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Data types and their representation
Jordi CortadellaDepartment of Computer Science
Outline
• Data representation
• Boolean expressions
• Real numbers
Introduction to Programming © Dept. CS, UPC 2
The memory
0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 10 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 0 01 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 00 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 11 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 00 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 10 0 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 00 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1
• • •
• • • Address
1036104010441048105210561060106410681072107610801084
byte
int
19
0
H e l lo w or l d
string(“Hello world”)
Introduction to Programming © Dept. CS, UPC 3
How much memory do we have?Our laptop has few Gbytes of memory:
Introduction to Programming © Dept. CS, UPC 4
Number representation
2100 10 1
1 7 base 10
325 5 1
3 21125
base 5
09 3 1
0 1227
281
base 3
04 2 1
0 118
116
0113264128
base 2
CCXVII Roman
Introduction to Programming © Dept. CS, UPC 5
1
Write the binary representation
Design a procedure that, given a number n, writes its binary representation (in reverse order).
// Pre: n 0// Writes the binary representation of n// in reverse order.void base2(int n);
04 2 1
0 118
116
0113264128
Write: 10011011
0 011011
n%2n/2
Introduction to Programming © Dept. CS, UPC 6
Write the binary representation// Pre: n 0// Writes the binary representation of n// in reverse order.
void base2(int n) { while (n > 1) { cout << n%2; n = n/2; } cout << n << endl;}
n cout217 1108 0
54 027 113 1
6 03 11 1
Introduction to Programming © Dept. CS, UPC 7
ExerciseDesign a procedure that, given a number n and a base b, writes the representation of n in base b(in reverse order).
// Pre: n 0, 2 b 9.// Writes the binary representation of n// in base b (in reverse order).void base(int n, int b);
Introduction to Programming © Dept. CS, UPC 8
Boolean expressions
Introduction to Programming © Dept. CS, UPC 9
Boolean Algebra
George Boole, 1815-1864
Introduction to Programming © Dept. CS, UPC 10
Maximum of three numbers// Returns max(a, b, c)int max3(int a, int b, int c);
a > banda > c
a
b
b > c
c
else
a,b,c
else
b,c
a >= band
a >= c
a
b c
else
b >= c
else
a,b,c
b,c
Introduction to Programming © Dept. CS, UPC 11
Maximum of three numbers// Returns max(a, b, c)int max3(int a, int b, int c) { if (a > b and a > c) { return a; } else { if (b > c) { return b; } else { return c; } }}
Choosebetweenb and c
a > banda > ca
b
b > c
c
else
a,b,c
else
b,c
Introduction to Programming © Dept. CS, UPC 12
Maximum of three numbers
// Returns max(a, b, c)int max3(int a, int b, int c) { if (a > b and a > c) return a; if (b > c) return b; return c;}
Introduction to Programming © Dept. CS, UPC 13
Boolean operators
if (a > b and a > c) …
while (i >= 10 or c == 0) …
if (not (a < b and a < c)) …
Introduction to Programming © Dept. CS, UPC 14
Truth tables
a b a and bfalse false falsefalse true falsetrue false falsetrue true true
a b a or bfalse false falsefalse true truetrue false truetrue true true
a not afalse truetrue false
Introduction to Programming © Dept. CS, UPC 15
…
Boolean expressions
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
x > 2 and x < 7
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
… …
x >= 3 and x <= 6
x <= 2 or x > 7
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
…x <= 2 not (x <= 2) x > 2
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
?x == 5
x
Introduction to Programming © Dept. CS, UPC 16
…
Boolean expressions
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
x > 2 and x < 7
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
… …
x >= 3 and x <= 6
x <= 2 or x > 7
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
…x <= 2 not (x <= 2) x > 2
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
not (x == 5) x != 5……
Introduction to Programming © Dept. CS, UPC 17
Exercise
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
x == 0 or x == 6
Introduction to Programming © Dept. CS, UPC 18
Exercise
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
x == 0 or x > 5
…
Introduction to Programming © Dept. CS, UPC 19
Exercise
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
x%2 == 0
……
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Exercise
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
(x > -1 and x < 3) or(x >= 6 and x < 11)
Introduction to Programming © Dept. CS, UPC 21
Complement of and/or
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
……
(x > 2 and x < 7)
not (x > 2 and x < 7)
x <= 2 or x >= 7
not not
De Morgan’s lawIntroduction to Programming © Dept. CS, UPC 22
De Morgan’s law
not (e1 and e2) not e1 or not e2
not (e1 or e2) not e1 and not e2
Exercise
Simplify: not (x >= y or y%2 == 0)
x < y and y%2 != 0
Introduction to Programming © Dept. CS, UPC 23
Operator precedence
a + bc (a + b)c
a or b and c (a or b) and c
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Real numbers
Introduction to Programming © Dept. CS, UPC 25
Intersection of circles
x1, y1, r1 x2, y2, r2
• Write a program that reads the center and the radius of two circles and prints “yes” or “no” depending on whether they intersect or not.
• Example:
2 5.3 1.34 0 0 2 no 1.5 2.5 10 0.5 3.6 4.3 yes
Introduction to Programming © Dept. CS, UPC 26
Intersection of circles
(
(𝑟1 𝑟2
Circles intersect if and only if the distance betweencenters is smaller than or equal to the sum of radii.
Introduction to Programming © Dept. CS, UPC 27
Intersection of circles
(
(𝑟1 𝑟2
√ (𝑥1−𝑥2 )2+( 𝑦1−𝑦 2)2≤𝑟1+𝑟 2
(𝑥1−𝑥2 )2+( 𝑦1−𝑦 2 )2≤ (𝑟1+𝑟2 )2
Introduction to Programming © Dept. CS, UPC 28
// Reads the center and radius of two circles// and prints whether they intersect or not.int main() { double x1, y1, r1, x2, y2, r2; // Real numbers cin >> x1 >> y1 >> r1 >> x2 >> y2 >> r2;
double dx = x1 – x2; dx = dxdx; // (x1 – x2)^2 double dy = y1 – y2; dy = dydy; // (y1 – y2)^2 double r = r1 + r2; r = rr; // (r1 + r2)^2
bool intersect = dx + dy <= r; // true or false
if (intersect) cout << "yes" << endl; else cout << "no" << endl;}
Intersection of circles
Introduction to Programming © Dept. CS, UPC 29
Boolean variablesbool b = false;b = i <= n; // true if i <= n, false if i > nb = x + y > z;b = (i < 10) and (j >= 20);b = (large and not found) or (x > y);b = true;
if (b) { … } // if (b == true) { … }
while (not b) {…} // while (b == false) { … }
Introduction to Programming © Dept. CS, UPC 30
Real numbers• Two types:
– float (single precision, 32 bits)– double (double precision, 64 bits)
• Arithmetic operations: + - / (no remainder)
• Real constants:2 -5.003 3.1416 1.4e9 0.6E-15
1.4 0.6 Introduction to Programming © Dept. CS, UPC 31
© Dept. CS, UPC 32
Type conversion
• Arithmetic operations between integer and real values usually imply an implicit type conversion.
• Be careful:
int i=3, j=2; double x; x = i/j; // x = 1.0 x = i/double(j); // x = 1.5 x = double(i)/j; // x = 1.5 x = double(i/j); // x = 1.0 x = i/2; // x = 1.0 x = i/2.0; // x = 1.5
i = x; // i = 1 j = 3.14159265; // j = 3
Introduction to Programming
// Returns x2
double sq(double x) { return xx;}
// Reads the center and radius of two circles// and prints whether they intersect or not.int main() { double x1, y1, r1, x2, y2, r2; cin >> x1 >> y1 >> r1 >> x2 >> y2 >> r2;
if (sq(x1 – x2) + sq(y1 – y2) <= sq(r1 + r2)) cout << "yes"; else cout << "no";
cout << endl;}
Revisiting intersection of circles
Introduction to Programming © Dept. CS, UPC 33
Summary• Variables are stored in memory locations and
internally represented as bit vectors.
• Boolean expressions and variables:– Can take two values: true or false.– Use negative thinking when simpler than positive
thinking (apply De Morgan’s law).
• Real values:– Can be represented with a limited precision.– Be careful with int double conversions.
Introduction to Programming © Dept. CS, UPC 34
