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LECTURE 7:COMPUTERS AS STORAGE
CSC 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, 810913
Numbers & arithmetic easy to understand Subtracting roman numerals is not for faint-
of-heart
Positional Notation for 58622 = 2 ones = 2 * 1 = 2
Positional Notation for 58622 = 2 ones = 2 * 1 = 2
6 = 6 tens = 6 * 10 = 60
Positional Notation for 58622 = 2 ones = 2 * 1 = 2
6 = 6 tens = 6 * 10 = 60
8 = 8 hundreds = 8 * 100 = 800
Positional Notation for 58622 = 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 58622 = 2 ones = 2 * 1 = 2
6 = 6 tens = 6 * 10 = 60
8 = 8 hundreds = 8 * 100 = 800
5 = 5 thousands = 5 * 1000 = + 5000
5862
Decimal Positional Notation
Formal equation for a number dn...d3d2d1d0 d0 is digit in ones place, d1 is in tens place,
…d0 * 100
d1 * 101
d2 * 102
d3 * 103
…
+ dn * 10n
Base-10 Positional Notation
d0 2 = 2 ones = 2 * 1 = 2
d1 6 = 6 tens = 6 * 10 = 60
d2 8 = 8 hundreds = 8 * 100 = 800
d3 5 = 5 thousands = 5 * 1000 = + 5000
5862
Base-10 Positional Notation
d0 2 = 2 ones = 2 * 100 = 2
d1 6 = 6 tens = 6 * 101 = 60
d2 8 = 8 hundreds = 8 * 102 = 800
d3 5 = 5 thousands = 5 * 103 = + 5000
5862
Base-10 Positional Notation
d0 2 = 2 ones = 2 * 100 = 2
d1 6 = 6 tens = 6 * 101 = 60
d2 8 = 8 hundreds = 8 * 102 = 800
d3 5 = 5 thousands = 5 * 103 = + 5000
5862
Base-10 Positional Notation
d0 2 = 2 ones = 2 * 100 = 2
d1 6 = 6 tens = 6 * 101 = 60
d2 8 = 8 hundreds = 8 * 102 = 800
d3 5 = 5 thousands = 5 * 103 = + 5000
5862
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 PS3 can play Guitar Hero 3 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
A16 = 1010 D16 = 1310
B16 = 1110 E16 = 1410
C16 = 1210 F16 = 1510
Positional Notation
To convert dn...d3d2d1d0 into decimal:
From base-10d0 * 100
d1 * 101
d2 * 102
d3 * 103
…
+ dn * 10n
Positional Notation
To convert dn...d3d2d1d0 into decimal:
From base-bd0 * b0
d1 * b1
d2 * b2
d3 * b3
…
+ dn * bn
Converting Binary to Decimal1010112 = d0
d1
d2
d3
d4
d5
Converting Binary to Decimal1010112 = d0 1 *
d1 1 *
d2 0 *
d3 1 *
d4 0 *
d5 1 *
Converting Binary to Decimal1010112 = d0 1 * 20 =
d1 1 * 21 =
d2 0 * 22 =
d3 1 * 23 =
d4 0 * 24 =
d5 1 * 25 =
Converting Hex to Decimal
2716 = d0
d1
3F16 = d0
d1
Converting Hex to Decimal
2716 = d0 716= 710
d1 216= 210
3F16 = d0 F16=1510
d1 316= 310
Converting Hex to Decimal
2716 = d0 716= 710 * 160 =
d1 216= 210 * 161 =
3F16 = d0 F16=1510 * 160 =
d1 316= 310 * 161 =
Positional Notation Review
To convert dn...d3d2d1d0 into decimal:
From base-bd0 * b0
d1 * b1
d2 * b2
d3 * b3
…
+ dn * bn
Converting Decimal To Binary Converting from decimal to binary
(base-2):While decimal number ≠ 0
Divide decimal number by 2Move remainder to left end of
answerReplace decimal number with
quotient
3410 =
Converting Decimal To Base-b More generally, convert from decimal
to base-b: While decimal number ≠ 0
Divide decimal number by bMove remainder to left end of
answerReplace decimal number with
quotient
33510 = 16
Your Turn
Get in groups & work on following activity
For Next Lecture
Read sections 7.1, 9.8 for Friday What is the boolean data type? How do the boolean operations work? How do C++ functions return boolean
values?
Week #3 weekly assignment due Tuesday Problems available via Angel If problem takes more than 10 minutes,
TALK TO ME!
