8.2.2. Sign Magnitude

So, the similar information I am representing in another number system which is your hexadecimal. Now, you just see what I am saying this is your 8 bit number all 0s to all 1s. So, all 1 in decimal it becomes 255 in hexadecimal it is your FF, FF means 15 F represents 15, 15 × 16^1 + 15 × 16^0. So, in that particular case we will get that this is nothing but 255 in decimal.

This chunk discusses how to convert between binary and hexadecimal systems. In binary, each digit (bit) is either 0 or 1, and an 8-bit binary number can have values ranging from all 0s (0 in decimal) to all 1s (255 in decimal). The hexadecimal system, which is base 16, uses the symbols 0-9 and A-F, where F represents 15. Thus, the hexadecimal representation of the decimal number 255 is 'FF'. This is calculated by multiplying the values of each hex digit by their respective powers of 16.

Think of binary numbers like a simple light switch that can be either off (0) or on (1). An 8-switch panel can create a variety of combinations - just like how your house can have a variety of lighting scenarios. The hexadecimal system, on the other hand, is like using color codes in design; it’s a more compact way to convey complex color information. For example, the color white in hex code is #FFFFFF, which tells web developers exactly what light combinations to use.

Now, when we are going to discuss that representation of number in computer or in your binary system or binary digit most of the time we are going to take help of hexadecimal number system because it is having one advantage. Say, when I am going to work with 32 bit number I have to write 32 bits ok all 32 bits 0 or maybe combination of 0s and 1 which is slightly difficult.

In computing, hexadecimal is often preferred because it simplifies binary data representation. Instead of writing long strings of 0s and 1s, which can be cumbersome and error-prone, hexadecimal provides a more compact format. For instance, a 32-bit binary number becomes an 8-character hexadecimal number. This makes it easier for programmers and engineers to read and understand data.

Imagine trying to send a long message using only dots and dashes (binary). It would be tedious! Instead, if you could use letters or shorthand (hexadecimal), you still convey the same message but far more efficiently. Hexadecimal serves as that shorthand, making programming feel less like interpreting a Morse code and more like a regular language.

Now, for that we are having 2 ways of representing this number one is your sign magnitude in that particular case we are talking about the number system. Now, say if I am having 8 bit numbers. ... So, in that particular case now what will happen, if you take this particular 7 bit then it can the value that it can have go up to 127.

Sign magnitude is a method used in computers to represent positive and negative integers. In this method, one bit is designated as the sign bit (0 for positive, 1 for negative), while the remaining bits represent the magnitude of the number. In an 8-bit number, this allows for values between -127 and +127, since you reserve one bit for the sign. Therefore, there are a total of 256 possible combinations of bit patterns, but only 255 unique representations since two patterns can represent zero (positive 0 and negative 0).

Think of sign magnitude like a scoring system in games. The highest score you can achieve is 127 points, and you can also lose points, leading to negative scores, indicating a loss. The sign bit acts as a flag indicating whether your score is a win (positive) or a loss (negative). This helps players quickly understand their standing—just like how computers use bits to determine the value of numbers.

So, we are going to use this particular bit pattern to represent another number and we are saying that we are representing it as a -128, but why not +128. ... So, in that particular case you will find that just concentrate on this particular most significant bit it is 1.

In sign magnitude representation, -128 is used instead of +128 due to the restricted range of values allowable within the system. This restriction occurs because one bit is used for the sign, leaving one less bit for actual value representation. Consequently, while we can theoretically have an extra pattern for +128, the presence of a leading 1 indicates a negative value. Thus, the number representation in sign magnitude supports a range from -128 to 127.

Consider a temperature gauge that can only read between -128°C and +127°C. You can show whether it's freezing (negative) or warm (positive), but you can't show beyond that range. Just like that temperature gauge, the sign magnitude representation has its limits, preventing the expression of values beyond what it is designed to handle.

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