PHY 107 – Programming For Science

Positional Notation  Used in nearly all modern numerical systems  Right-to-left ordering of digits within larger number  Expresses value using value of each digit (0, 1, 2, … 9)  Value of position in which the digit is places  e.g., 3, 13, 913, 0913, 10913,  Numbers & arithmetic easy to understand  Subtracting roman numerals is not for faint-of-heart

Positional Notation for = 2 ones= 2 * 1 =2

Positional Notation for = 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60

Positional Notation for = 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800

Positional Notation for = 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800 5= 5 thousands= 5 * 1000 =5000

Positional Notation for = 2 ones= 2 * 1 =2 6= 6 tens= 6 * 10 =60 8= 8 hundreds= 8 * 100 =800 5= 5 thousands= 5 * 1000 =

Decimal Positional Notation  Formal equation for a number d n...d 3 d 2 d 1 d 0  d 0 is digit in ones place, d 1 is in tens place, … d 0 * 10 0 d 1 * 10 1 d 2 * 10 2 d 3 * 10 3 … + d n * 10 n

Base-10 Positional Notation d0d0 2= 2 ones= 2 * 1 =2 d1d1 6= 6 tens= 6 * 10 =60 d2d2 8= 8 hundreds= 8 * 100 =800 d3d3 5= 5 thousands= 5 * 1000 =

Base-10 Positional Notation d0d0 2= 2 ones= 2 * 10 0 =2 d1d1 6= 6 tens= 6 * 10 1 =60 d2d2 8= 8 hundreds= 8 * 10 2 =800 d3d3 5= 5 thousands= 5 * 10 3 =

Base-10 Positional Notation 0d00d0 2= 2 ones 0 = 2 * 10 0 =2 1d11d1 6= 6 tens 1 = 6 * 10 1 =60 2d22d2 8= 8 hundreds 2 = 8 * 10 2 =800 3d33d3 5= 5 thousands 3 = 5 * 10 3 =

Base-10 Positional Notation d0d0 2= 2 ones= 2 * 10 0 =2 d1d1 6= 6 tens= 6 * 10 1 =60 d2d2 8= 8 hundreds= 8 * 10 2 =800 d3d3 5= 5 thousands= 5 * 10 3 =

Computer Number Systems  Previous equation worked in decimal (base-10)  Usual number system used in day-to-day life  System requires representing 10 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9  Computers always in one of two states  Turned on, your PS4 can play Bloodborne  Cell phones great paperweights when turned off  Binary digits ( 0,1 ) only used by computers  To use them, helps to know powers-of-two bases

Digits In Other Bases  Binary (base-2) uses 2 digits: 0, 1  Octal (base-8) uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7  Hexadecimal (base-16) has 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F A 16 = D 16 = B 16 = E 16 = C 16 = F 16 = 15 10

Positional Notation  To convert d n... d 3 d 2 d 1 d 0 into decimal: From base-10 d 0 * 10 0 d 1 * 10 1 d 2 * 10 2 d 3 * 10 3 … + d n * 10 n

Positional Notation  To convert d n... d 3 d 2 d 1 d 0 into decimal: From base-b d 0 * b 0 d 1 * b 1 d 2 * b 2 d 3 * b 3 … + d n * b n

Converting Binary to Decimal = d0d0 d1d1 d2d2 d3d3 d4d4 d5d5

Converting Binary to Decimal = d0d0 1* d1d1 1* d2d2 0* d3d3 1* d4d4 0* d5d5 1*

Converting Binary to Decimal = d0d0 1* 2 0 = d1d1 1* 2 1 = d2d2 0* 2 2 = d3d3 1* 2 3 = d4d4 0* 2 4 = d5d5 1* 2 5 =

Converting Hex to Decimal = d0d0 d1d1 3F 16 = d0d0 d1d1

Converting Hex to Decimal = d0d = 7 10 d1d = F 16 = d0d0 F 16 =15 10 d1d = 3 10

Converting Hex to Decimal = d0d = 7 10 * 16 0 = d1d = 2 10 * 16 1 = 3F 16 = d0d0 F 16 =15 10 * 16 0 = d1d = 3 10 * 16 1 =

Positional Notation Review  To convert d n... d 3 d 2 d 1 d 0 into decimal: From base-b d 0 * b 0 d 1 * b 1 d 2 * b 2 d 3 * b 3 … + d n * b n

Converting Decimal To Binary  Converting from decimal to binary (base-2): While decimal number ≠ 0 Divide decimal number by 2 Move remainder to left end of answer Replace decimal number with quotient =

Converting Decimal To Base-b  More generally, convert from decimal to base-b: While decimal number ≠ 0 Divide decimal number by b Move remainder to left end of answer Replace decimal number with quotient = 16

Your Turn  Get in groups & work on following activity

For Next Lecture  Read pages 91 – 98 for Monday  What is the boolean data type?  How do the boolean operations work?  What happens when we execute an if statement  Week #3 weekly assignment due Tuesday  Problems available via D2L  If problem takes more than 10 minutes, TALK TO ME!