H 
12 

0000 
1101 


13 
0000 
0000 
0 
E 


0000 
1111 


15 
1010 
0000 
A 
0 




6 
4 
100 




27 



5 
C 

0011 
0111 


55 

0001 
1101 
1 
d 

0011 
0011 
3 
3 
33 


0000 

0 
192 


a 
a 




B 
B 



c 
c 




D 
D 



E 
E 




f 
f 

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