1.2. Biased Exponent

Now, just look for this particular representation. So, what is the size of this particular representation this is your 32 bits 1 bit is for sign bit, 8 bit is for your exponent and 23 bit is for significant.

In floating point representation, the total bit size is typically 32 bits. Of these, 1 bit is used to represent the sign of the number (positive or negative). The next 8 bits are designated for the exponent, which helps scale the number. Finally, the remaining 23 bits are allocated for the significand (or mantissa), which holds the significant digits of the number.

Think of this representation like a package for delivery. The 1-bit sign is a sticker on the package that tells whether the contents are delicate (negative) or can handle rough handling (positive). The 8-bit exponent is like the delivery address indicating how far and how to get the package there. Lastly, the 23-bit significand is the actual merchandise inside the box, representing the important details that need to be delivered.

So, this is basically same numbers positive and negative positive exponent and negative exponent now what is the exponent over here you just see here I am talking about 2 to the power. So, this is your 1 2 4 8 16, so 16 + 4, 20. So, basically we are talking about 2^20. So, this is the representation 2^20 and if I convert it it will come like that in decimal equivalent the value.

An exponent in floating point representation is used to indicate the scale of the number. In the example, the exponent is calculated as 2^20. To actually store this value, a biased exponent technique is used, which allows for both positive and negative exponents to be represented positively.

Imagine a seesaw. If one side goes up (positive exponent), it indicates more weight or presence while the other side is down (negative exponent), with less weight. The biased exponent is like a counterweight that ensures the seesaw is balanced, only actually showing the weight in a positive manner.

So, in this representation it is biased by 128; that means, whatever number we are storing over here that will be that to find out the exact exponent that 127 will be subtracted from that. So, 147 - 127 what I am getting this is 20, so this is the 20 that we are getting. So, this is the biased exponent.

The biased exponent allows us to work exclusively with positive integers to represent both negative and positive exponents. In a system where the bias is 128, you will take the actual exponent in binary, add 128 to it when storing, allowing for a wider range of representations. For instance, if you record 147, and the bias is 127, you subtract 127 to retrieve the original exponent, which is 20.

Consider a bank account system. The bank issues tokens that represent money stored. If the tokens only start from 128, you can always rest assured that even if someone withdraws their balance (negative exponent), it will still show an amount in the token system (positive integers) to avoid confusion.

So, this is the normalization. In the decimal number system also we do the normalization. So, these are the things that we are having. So, mantissa will be stored in your 2’s complement form exponent will be stored in your biased exponent. So, this is basically it will be stored in a biased exponent and after that what happen some values we need to subtract it may be from 128 or 127 for 8 bit numbers.

Normalization ensures that the floating point number is stored in a consistent format, where typically only one non-zero number precedes the decimal point. This approach simplifies and standardizes the representation while maximizing the range of values that we can represent. After normalization, offsets like 127 or 128 are applied depending on the representation used.

Consider formatting text for a book. Before printing, you would normalize the text so that the margins and spacing look consistent across all pages. This ensures that everything is neatly aligned and professional, making the content easy to read, similar to how normalization aligns our number representations.

Now, we must understand this 3 component while we are going to find out what is the exact value that we are storing in this particular representation.

Understanding the components of floating point representation is critical for converting the number back to its actual value. This involves knowing how to interpret the sign bit, biased exponent, and significand to recover the original number accurately.

Think of solving a puzzle where you need to assemble the frame first (the sign bit), find the edges (the biased exponent), and fit the pieces in the middle (the significand). Knowing where each piece fits together is essential for completing the whole picture successfully.

So, these are the two issues that we are having range and accuracy in floating point number. So, we are having a standard call IEEE standard and in most of the cases we use this particular standard because we should not come up with our own number system because globally it should be accepted.

The floating point representation has two main characteristics: range (the wide set of values it can represent) and accuracy (how precisely those values are represented). The IEEE standard ensures consistency across different systems, allowing for interoperability and reliability when handling floating point numbers.

Consider a common language everyone speaks to trade goods. Having a universal language ensures that sellers and buyers understand each other without confusion, similar to how the IEEE standard maintains clarity and interoperability in computing systems globally.

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