9.3.3. Precision of Real Numbers

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, numbers are encoded in a specific format. A standard 32-bit representation consists of three parts: 1 bit for the sign (indicating positive or negative), 8 bits for the exponent, and 23 bits for the significand (or mantissa). The sign bit determines if the number is positive (0) or negative (1). The exponent and significand together represent the actual number in scientific notation, similar to how we express numbers like 1.23 x 10^2.

Think of this representation as a way to package a letter. The sign bit is like a sticker on the envelope that says if it’s a happy letter or a sad letter. The exponent tells you how to arrange the letter for mailing, and the significand is the actual letter's content. Just as an envelope is divided into sections to carry different information about a letter, a number's representation is also divided to differentiate its sign, scale, and value.

Now, in this particular case we are having 2^20, but what we have stored over here you just see what is this equivalent 1 2. So, basically this is your 1 2 4 8 16 + 32 64 128. So, finally, this is the number that you are getting, so 128 + 19. So, we are getting 147. So, what we are storing basically in the exponent part we are storing 147, but basically we are looking for 20, but instead 20 we have storing it your 147. So, this is called biased exponent because now exponent may go positive as well as negative.

In floating point representation, the exponent is often stored in a 'biased' form to allow for both positive and negative exponents. For instance, if we want to store an exponent of 20, we might represent it in the biased form as 147 by adding a bias of 127 to 20. This approach allows the computer to encode both negative and positive values in a way that supports calculations without needing to handle negatives directly.

Imagine you are using a temperature scale and want to represent both hot and cold temperatures. Instead of having a thermometer showing negative values for cold temperatures, you might shift all readings up by a fixed number. For instance, if -20°C is represented as an index of 80, a zero temperature becomes 100. This way, everything is in a 'positive' range, making it simpler to understand and process.

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. The normalization will come like that decimal point will be always after 1 digit.

Normalization in floating point representation ensures that the decimal point is positioned just after the first non-zero digit. This is essential because it standardizes the representation of numbers and allows for greater precision. In binary, this means that the mantissa (significand) is adjusted to ensure it is stored in a format like 1.xxxxx, where 'x' represents the significant bits.

Think of normalization as putting a book on a shelf. If the book is not upright, it takes up more space and looks cluttered. By adjusting the book so that it stands up correctly, you minimize the shelf space used and keep everything organized. Just like this, normalization keeps the number representation tidy and efficient.

So, for floating point representation also that IEEE has given a format which is known as your 754, IEEE 754 format and in that particular format they are having two formats one is your 32 bit and another one is a 64 bit.

The IEEE 754 standard defines how floating point numbers are represented in computing. It standardizes two main formats for number representation: the 32-bit single precision and the 64-bit double precision. The choice between these formats affects the range and precision of the numbers that can be represented, with 64-bit providing greater accuracy and wider range.

Choosing between 32-bit and 64-bit representation is like choosing between different sizes of containers. If you have a small jar (32-bit), it can only hold a limited amount of liquid accurately. However, if you switch to a much larger container (64-bit), you can store more liquid without worrying about spills or overflow. Similarly, using a larger bit representation allows for storing larger numbers more accurately.

So, these are the two issues that we are having range and accuracy in floating point number.

Accuracy in floating point representation refers to how close the stored value is to the actual value, while range refers to the extent of values that can be represented. With more bits allocated to the mantissa, accuracy increases, while a larger exponent increases the range of representable numbers. The IEEE standard helps optimize both of these attributes.

Think of accuracy and range like playing darts. Your accuracy is how close you hit the target, while the range refers to how far away you can throw the darts. If you can only throw short distances accurately, you might not hit distant targets. However, if you improve your technique, you can increase both the distance you can throw (range) and how closely you can hit the target (accuracy).

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