Computer Math

CPS120: Binary Representations

Binary computers have storage units called binary digits or bits:

Low Voltage = 0High Voltage = 1 all bits have 0 or 1

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Binary and Computers

8 bits = 1 byte

The number of bytes in a word determines the word length of the computer:

32-bit machines 64-bit machines etc.

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Binary and Computers

Binary Representations One bit can be either 0 or 1.

Therefore, one bit can represent only two things.

To represent more than two things, we need multiple bits. Two bits can represent four things because there are four combinations of 0 and 1 that can be made from two bits: 00, 01, 10,11.

Binary Representations (Cont’d)

If we want to represent more than four things, we need more than two bits. Three bits can represent eight things because there are eight combinations of 0 and 1 that can be made from three bits.

Binary & Alphanumeric Codes A binary code is a group of n bits that

assume up to 2n distinct combinations of 1’s and 0’s with each combination representing one element of the set that is being coded- i.e. permutations

With two bits we can form a set of four elements

With three bits we can represent 8 elements With four bits we can represent 16 elements

Binary Representations (Cont’d)

Figure 3.4 Bit combinations

Binary Representations (Cont’d)

In general, bits can represent 2 things because there are 2 combinations of 0 and 1 that can be made from n bits. Note that every time we increase the number of bits by 1, we double the number of things we can represent.

Memory Units 1 nibble 1 byte 1 word 1 long word 1 quad word 1 octa-word

4 consecutive bits 8 consecutive bits 2 consecutive bytes 4 consecutive bytes 8 consecutive bytes 16 consecutive

bytes

Larger Units of Memory 1 Kilobyte 1 Megabyte 1 Gigabyte 1 Terabyte 1 Petabyte 1 Exabyte

1024 bytes ~106 bytes ~109 bytes ~1012 bytes ~1015 bytes ~1018 bytes

32 Mb = 32*103 Kb = 32 * 103 *1024 bytes = 32,768,000 bytes