CMPSC210 Lecture 7: Hexadecimal Numbers. Prof. John Wenskovitch 09/16/2016
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1 CMPSC210 Lecture 7: Hexadecimal Numbers Prof. John Wenskovitch 09/16/2016
2 Last Time Binary numbers Converting from binary to decimal Converting from decimal to binary Introducing ways to represent negative numbers 09/16/2016 Hexadecimal Numbers 2
3 Reminder Q&A Lecture on Monday 9/19. Send questions! 09/16/2016 Hexadecimal Numbers 3
4 What is Hexadecimal? Base-16 counting. Now we need 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (A) 16 = (10) 10 (B) 16 = (11) 10 (C) 16 = (12) 10 (D) 16 = (13) 10 (E) 16 = (14) 10 (F) 16 = (15) 10 (10) 16 = (16) 10 09/16/2016 Hexadecimal Numbers 4
5 Powers of 16 Just multiply the previous value by 16! 16 0 = = = 16 1 = = = = = = = = = = = = = /16/2016 Hexadecimal Numbers 5
6 Converting Hexadecimal to Decimal What does (DEFACE) 16 mean? (D 16 5 ) + (E 16 4 ) + (F 16 3 ) + (A 16 2 ) + (C 16 1 ) + (E 16 0 ) D E F A C 16 + E 1 ( ) + ( ) + ( ) + (10 256) + (12 16) + (14 1) ,603,198 09/16/2016 Hexadecimal Numbers 6
7 Converting Decimal to Hexadecimal Convert ( ) 10 into hexadecimal. With binary, we started by finding a power bigger than what we were converting = = With binary, we subtracted off that value: = Problem: We can subtract off again! And again! And again! 09/16/2016 Hexadecimal Numbers 7
8 Converting Decimal to Hexadecimal Convert ( ) 10 into hexadecimal. New procedure: Once we find the power bigger and step back, we also need to find a coefficient in the [1,F] range that is bigger = = = = = = = = = A coefficient of 9 is too big, so we want to use 8. Write an 8, and subtract off = /16/2016 Hexadecimal Numbers 8
9 Converting Decimal to Hexadecimal Repeat the process. We need a coefficient in front of 16 4 that is bigger than , then to step back one. Easy method: divide by 65536! = So our coefficient is 11 in hexadecimal that s B. Multiply B = to find what to subtract off. Do the subtraction: = Write a B: 8B 09/16/2016 Hexadecimal Numbers 9
10 Converting Decimal to Hexadecimal Repeat the process. We need a coefficient in front of 16 3 that is bigger than 52512, then to step back one. Easy method: divide by 4096! = So our coefficient is 12 in hexadecimal that s C. Multiply C 4096 = to find what to subtract off. Do the subtraction: = 3360 Write a C: 8BC 09/16/2016 Hexadecimal Numbers 10
11 Converting Decimal to Hexadecimal Repeat the process. We need a coefficient in front of 16 2 that is bigger than 3360, then to step back one. Easy method: divide 3360 by 256! = So our coefficient is 13 in hexadecimal that s D. Multiply D 256 = 3328 to find what to subtract off. Do the subtraction: = 32 Write a D: 8BCD 09/16/2016 Hexadecimal Numbers 11
12 Converting Decimal to Hexadecimal Repeat the process. We need a coefficient in front of 16 1 that is bigger than 32, then to step back one. Easy method: divide 32 by 16! = 2 So our coefficient is 2 in hexadecimal that s 2. Multiply 2 16 = 32 to find what to subtract off. Do the subtraction: = 0 Write a 2: 8BCD2 09/16/2016 Hexadecimal Numbers 12
13 Converting Decimal to Hexadecimal Repeat the process. We need a coefficient in front of 16 0 that is bigger than 0, then to step back one. Easy method: divide 0 by 1! 0 1 = 0 So our coefficient is 0 in hexadecimal that s 0. Multiply 0 1 = 0 to find what to subtract off. Do the subtraction: 0 0 = 0 Write a 0: 8BCD20 Done! ( ) 10 = (8BCD20) 16 09/16/2016 Hexadecimal Numbers 13
14 Try These Convert (DECAF) 16 to decimal. Convert ( ) 10 to hexadecimal. 09/16/2016 Hexadecimal Numbers 14
15 Hexadecimal Conversion Convert (DECAF) 16 to decimal. (DECAF) 16 = D E C A F 16 0 = = (912559) 10 Convert ( ) 10 to hexadecimal ( ) = = = = = = (12D687) 16 09/16/2016 Hexadecimal Numbers 15
16 Hexadecimal and Binary If computers use binary, then why is MIPS/MARS showing us hexadecimal? Hexadecimal is binary. (54321) 10 = ( ) 2 (54321) 10 = (D431) 16 09/16/2016 Hexadecimal Numbers 16
17 Hexadecimal and Binary (54321) 10 = ( ) 2 (54321) 10 = (D431) 16 D = 13 = = = = 0001 (54321) 10 = ( ) 2 09/16/2016 Hexadecimal Numbers 17
18 What Does This Work Remember that 2 4 = 16 That means that 16 2 = 2 8, 16 3 = 2 12, etc. So, we can take every 4 bits from a binary string and convert them into a hexadecimal string Saves space Easier to not have a typo in A52A9 than in /16/2016 Hexadecimal Numbers 18
19 Always Right to Left! Convert ( ) 2 to hex. If we go left-to-right, we get: 1001 = = A 1001 = 9 01 = 1 If we go right-to-left, we get: 10 = = = A 0101 = 5 Which is correct? 09/16/2016 Hexadecimal Numbers 19
20 Order Doesn t Matter So Much with Hex -> Binary but you still need 4 bits per hex digit (hexit?) Convert (1ABCDEF) 16 into binary. 1 = 0001 A = 1010 B = 1011 C = 1100 D = 1101 E = 1110 F = /16/2016 Hexadecimal Numbers 20
21 Try These Convert (3A8D2) 16 to binary Convert ( ) 2 to hex. 09/16/2016 Hexadecimal Numbers 21
22 Solutions Convert (3A8D2) 16 to binary Convert ( ) 2 to hex A5 09/16/2016 Hexadecimal Numbers 22
23 Rough Binary-Decimal Estimations 2 10 is about 10 3 (1024 vs. 1000) 2 20 (or 16 5 ) is about 10 6 ( vs ) 2 30 is about 10 9 What power of 2 roughly corresponds to ? 2 19 Remember, halfway between 0 and just means dropping by a single power of 2. 09/16/2016 Hexadecimal Numbers 23
24 Any Questions? 09/16/2016 Hexadecimal Numbers 24
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