Introduction to Integer Data Type
Integers are a fundamental data type used in programming to represent whole numbers. Depending on the machine architecture, an integer can occupy either 2 bytes (16 bits) or 4 bytes (32 bits) of memory. The size impacts how much numerical content the integer can hold.
Determining Integer Size Programmatically
- The
sizeofoperator (a unary operator, not a function) is used to find the size of any data type at runtime. - Example: On many modern systems,
sizeof(int)returns 4, indicating 4 bytes. For a deeper understanding of data types and operators, see Understanding Variable Data Types and Operators in C++.
Understanding Range
- Range is the span between the minimum and maximum values a data type can represent.
- For example, a data set {0, 1, 2, 3, 4} has a range of 0 to 4.
Decimal Number System (Base 10)
- Humans naturally use the decimal system, which has digits from 0 to 9.
- Each digit's value depends on its place, weighted as powers of 10 starting from the right (100, 101, 102, etc.).
- Example: Number 568 equals (5×102) + (6×101) + (8×100) = 500 + 60 + 8.
- For fundamental concepts about numbers, consider reviewing Understanding the Real Number System: Key Concepts and Definitions.
Binary Number System (Base 2)
- Machines use binary, consisting only of 0s and 1s.
- Each bit's value is weighted as powers of 2, starting from the right (20, 21, 22, etc.).
- Example: Binary 1100 equals (1×23) + (1×22) + (0×21) + (0×20) = 8 + 4 + 0 + 0 = 12.
Calculating Range Based on Bits
- For n-bit unsigned binary data, the minimum value is 0 and the maximum is 2^n - 1.
- Example: 4-bit unsigned data ranges from 0 to 15.
- Formula for maximum value:
Max = 2^n - 1.
Integer Range Examples
For 2-byte (16-bit) integers:
- Unsigned range: 0 to 65,535 (
2^16 - 1). - Signed range (using two's complement): from -32,768 to 32,767 (
-2^{15}to2^{15} - 1).
For 4-byte (32-bit) integers:
- Unsigned range: 0 to 4,294,967,295 (
2^{32} - 1). - Signed range: calculated via two's complement method: from -2,147,483,648 to 2,147,483,647 (
-2^{31}to2^{31} - 1).
Representing Negative Numbers
- To store signed integers, computers use number systems like signed magnitude, one's complement, and two's complement.
- Two's complement is the most widely used method.
- It expands the range to include negative numbers while maintaining arithmetic operations' consistency.
Summary
- The integer data type size varies by system (commonly 2 or 4 bytes).
- Understanding decimal and binary systems is essential to grasp integer representation and range.
- The
sizeofoperator helps determine integer size programmatically. - The range of integers depends on bit length and whether integers are signed or unsigned.
- Two's complement representation is standard for handling negative integers in computing.
For more on how computers represent different data, including arrays, refer to Understanding Arrays in Programming: Declaration, Initialization, and Memory Representation. For foundational insight into data representation in C programming, see Understanding Data Representation in C Programming.
For more in-depth knowledge on number representations, exploring digital electronics resources such as Neso Academy's "Digital Electronics" playlist is recommended.
Today we are going to talk about a fundamental Data Type called integer. we have various data types to study. And today we begin our session with integer
Data Type. Recall from "lecture 3" i.e. introduction to variables. Integer can take either 2 bytes of memory space or 4 bytes of memory space,
depending upon the machine you're working with. As we know 1 byte is equal to 8 bits, therefore 2 bytes equals to 16 bits and
4 bytes is equal to 32 bits. More the size more content it can hold. Obviously, if you want to know the size of integer
programmatically, then you can by using "sizeof" operator. "sizeof" is a unary operator that helps us
in determining the size of any data type that we provide to it. As you can see here in my computer, "sizeof" integer is actually equal to four bytes
as the output is 4 of this "sizeof". I provided the output with the help of this printf function. Note: "sizeof" is a unary operator and not
a function. You might think of it as a function, but it is not a function. Now let's define "Range".
Range is nothing but upper and lower limit of some set of data. For example, here you can see this set consists of some values {0, 1,
2, 3, 4}. If you want to determine the range of this set of data, then you have to determine the minimum value as well as the maximum value.
Here, you can see the minimum value is 0 and the maximum value is equals to 4. Therefore, the ranges from 0 to 4. As you can see
in this set of data, there is no value which is less than 0, and there is no value which is greater than 4. And that is what the definition of "Range" is.
Now, before diving deep into the concept of range of integers, it is better to get into some prerequisites. And one of the prerequisites that we have to pass through is
knowing Decimal number system. In our day to day life, we use Decimal number system for many purposes. Maybe for counting purpose, or maybe when we
need to quantize some information like 2 apples, 30 bananas, and so on. Decimal number system is human understandable number system.
and is also called as base 10 number system. "base 10 simply means the range of number that we are available with is from 0 to 9.
Let's take one example of a decimal number. We pronounce this number as five hundred and sixty-eight. Instead of simply saying 568.
Have you ever wonder why do we pronounce it like that way? Because in reality, we have numbers ranging from 0 to 9.
And if you want to represent such a large number, we have to multiply each digit with their respective place values. Here place value are starting from the
rightmost end, and it is 10 raise to the power 0, then 10 raise to the power 1, then 10 raise to the power 2, and so on.
When we multiply 8 with 10 raise to the power 0, we get 8 only. When we multiply 6 with 10 raise to the power 1 we get 60.
When we multiply 5 with 10 raise to the power 2, we get 500. And now, in order to get the desired result, we add them up to get
568 as our result. Isn't that easy? Okay! But this is not something that machine could understand. It has its
own number system called Binary number system. Binary number system is machine understandable number system, and is also called as base 2 number system.
Means range is from 0 to 1. Only two numbers. As you can see, here, I've written a four bit data
1 1 0 0. In this case, also each digit must also have some place values associated with them. Because binary number system has base
is equals to 2. Therefore starting from the rightmost end we have 2 raise to the power 0, 2 raise to the power 1, 2 raise to the power 2
2 raise to the power 3 as their place values. And if you want to represent this number in decimal, then you have to follow the same method that we had adopted
in case of decimal number system i.e multiplying the digits with their respective place values, and finally add them up.
As you can see here, if I multiply 2 raise to the power 0 with 0, I get 0. 2 raise to the power 1 with 1, I get 0. 2 raise to the power 2 with 1 I get 2 raise to the power 2
which is equals to 4. 2 raise to the power 3 multiplied with 1 I get 2 raise to the power 3 and that is equals to 8. By
adding all these numbers, we get 12 as our answer. Now what would be the range of 4 bit data? As we know, know to calculate range of any number,
we have to know its minimum, as well as maximum value. For 4 bit data, minimum value is equals to 0. When all
digits are 0. And maximum value is equals to 15 when all digits are equals to 1. You can get this value by using the same method
by multiplying each digit with their respective place values and finally add them up. In this case of maximum value you're multiplying 2 raise to the power 3 with 1,
then 2 raise to the power 2 with 1, 2 raise to the power 1 with 1 and 2 raise to the power 0 with 1 and finally add them up to get the answer 15.
And here because all of them are zero, multiplying it with their respective place values would finally give the answer is equals to 0.
Therefore, range of all the data is from 0 to 15 only. But there's one more formula available to calculate the maximum value. And that is,
2 raise to the power n minus 1. Suppose you have a 32-bit data, then adding all these place values multiplying with their digits is quite
difficult, then this formula is quite handy to apply. Here, by replacing n is equals to 4 for 4 bit data, you get 2 raise to the power 4
minus 1 which is 16 minus 1, and that finally gives the values equals to 15, which is our maximum value. Let's calculate the range
of integer. If in your machine, the size of integer is of 2 bytes, then the unsigned range is from 0 to
65 535. By applying the same formula for calculating the maximum value. By replacing n with 16, you get 2 raise to
the power 16 minus 1 which is nothing but 65 535. Usually, you also have to represent the negative values.
And that we can do with the help of 2's compliment representation. We actually have three different representations to represent negative numbers, and they are
signed magnitude, 1's compliment and 2's compliment. But in most of the computers 2's compliment representation is used. 2's compliment range is from
-2 raise to the power n-1 to +2 raise to the power n-1 minus 1. If you want to know more about 2's compliment representation
and other representations, then you can refer the Neso Academy playlist of "Digital Electronics" here on YouTube. Here, you can see -2 raise to the power
n-1. By replacing n is equals to 16 I get 2 raise to the power 15 which is -32 768
and +2 raise to the power n-1 i.e. 2 raise to the power 15 minus 1 that gives 32 767. On the other hand if your PC supports
4 bytes integer, then Unsigned range is from 0 to 4 294 967 295. By applying the same formula for the
maximum value, Signed range can be obtained by using this 2's compliment range. That's it. See you in the next lecture. Bye. Bye.
You can determine the size of an integer using the sizeof operator in your programming language, such as C or C++. For example, sizeof(int) returns the number of bytes occupied by an integer on your system, which is commonly 4 bytes on modern architectures. This method allows you to programmatically check integer size at runtime.
A 4-byte (32-bit) signed integer uses two's complement representation and can range from -2,147,483,648 to 2,147,483,647. This range is calculated as from -2^(31) to 2^(31) - 1, allowing both negative and positive whole numbers to be represented efficiently.
Computers use the binary number system (base 2) because it corresponds naturally to the digital circuit states—on (1) and off (0). Each bit represents a power of 2, enabling straightforward hardware implementation for processing and storing information, unlike the decimal system which is base 10 and less efficient for electronic devices.
The maximum value of an n-bit unsigned integer is calculated using the formula 2^n - 1. For example, a 4-bit unsigned integer can represent values from 0 to 15 because 2^4 - 1 equals 15. This formula determines the highest number you can store when only positive integers (including zero) are allowed.
Two's complement is a method for encoding signed integers in binary allowing both positive and negative numbers to coexist. It is important because it simplifies arithmetic operations like addition and subtraction, and it avoids having two representations for zero, thereby making computations more efficient and consistent in digital systems.
The size of integer data types depends on the machine architecture and compiler implementation; common sizes are 2 bytes (16 bits) or 4 bytes (32 bits). The range is influenced by the number of bits and whether the integer is signed or unsigned. Signed integers use methods like two's complement to represent negative values, adjusting the range accordingly.
In the decimal system (base 10), each digit's place value is a power of 10, such as 10^0, 10^1, 10^2, etc., making the digits range from 0 to 9. In the binary system (base 2), each bit's place value is a power of 2 (2^0, 2^1, 2^2, etc.), with digits only 0 or 1. This difference underlies the fundamental way numbers are represented and processed in human and machine contexts.
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