1. Dale & Lewis Chapter 2 Binary Numbers and Number Systems

2. Number categories Definition of −Number  Unit of a mathematical system subject to arithmetic −Natural numbers  0, 1, 2, 3, 4… −Negative numbers  Number less then 0, i.e. -1, -2, -3, -4… −Integers  …-4, -3, -2, -1, 0, 1, 2, 3, 4… −Rational numbers  Fraction of integers, except division by 0, i.e. ¼, ½, -7/13…

3. The idea of a positional number system 4357 four thousand, three hundred and fifty seven four units of a thousand ( 4 x 1000) 4000 three units of a hundred ( 3 x 100) 300 five units of ten ( 5 x 10) 50 seven units of one ( 7 x 1) 7 4 x 10 3 + 3 x 10 2 + 5 x 10 1 + 7 x 10 0 5743 same digits, different positions, different number The position of each digit determines that digit’s contribution to the number.

4. The idea of a positional number system base: b any integer > 1 digits: 0, 1,..., b−1 number: d n-1 d n-2 …d 2 d 1 d 0 its definition d n-1 x b n-1 + d n-2 x b n-2 + … + d 2 x b 2 + d 1 x b 1 + d 0 x b 0 Examples: BaseDigits 20, 1 50, 1, 2, 3, 4 80, 1, 2, 3, 4, 5, 6, 7 100, 1, 2, 3, 4, 5, 6, 7, 8, 9 160, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

5. Binary, Octal and Hexadecimal Comparison of first natural numbers in four different bases etc...

6. …So what base was used in the data in the Matrix movies?

7. Arithmetic in other bases All the familiar rules of pencil-and-paper decimal arithmetic carry over to any other base Addition 46 101110 + 27 + 11011 Subtraction 5037 - 95 57 111001 - 6 - 110 Decimal Grid Binary Grid

8. Octal Grid Hexadecimal Grid

9. Examples of arithmetic In Octal: 375476312 +6317 + 4634 In Hexadecimal: 3B6F89FCD +5743 + 9FD In Decimal: 123 + 45 +3682 + 12 + 654

10. Conversion between Decimal and Binary Binary to Decimal −Use the definition of a number in a positional number system with base 2 −Evaluate the definition formula using decimal arithmetic Example 101011 = 1 x 2 5 + 0 x 2 4 + 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 = 43 (decimal)

11. Conversion between Decimal and Binary Decimal to Binary −Repeatedly divide by 2 −Quotient Carries −Remainder is the next digit −Binary number is developed right to left Example (Quotient) (Remainder) (Binary) 173 ÷ 286 1 1 86 ÷ 2 43 0 01 43 ÷ 2 21 1 101 21 ÷ 2 10 1 1101 10 ÷ 2 5 0 01101 5 ÷ 2 2 1 101101 2 ÷ 2 1 0 0101101 1 ÷ 2 0 1 10101101

12. Generalization: Conversion between Decimal and base b Base b to Decimal −Use the definition of a number in a positional number system with base b −Evaluate the definition formula using decimal arithmetic Decimal to base b −Repeatedly divide by b −Quotient carries −Remainder is the next digit −Base b number is developed right to left

13. Conversion between Binary and Octal/Hexadecimal Binary to Octal −Group bits into threes, right to left −Convert each group into an octal digit Example 1011010111 = 001 011 010 111 = 1327 (octal) Binary to Hexadecimal −Group bits into fours, right to left −Convert each group into a hexadecimal digit Example 1011001011 = 0010 1100 1011 = 2CB (hexadecimal)

14. Conversion between Binary and Octal/Hexadecimal Octal to Binary −Convert each digit to a three-bit binary representation Example 705 = 111 000 101 = 111000101 (binary) Hexadecimal to Binary −Convert each digit to a four-bit binary representation Example 10AF = 0001 0000 1010 1111 = 0001000010101111 (binary) Conversion tables can be reconstructed in the margins of a test paper when needed

15. What about converting between Octal and Hexadecimal? Hexadecimal is not base 8, so grouping won’t work as in binary  octal or binary  hexadecimal Octal to Hexadecimal −Convert octal to binary and then binary to hexadecimal Hexadecimal to Octal −Convert hexadecimal to binary and then to octal

17. DecimalBinaryOctalHexadecimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 8 100010 8 9 100111 9 10 101012 A 11 101113 B 12 110014 C 13 110115 D 14 111016 E 15 111117 F 16100002010 17100012111 18100102212 19100112313 20101002414 21101012515 22101102616 23101112717

18. 0123456789 000010203040506070809 101020304050607080910 202030405060708091011 303040506070809101112 404050607080910111213 505060708091011121314 606070809101112131415 707080910111213141516 808091011121314151617 909101112131415161718 01 00001 1 10

19. 01234567 00001020304050607 10102030405060710 20203040506071011 30304050607101112 40405060710111213 50506071011121314 60607101112131415 70710111213141516

20. 0123456789ABCDEF 0000102030405060708090A0B0C0D0E0F 10102030405060708090A0B0C0D0E0F10 202030405060708090A0B0C0D0E0F1011 3030405060708090A0B0C0D0E0F101112 40405060708090A0B0C0D0E0F10111213 505060708090A0B0C0D0E0F1011121314 6060708090A0B0C0D0E0F101112131415 70708090A0B0C0D0E0F10111213141516 808090A0B0C0D0E0F1011121314151617 9090A0B0C0D0E0F101112131415161718 A0A0B0C0D0E0F10111213141516171819 B0B0C0D0E0F101112131415161718191A C0C0D0E0F101112131415161718191A1B D0D0E0F101112131415161718191A1B1C E0E0F101112131415161718191A1B1C1D F0F101112131415161718191A1B1C1D1E