Introduction

When we think about writing a program in C, the first step is understand how variables should be assigned. There are several variable’s type in C, and here we are introducing the type int, which is used for integer data types. Basically, we can define a variable as an integer in two ways:

Additionally, there are a large set of storage size-specific declarations for a integer, and here we will explain just an initial idea about it. Figure 2 showns the Integer representation of whole numbers or fixed-point numbers (fixed number of digits). Generally, computers use a fixed number of bits to represent them, where commonly used bit-lengths for integers are 8-bit, 16-bit (short), 32-bit (long) or 64-bit (long long). There are two representation schemes for integers called signed integer type (signed int) capable of containing the range of values from -32,767 to 32,767, and unsigned integer type (unsigned int) containing the range of values from 0 to 65,535 ($32767 \times 2+1$). Therefore, unsigned qualifier should be used when we are working with only positive values.

Furthermore, there are three representation schemes for signed integers called Sign-Magnitude representation, 1’s Complement representation, and 2’s Complement representation. The 1’s and the 2’s complements of a binary number are important because they permit different representation for negative numbers. In all of these schemes, positive signed binary numbers starts with value 0 while negative ones starts with value 1 (Figure 3).

Consequently, the disadvantage of signed binary numbers is that there is 1 bit used to store the sign positive or negative while the remaning $n-1$ bits are assign to the range of digits from $-2^{n-1}$ to $2^{n-1}$. If we have 8 bits to represent a signed binary number, we have to use 1 bit for the **sign bit** and 7 bits for the **magnitude bits**: Thus, we can representing the numbers ranging from -128 to 127 using 2's Complement Representation. Probably now you are asking why there is one extra number being accounted when using 2's Complement Representation. The answer can be found in the Figure 4.

Examples

Unsigned int

Supose we are interested in representing a sequence of number $x$ where $x \in \lbrace 0, 1, \ldots, 15\rbrace$. We can assign these numbers as unsigned numbers of 4 bits. Consequently, we have 4 zero bits associated to describe this numbers because our variable belongs to the interval $[0, 2^{4}−1] \in \mathcal{N}_{0}$.

Representation of numbers from 0 to 15 in 4 bits

bits	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
x	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Signed int

Supose now we are interested in representing a sequence of number $y$ where $y \in \lbrace -7, -6, \ldots,6, 7\rbrace$. We have to assign them as signed numbers using 4 bits because 1 bit will be used for sign bit and 3 bits for the magnitude bits to describe $y \in \left[-|2^3-1|,2^3-1\right] \in \mathcal{Z}$.

Sign-Magnitude Representation of numbers from -7 to 7 using 4 bits

bits	0111	0110	0101	0100	0011	0010	0001	0000	1000	1001	1010	1011	1100	1101	1110	1111
y	7	6	5	4	3	2	1	0	-0	-1	-2	-3	-4	-5	-6	-7

2's Complement Representation of numbers from -8 to 7 using 4 bits

bits	1000	1001	1010	1011	1100	1101	1110	1111	0000	0001	0010	0011	0100	0101	0110	0111
y	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7

References

Barnett R.; O’Cull L.; Cox, S. Embedded C Programming and the Microship PIC. Delmar Learning, ed. 1, 2004.

Cadenhead, R.; Liberty, J. Sams Teach Yoirself C++. Pearson Education, ed. 6, 2017.

C Data Types - https://en.wikipedia.org/wiki/C_data_types

Did you find this page helpful? Consider sharing it 🙌