1. 4.2 Number Bases in Positional Systems

2. Ｒｅｃａｌｌ： Ｄｅｃiｍａｌ (base １０) System
１００００
１０００
１００ １０ １
１０４
１０３
１０２
１０１ ４ ３ ５ ２
４　ｘ　１００００
3　ｘ１０００
5　ｘ１００
1　ｘ１０ ２
= 43512
Uses 10 digits (symbols): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each position implies power of 10

3. Binary (base 2) System
Computer signals in terms of ‘yes’/’no’, ‘on’/’off’, ‘high voltage’/’low voltage’, 1/0.
Computer data storage also in terms of ‘on’/’off’, ‘north pole’/’south pole’, 1/0.
Binary system uses 2 symbols: 0, 1
E.g., 1two = 1ten two = 2ten two = 3ten two = 4ten
101two = 5ten

4. Binary (base 2) System 64 +0 +1 = 65dec Uses 2 digits (symbols): 0, 1
Each position implies power of 2

5. Binary (base 2) System
Decimal
Binary 1 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010
1011 12
1100 13
1101

6. Converting Binary to Decimal
Express the following binary notation in decimal notation.
0110binary
= (0 x 23) + (1 x 22) + (1 x 21) + (0 x 20) = = 6
1111binary
= (1 x 23) + (1 x 22) + (1 x 21) + (1 x 20) = = 15

7. Your Turn 111binary = (1 x 22) + (1 x 21) + (0 x 20) = 4 + 2 + 1 = 7
Express the following binary notation in decimal notation.
111binary
= (1 x 22) + (1 x 21) + (0 x 20) = = 7
10001binary
= (1 x 24) + (0 x 23) + (0 x 22) + (0 x 21) + (1 x 20) = = 17

8. Hexadecimal Notation (base 16)
C++ Code
Find the gcd of two numbers E.g., given: 12 and 36 gcd = 12

9. Computer Program Code Machine Language Code (What? No binary code?)
Hexadecimal Notation can serve as a short hand for Binary Notation.
In the last line: B
A C F F D

10. Hexadecimal (base 16) Notation
Uses 16 symbols:
E.g., 2Bhexadecimal = 43ten A3hexadecimal = 163ten
Decimal 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hexadec A B C D E F

11. Hexadecimal (base 16) Notation
4096
256 １6 １
１6３
１6２
１6１
１6０ ３ A C ２
3　ｘ 4096
10　ｘ 256
12　ｘ 16 ２
12288
2560
192 2
= 15042dec
Uses 16 digits (symbols): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Each position implies power of 16

12. Ｃｏｎｖｅｒｔｉｎｇ Ｈｅｘａｄｅｃｉｍａｌ ｔｏ Ｄｅｃｉｍａｌ Ｎｏｔａｔｉｏｎ
Ｃｏｎｖｅｒｔｉｎｇ　Ｈｅｘａｄｅｃｉｍａｌ　 ｔｏ　Ｄｅｃｉｍａｌ　Ｎｏｔａｔｉｏｎ
Dec
Hex 1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F
Ｃｏｎｖｅｒｔ from hexadecimal to decimal notation:
2Bhex
Recall: Pos. val Digit B Dec. val. (2 x 16) + (11 x 1) = 43dec
A3hex
(10 x 16) + (3 x 1) = 163dec

13. Your Turn Ｃｏｎｖｅｒｔ from hexadecimal to decimal notation: 8B hex1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F
Ｃｏｎｖｅｒｔ from hexadecimal to decimal notation:
8B hex
Recall: Pos. val Digit B Dec. val. (8 x 16) + (11 x 1) = 138dec
45 hex
(4 x 16) + (5 x 1) = 69dec

14. Ｃｏｎｖｅｒｔｉｎｇ Ｈｅｘａｄｅｃｉｍａｌ ｔｏ Ｄｅｃｉｍａｌ Ｎｏｔａｔｉｏｎ
Ｃｏｎｖｅｒｔｉｎｇ　Ｈｅｘａｄｅｃｉｍａｌ　 ｔｏ　Ｄｅｃｉｍａｌ　Ｎｏｔａｔｉｏｎ
(15 x 256+ (15 x 16) + (15 x 1)
= = 3095
Find the decimal numeral for the following hex numeral.
12A
FFF
(15 x 256+ (15 x 16) + (15 x 1) = = 3095
Pos val.
162
161
160
256 16 1
Digit 2 A
Value
1 x 256
2 x 16
10 x 1 32 10
= 298

15. Your Turn Convert the Hexadecimal Numeral to Decimal Numeral: 13E.1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F
Convert the Hexadecimal Numeral to Decimal Numeral: 13E.
Pos val.
162
161
160
256 16 1
Digit 3 E
Value
1 x 256
3 x 16
14 x 1 48 14
= 308

16. How Is Binary Notation Related to Hex Notation?
128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20 64
+16 +8 +2
= 90ten 16 1
161
160
128 64 32 8 4 2 27 26 25 24 23 22 21 20 5 A
5 x 16
10 x 1
= 90ten

17. Dec
Binary
Hex 1 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010 A
1011 B 12
1100 C 13
1101 D 14
1110 E 15
1111 F
Your Turn
Convert from binary to hexadecimal
Binary: Hex:
Binary: Hex: A B
Binary: Hex: B C

18. Your Turn A3 Hex: A3 Binary: 1010 0011 FC Hex: FC Binary: 1111 1100
Dec
Binary
Hex
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10
1010 A 11
1011 B 12
1100 C 13
1101 D 14
1110 E 15
1111 F
Your Turn
Convert from hexadecimal to binary. A3
Hex: A3 Binary: FC
Hex: FC Binary: