# Chapter 2. Binary Values and Number Systems

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1 Chapter 2 Binary Values and Number Systems

2 Numbers Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative numbers A value less than 0, with a sign Examples: -24, -1, ,

3 Integers A natural number, a negative number, zero Examples: 249, 0, , - 32 Rational numbers An integer or the quotient of two integers Examples: -249, -1, 0, 3/7, -2/5 Real numbers In general cannot be represented as the quotient of any two integers. They have an infinite # of fractional digits. Example: Pi =

4 2.2 Positional notation How many ones (units) are there in 642? ? Or is it ? Or maybe ? 4 4

5 Positional Notation 642 is in BASE 10 The base of a number determines how many digits are used and the value of each digit s position. To be specific: In base R, there are R digits, from 0 to R-1 The positions have for values the powers of R, from right to left: R 0, R 1, R 2, 5 5

6 Positional Notation In our example: 642 in base 10 positional notation is: 6 x 10 2 = 6 x 100 = x 10 1 = 4 x 10 = x 10º = 2 x 1 = 2 = 642 in base 10 This number is in base 10 The power indicates the position of the digit inside the number 6 6

7 Positional Notation Formula: R is the base of the number d n * R n-1 + d n-1 * R n d 2 * R + d 1 n is the number of digits in the number d is the digit in the i th position in the number 642 is 6 3 * *

8 Positional Notation reloaded The text shows the digits numbered like this: d n * R n-1 + d n-1 * R n d 2 * R + d 1 but, in CS, the digits are numbered from zero, to match the power of the base: d n-1 * R n-1 + d n-2 * R n d 1 * R 1 + d 0 * R 0 8 7

9 Positional Notation What if 642 has the base of 13? + 6 x 13 2 = 6 x 169 = x 13 1 = 4 x 13 = x 13º = 2 x 1 = 2 = 1068 in base in base 13 is equal to 1068 in base =

10 Positional Notation In a given base R, the digits range from 0 up to R 1 R itself cannot be a digit in base R Trick problem: Convert the number 473 from base 6 to base 10 10

11 Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 11 9

12 12

13 Converting Binary to Decimal What is the decimal equivalent of the binary number ? =???

14 Converting Binary to Decimal What is the decimal equivalent of the binary number ? 1 x 2 6 = 1 x 64 = x 2 5 = 1 x 32 = x 2 4 = 0 x 16 = x 2 3 = 1 x 8 = x 2 2 = 1 x 4 = x 2 1 = 1 x 2 = x 2º = 0 x 1 = 0 = 110 in base

15 QUIZ: =??? 10 15

16 Bases Higher than 10 How are digits in bases higher than 10 represented? Base 16 (hexadecimal, a.k.a. hex) has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F 16 10

17 Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number DEF? D x 16 2 = 13 x 256 = E x 16 1 = 14 x 16 = F x 16º = 15 x 1 = 15 = 3567 in base 10 17

18 QUIZ: 2AF 16 =??? 10 18

19 Converting Octal to Decimal What is the decimal equivalent of the octal number 642? =???

20 Converting Octal to Decimal What is the decimal equivalent of the octal number 642? 6 x 8 2 = 6 x 64 = x 8 1 = 4 x 8 = x 8º = 2 x 1 = 2 = 418 in base

21 Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics? 21

22 Today we ve covered pp of the text (stopped before Arithmetic in Other Bases) Solve in notebook for next class: 1, 2, 3, 4, 5, 20, 21, 22 22

23 QUIZ: Convert to decimal =??? 10 AB7 16 =??? =??? =??? 10 23

24 Addition in Binary Remember that there are only 2 digits in binary, 0 and is 0 with a carry Carry Values 24 14

25 Addition QUIZ Carry values go here Check in base ten! 25 14

26 Subtraction in Binary Remember borrowing? Apply that concept here: Borrow values Check in base ten! 26 15

27 Subtraction QUIZ Borrow values Check in base ten! 27 15

28 Converting Decimal to Other Bases Algorithm for converting number in base 10 to any other base R: While (the quotient is not zero) Divide the decimal number by R Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient A.k.a. repeated division (by the base): 28 19

29 Converting Decimal to Binary Example: Convert to binary = 89 rem. 1 2 = 44 rem. 1 2 = 22 rem. 0 2 = 11 rem. 0 MSB LSB 2 = 5 rem. 1 2 = 2 rem. 1 2 = 1 rem = = 0 rem. 1 Notes: The first bit obtained is the rightmost (a.k.a. LSB) The algorithm stops when the quotient (not the remainder!) becomes zero 29 19

30 Repeated division QUIZ Convert to binary 42 2 = rem =

31 The repeated division algorithm can be used to convert from any base into any other base, but we use it only for 10 2 SKIP Converting Decimal to Octal Converting Decimal to Hex 31

32 Converting Binary to Octal Mark groups of three (from right) Convert each group is 253 in base

33 Converting Binary to Hexadecimal Mark groups of four (from right) Convert each group A B is AB in base

34 Counting Note the patterns! 34

35 Converting Octal to Hexadecimal End-of-chapter ex. 25: Explain how base 8 and base 16 are related A B 253 in base 8 = AB in base

36 Binary Numbers and Computers Computers have storage units called binary digits or bits Low Voltage = 0 High Voltage = 1 All bits are either 0 or

37 Binary and Computers Word= group of bits that the computer processes at a time The number of bits in a word determines the word length of the computer. It is usually a multiple of 8. 1 Byte = 8 bits 8, 16, 32, 64-bit computers 128? 256? 37 23

38 Individual work Read Grace Hopper s bio, the trivia frames, chapter review questions, and ethical issues 38

39 Who am I? I wrote the world s first compiler in 1952! 39

40 From the history of computing: bi-quinary Roman abacus (source: MathDaily.com) The front panel of the legendary IBM 650 IBM 650 (source: Wikipedia) 40

41 Chapter Review questions Describe positional notation Convert numbers in other bases to base 10 Convert base-10 numbers to numbers in other bases Add and subtract in binary Convert between bases 2, 8, and 16 using groups of digits Count in binary Explain the importance to computing of bases that are powers of

42 Homework Due next Friday, Feb. 3: End-of-chapter ex. 23, 26, 28, 29, 30, 31, 38 End-of-chapter thought question 4 (paragraph-length answer required) The latest homework assigned is always available on the course webpage 42

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