Binary numbers, bits, and Boolean operations

CSC 2001

Overviewsections 1.1, 1.5bitsbinary (base 2 numbers)

conversion to and fromaddition

Boolean logic

“Bits” of informationbinary digits

0 or 1why?

not necessarily intuitive, but…easy (on/off)powerful (more in a later lecture)

Number basesWhen we see that number 10, we

naturally assume it refers to the value ten.

So, when we read this…There are 10 kinds of people in this

world: those who understand binary, and those who don't.

It might seem a little confusing.

Number basesIn the world today, pretty much

everyone assumes numbers are written in base ten.Originated in IndiaThis cultural norm is very useful!

But 10 does not necessarily mean ten.What it really means is…

(1 x n1) + (0 x n0), where n is our base or our number system.

Base ten (decimal)So in base ten, we’ll set n = ten.Thus…

10 =(1 x n1) + (0 x n0) =(1 x ten1) + (0 x ten0) =(1 x ten) + (0 x one) =ten

Base ten (decimal)

527 =(5 x ten2) + (2 x ten1) + (7 x ten0) =(5 x one hundred) + (2 x ten) + (7 x

one) =five hundred and twenty seven

Other basesIn computer science, we’ll see that

base 2, base 8, and base 16 are all useful.

Do we ever work in something other than base 10 in our everyday life?

Other bases

Base twelveSumerian?smallest number divisible by 2, 3, & 4time, astrology/calendar, shilling,

dozen/gross, foot10 (base twelve) = 12 (base ten) [12 + 0]527 (base twelve) = 751 (base ten) [(5x144)

+ (2x12) + (7x1)]

Other bases

Base sixtyBabylonianssmallest number divisible by 2, 3, 4, & 5time (minutes, seconds),

latitude/longitude, angle/trigonometry10 (base sixty) = 60 (base ten) [60 + 0]527 (base sixty) = 18,127 (base ten)

[(5x602) + (2x60) + 7 = (5x3600) + 120 + 7]

Base two (binary)Just like the other bases…number abc = (a x two2) + (b x two1) +

(c x two0) = (a x 4) + (b x 2) + (c x 1)So..

There are 10 kinds of people in this world: those who understand binary, and those who don't.

means there are 2 kinds of people (1x2 + 0x1)

binary -> decimal practice11

1010

1000001111

Answers11 =

(1x2) + (1x1) = 31010 =

(1x23)+(0x22)+(1x2)+(0x1) =8 + 0 + 2 + 0 = 10

1000001111 =(1x29) + (1x23) + (1x22) + (1x2) + (1x1) =512 + 8 + 4 + 2 + 1 = 527

Powers of two20 = 121 = 222 = 423 = 824 = 1625 = 32

26 = 6427 = 12828 = 25629 = 512210 = 1024

decimal -> binaryAlgorithm (p. 42) figure 1.17Step 1: Divide the value by two and record

the remainderStep 2: As long as the quotient obtained is not

zero, continue to divide the newest quotient by two and record the remainder

Step 3: Now that a quotient of zero has been obtained, the binary representation of the original value consists of the remainders written from right to left in the order they were recorded.

Example 1:13 (base ten) = ?? (base 2)Step 1: Divide the value by two

and record the remainder13/2 = 6 (remainder of 1)

1

Example 1:13 (base ten) = ?? (base 2)13/2 = 6 (remainder of 1) 1Step 2: As long as the quotient obtained is

not zero, continue to divide the newest quotient by two and record the remainder

6/2 = 3 (remainder of 0) 03/2 = 1 (remainder of 1) 11/2 = 0 (remainder of 1) 1

Example 1:13 (base ten) = ?? (base 2)13/2 = 6 (remainder of 1) 16/2 = 3 (remainder of 0) 03/2 = 1 (remainder of 1) 11/0 = 0 (remainder of 1) 1Step 3: Now that a quotient of zero has

been obtained, the binary representation of the original value consists of the remainders written from right to left in the order they were recorded.

101 1

Example 2: 527527/2 = 263 r 1 1263/2 = 131 r 1 1131/2 = 65 r 1 165/2 = 32 r 1 132/2 = 16 r 0 016/2 = 8 r 0 08/2 = 4 r 0 04/2 = 2 r 0 02/2 = 1 r 0 01/2 = 0 r 1 1

0 11 11 10 0 0 0

In-class practice37

18

119

Answers 37:

37/2=18r1; 18/2=9r0; 9/2=4r1; 4/2=2r0; 2/2=1r0; 1/2=0r1

100101 = 1 + 4 + 32 = 37 18:

18/2=9r0; 9/2=4r1; 4/2=2r0; 2/2=1r0; 1/2=0r1 10010 = 2 + 16 = 18

119: 119/2=59r1; 59/2=29r1; 29/2=14r1; 14/2=7r0;

7/2=3r1; 3/2=1r1; 1/2=0r1 1110111 = 1 + 2 + 4 + 16 + 32 + 64 = 119

Binary operationsBasic functions of a computer

ArithmeticLogic

Binary additionAddition

Useful binary addition facts:0 + 0 = 01 + 0 = 10 + 1 = 11 + 1 = 10

Example

101011+011010

10

1

10

1

0

1

01

Multiplication and division by 2

Multiply by 2add a zero on the right side1 x 10 = 1010 x 10 = 100

Integer division by 2 (ignore remainder)drop the rightmost digit100/10 = 101000001111/10 = 100000111

(527/2 = 263)

Binary numbers & logicAs we have seen, 1’s and 0’s can

be used to represent numbersThey can also represent logical

values as well.True/False (1/0)George Boole

Logical operations and binary numbers

Boolean operatorsANDORXOR (exclusive or)NOT

Truth tablesAND F T

F F F

T F T

XOR F T

F F T

T T F

OR F T

F F T

T T T

NOT F T

- T F

- - -

Truth tables (0 = F; 1 = T)AND 0 1

0 0 0

1 0 1

XOR 0 1

0 0 1

1 1 0

OR 0 1

0 0 1

1 1 1

NOT 0 1

- 1 0

- - -

In-class practice(1 AND 0) OR 1

(1 XOR 0) AND (0 AND 1)

(1 OR ???)

(0 AND ???)

Answers(1 AND 0) OR 1 =

0 OR 1 = 1(1 XOR 0) AND (0 AND 1) =

1 AND 0 = 0(1 OR ???) =

1(0 AND ???) =

0

Summary Binary representation and arithmetic and

Boolean logic are fundamental to the way computers operate.

Am I constantly performing binary conversions when I program? Absolutely not (actually hardly ever!) But understanding it makes me a better programmer.

Am I constantly using Boolean logic when I program? Definitely! A good foundation in logic is very helpful when

working with computers.