تكنولوجيا المعلومات والاتصالات (بالإنجليزية) الصف الثالث الثانوي، الأسبوع 15: Review: number systems and conversions

Review and your own mini-project

الأسبوع 15 · مراجعة الفصل الدراسي الأول (الوحدتان ١ و٢) ومشروع صغير من تصميمك · 3 دروس

Review: number systems and conversions

ICT Secondary 3 Term 1, Review: pp. 5–86

GoalReview digits, place values, binary, hexadecimal and the six conversions with new numbers, and check each answer by going back.

New words

place value · the value of a place in a number: a power of the basebase · the number of symbols of a system: 2, 10 or 16check by going back · convert the answer back; you must reach the starting numberbyte · 8 bits, which is two hexadecimal symbols

Explanation

Here is Unit 1 in one page. A digit is one symbol and a number is made of digits; its value comes from place values, which are powers of the base (Week 1). Computers use only 0 and 1 because circuits have two states. A bit is one binary digit, a byte is 8 bits, n bits make 2ⁿ patterns, and 1 KB is 1,024 bytes (Week 2). In binary the place values are 1, 2, 4, 8, and so on; to go from decimal to binary we divide by 2 and read the remainders from the last to the first (Week 3). Hexadecimal has sixteen symbols, 0 to 9 and A to F, and place values 1, 16, 256; to go from decimal we divide by 16 and write the remainders 10 to 15 as A to F (Week 4). Because 16 = 2⁴, each hexadecimal symbol is four bits, so we convert between hexadecimal and binary in groups of four from the right (Week 5). The converter programs of Weeks 6 and 7 call ready functions that do these conversions, give hexadecimal in lower case, and fail on bad input, so a good program checks the text first. Every answer on this page is checked by going back.

Conversion	Method
binary to decimal	multiply each bit by its place value 1, 2, 4, 8 … and add
decimal to binary	divide by 2 again and again; read the remainders from last to first
hexadecimal to decimal	multiply each symbol by 1, 16, 256 … and add
decimal to hexadecimal	divide by 16; the remainders 10 to 15 are A to F; read last to first
hexadecimal to binary	four bits for each symbol
binary to hexadecimal	groups of four bits from the right, padded on the left

Worked examples

Division	Quotient	Remainder
52 ÷ 2	26	0
26 ÷ 2	13	0
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Decimal 52 to binary, step by step. Read the remainders from the last to the first: 110100.

Conversion	Answer	Check by going back
decimal 52 to binary	110100	32 + 16 + 4 = 52
binary 100111 to decimal	39	39 divided by 2 again and again gives 100111
hexadecimal 3B7 to decimal	951	951 ÷ 16 = 59 remainder 7; 59 ÷ 16 = 3 remainder 11 (B); this gives 3B7
decimal 1500 to hexadecimal	5DC	5 × 256 + 13 × 16 + 12 = 1,500
hexadecimal 9E to binary	1001 1110	9 is 1001 and E is 1110
binary 1110101 to hexadecimal	75 (from 0111 0101)	7 × 16 + 5 = 117 = 64 + 32 + 16 + 4 + 1

Six conversions with new numbers, each with its check.

Php

1echo decbin(52), " ", bindec("100111"), " ", hexdec("3B7"), " ", strtoupper(dechex(1500)), "\n";
2echo str_pad(decbin(hexdec("9E")), 8, "0", STR_PAD_LEFT), " ", strtoupper(dechex(bindec("1110101")));

Output

110100 39 951 5DC
10011110 75

The same checks in PHP. The ready functions give hexadecimal in lower case, so strtoupper makes capitals, and str_pad restores the leading zeros. The output lines come from running it.

1Console.WriteLine(Convert.ToString(52, 2))
2Console.WriteLine(Convert.ToInt32("100111", 2))
3Console.WriteLine(Convert.ToInt32("3B7", 16))
4Console.WriteLine(Convert.ToString(1500, 16).ToUpper())

Output

The same checks in VB.NET, inside a small console program. Convert.ToString(1500, 16) gives lower case, so ToUpper makes capitals. Bad input fails: Convert.ToInt32("1021", 2) raises a FormatException because 2 is not a binary symbol, and PHP’s bindec("1021") quietly ignores the 2 (with a notice) and gives 5.

Class activity

Convert on paper: decimal 75 to binary, binary 101101 to decimal, hexadecimal 2F8 to decimal, decimal 900 to hexadecimal, and hexadecimal C5 to binary. Check each answer by going back.

Practice

How many patterns can 7 bits make?

How many bits does a menu of 20 choices need, if every choice gets its own pattern?

How many hexadecimal symbols make one byte?

Why should the converter program check its input?