Bits and Bytes and the Binary System

Overview

Bits are the most basic unit of information in computing. Bit is short for binary digit. They represents a logical state, which is one of two possible values. It is either 1 or 0, but can also be represented as true or false, yes or no, on or off.

A Byte is meassuring size of Bits. Most of the time one byte represents 8 Bits.

Content

Data Type Sizes

In computing and programming, specific data types have a fixed size, representing the memory they occupy in the computer. To write efficient software, you should choose the right data type to avoid unnecessary memory usage.

For example: If you need a variable for the age of a person, you just need a 8 bit data type. Because:

$2^8=256$

which is sufficient.

You also have to take into account that if you need a negative number to be represented, you need to use a signed data type, which cuts the available range in half. Instead of representing values from 0 to 255, a signed 8-bit integer can store values from -128 to 127.

How to represent a (positiv) integer number in binary

Imagine you want to represent the number 123 as a byte, meaning as 8 Bits. To do this, you can create a table with 8 columns, one for each bit, along with his corresponding exponent. Below each column, write the value of the bit’s exponent. Then starting from left to right, check whether the exponent value fits into the number you want to represent.

• If it does not, the bit value is 0
• If it fits, set the bit to 1 and subtract the exponent value from the remaining number
• Repeat this process for the next exponent, always checking whether it fits into the current remainder

Here’s how the number 123 is represented in binary:

Exponents	2^7	2^6	2^5	2^4	2^3	2^2	2^1	2^0
Exponent Value	128	64	32	16	8	4	2	1
Bit	0	1	1	1	1	0	1	1
Calculation	123	123	123-64=59	59-32=27	27-16=11	11-8=3	3	3-2=1

How to represent a signed integer dtype in binary

If you want to represent a signed dtype the leftmost bit, which is the Most Significant Bit (short: MSB), is reserved for the sign

• 0 → positiv number
• 1 → negativ number

Since the first bit is reserved for representing the sign, you lose a whole bit for number representation. This meands that signed data types have a smaller numerical range compared to unsigned data types.

If you want to represent a negative number, it is common that the twos complement is used. For this, the number is represented as described for unsigned numbers. But after this every bit is negated (1 to 0 and 0 to 1) and is added with a 1 as binary.

First represent 123 as above: $123=01111011$ negate it’s values: $10000100$

Then add a 1 as binary $1=00000001$

$10000100+00000001=10000101$

Representing floating point numbers in binary

Most floating-point numbers cannot be represented exactly in a computer. Exact representation would only be possible if we had an infinite number of bits. However, for the float datay type we are limited to 32 bits. Because of this, there is usually a rounding error during representation, and the floating-point number is approximated rather than stored exactly.

These 32 bits are divided into three parts to represent a floating-point number:

• 1 bit for the sign
• 8 bit for the exponent
  • the exponent determines the position of the decimal point
• 23 bit for the mantissa
  • the mantissa holds the significant digits of the number
  • these bits correspond to negative exponents in base 2

a shortend example:

Exponent	sign bit	2^8	…	2^2	2^1	2^0	point	2^-1	2^-2	2^-3	2^-4	…	2^-23
Exponent Values		256	…	4	2	1	point	0.5	0.25	0.125	0.0625	…	0.0000…

To represent a floating-point number in binary, follow the same procedure as for integer numbers, but consider the exponent and mantissa structure.

Binary Counter

With increasing decimal numbers, the binary representation follows these rules:

• check the last binary digit: if it is 0, change it to 1
• if the last binary digit is 1, change it to 0 and check the next digit to the left
• repeat this process untill you find a 0, which you then change to 1

The initial binary state is 0, so the decimal number 0 is represented as binary 0. Since the smallest common datatype is 8 bits, it is actually represented as 00000000.

Example

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111
16	10000
17	10001
18	10010
19	10011
20	10100
21	10101
22	10110
23	10111
24	11000
25	11001
26	11010
27	11011
28	11100
29	11101
30	11110

Footer

Links