Number System

You count with ten digits because people first counted on ten fingers. A computer has no fingers: inside it, every tiny switch is either off or on, so it counts with only two digits, 0 and 1. This lesson teaches the four number systems used in computing — decimal, binary, octal and hexadecimal — how to change a number from one system to another, and how a computer's memory is measured in bits and bytes. You met base 2 and base 5 in Mathematics; the same ideas work here. Almost every mark in this lesson is earned by following a procedure step by step, so practise until the steps are automatic.

15.1 Number systems: decimal, binary, octal and hexadecimal

All four systems in this lesson are positional: the value of a digit depends on its position (place) in the number. The place at the right end is worth 1; every place to the left is worth base times the place on its right.

Decimal (base 10). Ten digits, 0 to 9. The places are worth 1, 10, 100, 1000 and so on, the powers of 10:

Binary (base 2). Only two digits, 0 and 1. The places are worth 1, 2, 4, 8, 16, 32, 64, 128 … — each place is double the one on its right. A binary digit is called a bit. Example: (110111)_2.

Octal (base 8). Eight digits, 0 to 7. The places are worth 1, 8, 64, 512, 4096 … Example: (375)_8. The digits 8 and 9 do not exist in octal.

Hexadecimal (base 16). Sixteen symbols: the digits 0 to 9 and the letters A to F, where A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15. The places are worth 1, 16, 256, 4096 … Example: (\mathrm{8F5})_{16}. "Hex" is the short name.

To show which system a number belongs to, write its base as a small number after it: (101)_2 is binary and means five, while (101)_{10} is the ordinary one hundred and one. A number written with no base is taken to be decimal.

Why computers use binary

A computer is built from millions of tiny electronic switches called transistors, and each one is either off or on. Two states are easy to make, easy to tell apart and hard to confuse even when a signal is weak: a low voltage stands for 0 and a high voltage for 1. The same two-state idea stores data everywhere in computing — a spot magnetised one way or the other on a hard disk, a pit or no pit on a DVD, a charge or no charge in a pen drive, light off or on in an optical fibre. Ten clearly different voltages would be far harder to build and to read reliably. So all data inside a computer — numbers, letters, pictures, sound — is stored as patterns of bits.

Why octal and hexadecimal are used

Long strings of 0s and 1s are hard for people to read and easy to copy wrongly. Because 8 = 2^3 and 16 = 2^4, one octal digit stands for exactly three bits and one hexadecimal digit for exactly four bits, so they are short, exact ways of writing binary. You will meet hexadecimal often:

• colour codes in web pages (Lesson 10): #FF0000 is pure red — FF of red, 00 of green, 00 of blue;
• MAC addresses of network cards (Lesson 14), such as 3C-52-82-9A-7F-10;
• memory addresses and error codes shown by programs.

Octal is used less today; you may see it in the file permissions of Linux, such as 755.

Look down the table: in each system, when the largest digit is used up, the count goes back to 0 and a 1 is carried to the next place — after 1 in binary, after 7 in octal, after 9 in decimal and after F in hexadecimal.

15.2 Converting decimal numbers to binary, octal and hexadecimal

To change a decimal number into another base, use repeated division by the new base:

1. Divide the decimal number by the base (2, 8 or 16). Write the quotient below it and the remainder at the side.
2. Divide that quotient by the base again, and write the new quotient and remainder.
3. Repeat until the quotient becomes 0.
4. Write the remainders from the bottom to the top. This is the answer.

Each division by the base splits off the last digit of the number in that base, just as dividing 4725 by 10 leaves the remainder 5, its last digit. So the first remainder is the rightmost digit and the last remainder is the leftmost digit — which is why the remainders are read upwards.

A second method for binary: subtracting place values

For binary there is a quick method that also checks the division method. Write the place values 1, 2, 4, 8, 16, 32, 64, 128 … Take the largest place value that fits into the number, write 1 under it and subtract; repeat with what is left; write 0 under every place value that is not used.

15.3 Converting binary, octal and hexadecimal numbers to decimal

To change a number from any base into decimal, use positional expansion:

1. Write the place value under each digit, starting with 1 at the right and multiplying by the base for each place to the left. (The place values are the powers of the base: \text{base}^0, \text{base}^1, \text{base}^2 …)
2. Multiply each digit by its place value. In hexadecimal, change the letters A to F into 10 to 15 first.
3. Add the products. The sum is the number in decimal.

These are the place values you will need most often:

15.4 Converting between binary, octal and hexadecimal

Since 8 = 2^3, every octal digit is exactly 3 bits; since 16 = 2^4, every hexadecimal digit is exactly 4 bits. So these conversions need no division at all — only grouping, using the two tables below.

Why does grouping work? Look at the place values of binary in threes from the right: 1, 2, 4 | 8, 16, 32 | 64, 128, 256. Each group of three is worth 1, 8 or 64 times the same pattern 1, 2, 4 — and 1, 8 and 64 are exactly the place values of octal. So each group of three bits, read as a number from 0 to 7, is one octal digit in the right place. In the same way, groups of four bits are worth 1, 16, 256 … times a number from 0 to 15: the hexadecimal digits.

Binary to octal. Starting from the right, cut the bits into groups of three. If the last group on the left is short, add 0s on its left. Write the octal digit for each group.

Binary to hexadecimal. The same, with groups of four.

Octal to binary. Replace each octal digit by its three bits, keeping any 0s inside a group. Hexadecimal to binary. Replace each hex digit by its four bits. Zeros at the far left of the final answer may be dropped.

Octal to hexadecimal, and back. There is no direct rule, because 8 and 16 are not powers of each other. Go through binary: write each octal digit as 3 bits, then regroup the bits in fours from the right (or the reverse).

print(bin(45))
print(oct(500))
print(hex(2024))
print(int("110101", 2))
print(int("1572", 8))
print(int("3AF", 16))
print(format(156, "b"), format(156, "o"), format(156, "X"))

0b101101
0o764
0x7e8
53
890
943
10011100 234 9C

15.5 Bits, bytes and the units of memory

One byte stores one character in the ASCII code used by English text: the letter A is the number 65, stored as the byte 01000001. So the word NEPAL needs 5 bytes, which is 40 bits. Nepali letters are not in ASCII; they are stored in Unicode, and in UTF-8, the form used on the web, a Devanagari character such as क takes 3 bytes. The word नेपाल is stored as five characters (न, े, प, ा, ल), so it takes 15 bytes. With n bits a computer can make 2^n different patterns: one byte (8 bits) has 2^8 = 256 patterns, the numbers 0 to 255. The largest byte, (11111111)_2, is 255, which is (377)_8 and (\mathrm{FF})_{16} — the FF of a colour code. The number of bits a processor handles at one time is called its word length: a "64-bit processor" works on 64 bits at once.

Bigger units are made by multiplying by 1024, not 1000, because computers count in powers of 2, and 2^{10} = 1024 is the power of 2 nearest to a thousand.

Practice questions

Group A — Objective (1 mark each)

1. The base of the hexadecimal number system is [1]a2b8c10d16
2. Which digit is not used in the octal system? [1]a0b5c7d8
3. (1011)_2 in decimal is [1]a11b13c9d1011
4. (25)_{10} in binary is [1]a11001b10011c11010d10101
5. In hexadecimal, the symbol D stands for [1]a11b12c13d14
6. How many bits make one byte? [1]a4b8c16d1024
7. 1 KB is equal to [1]a1000 bitsb1024 bitsc1024 bytesd1000 MB
8. (17)_8 in decimal is [1]a15b17c23d7
9. The binary equivalent of (\mathrm{A})_{16} is [1]a1010b1100c1001d1110
10. To convert a binary number into octal, its bits are grouped in [1]atwosbthreescfoursdeights
11. Which of these units is the largest? [1]aTBbPBcGBdEB
12. (64)_8 in binary is [1]a110100b1101111c1110001d11000101
13. (\mathrm{FF})_{16} in decimal is [1]a225b255c256d1515
14. A nibble is [1]a2 bitsb4 bitsc8 bitsd16 bits

Group B — Short answer (2 marks each)

1. What is a number system? Define base (radix) with one example. [2]
2. Why do computers use the binary number system? [2]
3. Convert (89)_{10} into binary. [2]
4. Convert (11100110)_2 into decimal. [2]
5. Convert (726)_8 into decimal. [2]
6. Convert (1000)_{10} into hexadecimal. [2]
7. Convert (\mathrm{2C9})_{16} into decimal. [2]
8. Convert (1011101)_2 into octal and hexadecimal. [2]
9. Convert 3 MB into KB and into bytes. [2]
10. Convert (352)_8 into hexadecimal. [2]
11. How many bits are there in 2 KB? Show the working. [2]
12. Find the decimal value of (\mathrm{FFF})_{16}, the largest three-digit hexadecimal number. [2]

Group C — Long answer (4 marks each)

1. Convert as indicated:(a) (175)_{10} into binary(b) (1101101)_2 into decimal(c) (634)_8 into decimal(d) (\mathrm{4D})_{16} into binary. [1+1+1+1]
2. Convert (420)_{10} into binary, octal and hexadecimal, and check one of your answers by converting it back. [4]
3. Why is each octal digit worth exactly 3 bits and each hexadecimal digit exactly 4 bits? Convert (\mathrm{A7F})_{16} into octal through binary. [1+3]
4. (a) Write the units of memory from the bit to the yottabyte in order, with the relation between each pair.(b) A 32 GB memory card stores songs of 4 MB each. How many songs can it hold?(c) Why does a "1 TB" hard disk show only about 931 GB on a computer? [2+1+1]
5. Compare the binary, octal, decimal and hexadecimal systems in a table (base, digits, largest digit, one use). Write (100)_{10} in each of the other three systems. [2+2]