CS Club Meeting 4 10/10/13

Meeting Topic: Integers● ints in Java and C++ are 4 bytes and

signed○ long long (C++) and long (Java) are 8 bytes.

● There are 2^32 different values an int can take on.○ The smallest is -2^31, the largest is 2^31-1

● ints are stored as binary numbers. Each of the 32 bits (binary digit) is either 0 or 1○ The first bit is a sign bit; if it is a 1, the number is

negative, otherwise it is zero or positive.○ The int 1 is represented as 31 0’s followed by a 1○ And the int 0 is 32 0’s.

Two’s Complement Representation● Binary numbers are implemented using “2’s

complement”○ -1 is 11111111…

● Why?○ Let’s try decrementing 8 (0000…1000)○ 8 - 1 = 7 (0000…0111)○ Decrementing 0 (0000…0000) gives you…○ 0 - 1 = -1 (1111…1111)

● In general, -x + (x-1) = -1○ -x and x-1 are complements; -x = ~(x-1)○ ~ is the NOT operator. It flips every bit of an integer○ NOT 01111001 is 10000110

Two’s Complement Representation● This makes addition easier and makes

sense too.● Let’s try 10’s complement in decimal.

○ 4 digit decimal integer: ~x is 9999 - x○ ~ 1234 = 8765

● Let’s try subtraction○ - 3587 = ~3587 + 1 = 6412 + 1 = 6413○ 7892 - 3587 = 7892 + 6413 = 14305○ Keep only last 4 digits: 7892 - 3587 = 4305!

● Allows you to do subtraction with addition.● Note that ~x + 1 = 9999 - x + 1 = 10000 - x

Integer Overflow

● Integer overflow is when arithmetic attempts to create a value that is too large

● Arithmetic with large integers often leads to overflow○ If you do 100000 * 100000, you won’t get

10000000000 because that doesn’t fit in an int

○ In fact, you’ll get 1410065408 (10^10 mod 2^32)○ Be careful, even if you use a long!

int x = 100000;long long y = x * x;This will still overflow!!!

Integer Overflow Continued

● For what value of x is abs(x) < 0?○ SPOILER ALERT! The answer is coming up.○ x = INT_MIN (-2^31).

int abs(int x) {

if (x > 0) return x;

else return -x;

}

• What is 011111… + 1?o This is the transition from INT_MAX to INT_MIN

Addition is Easy

● Given two base 10 integers a and b, find their sum.

● Input Format:○ Line 1: Two numbers, a and b.

● Output Format:○ Line 1: One number, a + b.

● Constraints○ 5 points: -10^9 <= a,b <= 10^9○ 10 points: -10^18 <= a,b <= 10^18○ 20 points: 0 <= a,b <= 10^100○ 25 points: -10^100 <= a,b <= 10^100

Sample Case

Input: -10 999Output: 989Sample Explanation: -10 + 999 = 989.