تكنولوجيا المعلومات والاتصالات (بالإنجليزية) الصف الثالث الثانوي، الأسبوع 5: Binary to hexadecimal: grouping bits in fours

Hexadecimal and binary together

الأسبوع 5 · الوحدة ١ «النظم العددية» (٥): مثال آخر على التحويل إلى السداسي عشر، ومن السداسي عشر إلى الثنائي ومن الثنائي إلى السداسي عشر · 3 دروس

Binary to hexadecimal: grouping bits in fours

ICT Secondary 3 Term 1, Unit 1, Topic 1: pp. 22–26

GoalConvert binary to hexadecimal by grouping the bits in fours from the right.

New words

grouping · splitting the bits into groups of fourpadding · adding zeros on the left of the first group so that it has four bitsbyte as two symbols · 8 bits are two groups of four, so a byte is always two hexadecimal symbols

Explanation

To convert binary to hexadecimal, split the bits into groups of four, starting from the right. If the group at the left end has fewer than four bits, add zeros on its left. Then replace every group by its hexadecimal symbol from the lookup table. For example 11010111 splits into 1101 and 0111, which are D and 7, so (11010111)₂ = (D7)₁₆. And 101110 splits from the right into 10 and 1110; the group 10 is padded to 0010, so the answer is (2E)₁₆. Always start from the right, because the right end holds the place worth 1. A byte has 8 bits, so it is always two hexadecimal symbols, from 00 to FF.

Step	Result
Bits	101110
Groups from the right	10 \| 1110
Pad the left group	0010 \| 1110
Symbols	2 \| E

Worked examples

Groups	1101	0111
Symbols	D	7

Convert 11010111. Two groups of four are 1101 = D and 0111 = 7, so the answer is D7. Check: D7 is 13×16 + 7 = 215, and the bits are 128 + 64 + 16 + 4 + 2 + 1 = 215.

Grouping	Groups	Answer	Value
from the right	0101 \| 1011	5B	91
from the left	1011 \| 0110	B6	182

Grouping from the wrong end gives a wrong answer. The bits 1011011 split from the right as 101 and 1011, padded to 0101 and 1011, which is 5B and equals 91. Splitting from the left as 1011 and 011, padded to 1011 and 0110, gives B6, which is 182, twice as much.

Php

1<?php
2echo strtoupper(dechex(bindec("11010111"))), "\n";
3echo decbin(hexdec("3B"));

Output

D7
111011

PHP has no direct function from binary to hexadecimal. This program goes through a decimal number: bindec reads the bits, dechex writes hexadecimal in lower case, and strtoupper makes capitals. Then hexdec and decbin go the other way.

Class activity

Take a byte of your own, eight bits. Convert it to hexadecimal by grouping, then to decimal with the binary place values, and to decimal again from the hexadecimal. The two decimal answers must be equal.

Practice

Convert 10011100 to hexadecimal.

Convert 111001 to hexadecimal.

Convert 1101011 to hexadecimal.

How many hexadecimal symbols are needed for a pattern of 16 bits?

Why is the grouping started from the right?