Integers

In java, an integer is a 32-bit signed two's complement integer. Integers have a minimum value of $-2^{31}$ or $-2147483648$ and a maximum of $2^{31} - 1$ or $2147483647$.

This might be a bit overwhelming, so lets step through this sentence slowly (And if it isn't overwhelming, read this page anyways, you might learn something new).

"32-bit"

What does it mean for a data type to have a certain number of bits? Well, lets go over the concept of a bit. In a computer, data is made up of bits, which are basically 1s and 0s. A bit is the smallest unit of memory in a computer. So, if a data type has 32 bits, it's represented using 32 bits in a row. So an integer can be represented as such: 11000010011011010110110111011111 . These bits basically read as binary, so we would convert this binary string to decimal.

If you don't know what binary is, binary is how you'd represent a number using 1s and 0s. It is also known as base-2. Lets talk about how to convert binary to decimal and back.

Normally, in base-10, also known as decimal, we have each digit be multiplied by a power of 10, and there are 10 digits (0-9). So if we have a number such as 457, it can be represented as the following: $4*(10^2) + 5*(10^1) + 7*(10^0)$. If you don't know what an exponent is, that's ok too, you can basically interpret the previous expression as the following:$4*(10*10) + 5*(10) + 7*(1)$.

In base-2, we have each digit be multiplied by a power of 2, and there are only 2 digits. Lets look into how we convert a number such as 19 to binary. 19 in base-10 is represented as the following: $1*(10^1) + 9*(10^0)$. Let's see how we'd represent this in base-2. In base-2, the powers of 2 are the following: $1, 2, 4, 8, 16, 32, 64, ...$ . So in order to have a number in base-2, it would be represented as 1s and 0s multiplied by powers of 2.

This table might come in handy:

We want to convert 19 to binary, so in order to do this we find the biggest power of 2 which can fit into 19. In this case, it's 16. Lets put a 1 at the 16 spot in the table. so now our binary representation (which is incomplete), is 00010000. Now we subtract 16 from 19, and repeat this process until we remain with 0. We have 3, and the highest power we can pack into that is 2. Now we subtract 2 from 3, and place a 1 in the 2 spot in the table. Our binary representation is now: 00010010.Now we are left with 1, and we'll place a 1 in the 1 spot in the table. We are left with 0, and our binary representation is the following: 00010011.

Exercises:

1. What is 75 in binary?
2. What is 13 in binary?
3. What is 36 in binary?
4. What is 10010101 in decimal?
5. What is 11010001 in decimal?
6. What is 00101001 in decimal?

"signed"

What does it mean for a data type (in almost all cases a numeric data type) to be signed? If an integer is stored in 32 bits, lets use our old example above: 11000010011011010110110111011111.

Lets concentrate our focus to integers, since we will only be dealing with integers in this section. A signed integer is an integer where the first bit decides the sign of the integer. In our integer above, our bits start off like this: 11000... The first bit is set to 1, so we know that this is a negative number. If the bit is set to 0, then it's a positive number. The other 31 bits then decide what the number actually is. I won't bore you with the computational details, but our 32 bit number above actually comes out to be $-1033015841$.

Keep in mind that if you have an 8 bit signed integer as such: 10000000, then it equals $-128$, even though the 7 bits after the 1 are 0. If alter the next 7 bits further, it increases the number, so would be 10000001 would be $-127$. If you keep going until 11111111, you'll get $-1$. If you want $0$, naturally it will be represented as such: 00000000.

If you want to find the negative representation of an integer, the method is as follows. You take your number of bits, so lets say we have 8 bits. Then we have $2^{8} = 256$. Next, we take our negative number, lets say we use $109$. Now we subtract it from $256$, and simply write the binary representation of the difference. In our case, the differnce is $147$, and the binary representation of that is 10010011 .

"two's complement"

The description also mentioned "two's complement," and that basically indicates another method of calculating a negative number. If lets say we have 109, which has the binary representation of 01101101, then in order to get the negative represntation, we can flip all the bits, to get 10010010, and then add 1 to this number, making the final representation 10010011.

Exercises:

1. Assuming an integer only uses 8 bits, what is -109 in binary?
2. Assuming an integer only uses 8 bits, what is -26 in binary?
3. Assuming an integer only uses 16 bits, what is -602 in binary?
4. Assuming the number is signed and uses 8 bits, what is 10010100 in decimal?
5. Assuming the number is signed and uses 8 bits, what is 11010001 in decimal?
6. Assuming the number is signed and uses 16 bits, what is 1010100101011010in decimal?