Measuring information: bits, bytes and units

Leyla’s phone says: “Storage: 128 GB, free: 5 GB”. One photo takes about 3 MB. How many more photos will fit? How many seconds will a 2 MB file from a friend take to download? To answer, you need to know the units information is measured in and how to convert between them.

In the lesson “Information, its kinds and information processes” we learned what information is; now we will measure it with numbers. Azerbaijan’s university entrance exam (DİM) always checks this topic: in each of the four entrance exams held in 2025 and 2026, 3 of the 30 informatics tasks were on encoding information and its units, and in three of them the coded-answer task 87 asked you to match information volumes.

The bit and the formula N = 2ⁱ

Inside a computer all information — text, pictures and music alike — is stored as signals with only two states: there is a voltage or there is not; a spot on a disk is magnetized one way or the other. We write these two states as the digits 1 and 0.

One bit tells 2 options apart: 0 and 1. Two bits give 4 options: 00, 01, 10, 11. Three bits give 8, from 000 to 111. Every new bit doubles the number of options, because each old code can be preceded once by 0 and once by 1. So i bits can make 2ⁱ different codes.

N is not always a power of two. To number 30 students, 4 bits are too few (2⁴ = 16) and 5 bits are enough (2⁵ = 32). In such cases i is the smallest whole number with 2ⁱ ≥ N, that is, 2ⁱ⁻¹ < N ≤ 2ⁱ. Some codes (32 − 30 = 2) stay unused, but there is no such thing as half a bit.

The same formula gives the amount of information in a message. If an event has N equally likely outcomes, a message telling which one happened carries i bits, where N = 2ⁱ. A coin toss has two equally likely outcomes, so the message “which side came up” is 1 bit (N = 2), while “the ball is in box 11 of 16” is 4 bits (16 = 2⁴).

From bytes to terabytes: units and powers of two

Eight bits make 1 byte: one byte can hold 2⁸ = 256 different values (from 0 to 255). Larger units differ from each other by a factor of 1024 = 2¹⁰: kilobyte, megabyte, gigabyte, terabyte (KB, MB, GB, TB). Azerbaijani textbooks and DİM tasks write them as Kbayt, Mbayt, Gbayt, Tbayt.

1. 1
   Write the numbers with powers of two

   For example: 512 = 2⁹, 48 = 3 · 2⁴, 160 = 5 · 2⁵, 96 = 3 · 2⁵.

2. 2
   Convert the unit into bits

   byte → · 2³, KB → · 2¹³, MB → · 2²³, GB → · 2³³.

3. 3
   Add or subtract the exponents

   When multiplying, add the exponents: 2ᵃ · 2ᵇ = 2ᵃ⁺ᵇ; when dividing, subtract them: 2ᵃ ÷ 2ᵇ = 2ᵃ⁻ᵇ.

4. 4
   Go to the unit you need

   Divide the number of bits by the size of that unit in bits: by 2²³ for MB, by 2¹³ for KB.

The alphabet approach: I = K · i

Here the computer ignores the meaning of a text and simply counts symbols: “hello” and “qwert” are both 5 symbols. If every symbol takes i bits, a message of K symbols takes K · i bits. This number is the information volume of the message.

In real texts every symbol is encoded with a code table: 1 byte in ASCII, 2 bytes in UNICODE. The next lesson, “Encoding text”, is about this.

The size of a document and transfer time

Documents contain pictures as well as text. The size of a document is the sum of its parts: the text part is found with I = K · i, and the size of a picture is either given in the task or calculated with the formula from the lesson “Computer graphics: encoding raster and vector images”. Bring all parts to the same unit before adding.

When you send or download a file, the time depends on the file size and on the transfer rate of the communication channel. The rate is the number of bits sent per second, measured in bit/s.

KB, MB, GB = 2**13, 2**23, 2**33   # bits in 1 KB, 1 MB, 1 GB

print((1*GB + 1536*MB + 2**21*KB) / GB)
print(2**26 / MB, 40 * 2**17 * 8 / MB, 3 * 2**12 * KB / MB)

pages, lines, chars, i = 40, 32, 64, 7
text = pages * lines * chars * i / KB
print(text, text + 8 * 12)

4.5
8.0 5.0 12.0
70.0 166.0