9.1. Floating Point Number Representation

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.

Floating point representation is a way to express real numbers in a computer. In a 32-bit floating point representation, the bits are divided into three parts: 1 bit is used for the sign, 8 bits are for the exponent, and 23 bits are for the significand (also known as the mantissa). The sign bit indicates whether the number is positive or negative. The exponent determines the scale of the number, and the significand holds the precision bits of the number.

Think of signing a document. The sign bit is like your signature indicating if it's a contract you agree to (positive) or refuse (negative). The exponent can be thought of as the size of the document—a larger size means more content, while the significand is akin to the actual text of the document that specifies details.

So, now, if we look into this number representation say 1.10100001 into 2^10100. So, if this is the number that we want to represent in our floating point, then we have to see what is the significant part. The significant part is your 1010001, 1010001, so 10010001. So, this is the significant part and it is having total 23 bits. So, remaining part will be all 0s.

When representing a number like 1.10100001, we express it in scientific notation as 1.10100001 multiplied by 2 raised to a power, which in this example is 2 raised to 20. The significant part (or mantissa) in binary is stored as 10010001. For floating point representation, to handle a variety of magnitudes (both very small and very large numbers), the exponent is stored using a bias, which allows for easier representation of negative exponents.

Imagine you are measuring the height of a mountain. The base height (significand) is how tall the mountain starts from sea level (1.10100001), and the exponent indicates how many times you've multiplied that height by a certain factor (like 2^20). The bias in this case helps to classify both tiny hills (negative exponents) and huge mountain ranges (positive exponents) into easily interpretable numbers.

This is called biased exponent because now exponent may go positive as well as negative. In this representation, it is biased by 128; that means, whatever number we are storing over here that to find out the exact exponent that 127 will be subtracted from that.

The concept of the biased exponent allows us to store both positive and negative exponents as positive numbers. In IEEE 754 floating point representation for 32-bit numbers, the bias is set at 128. When we need to retrieve the actual exponent value, we subtract 127 from the stored exponent to get the true value. For example, if we store 147, the actual exponent would be 147 - 127 = 20.

Consider a thermometer that only uses positive numbers to describe temperatures. If 0°C represents a baseline, we can treat colder temperatures as negative values but express them by adding a base number (the bias). So -20°C becomes 100 when we add 120 to it for ease of handling, even though we mentally know it’s still a negative temperature.

So, this is the way we are storing our floating point numbers. It is having 3 components: one is your sign bit, then biased exponent, and significand.

In floating point representation, we use three key components to store a number: the sign bit, the biased exponent, and the significand. The number is first normalized so that the decimal point is placed right after the first non-zero digit. This way of storing numbers helps in maximizing the precision of the significand while still utilizing the limited number of bits effectively.

Think of normalizing as filing documents in a cabinet. You place a paper in such a way that the most important information (the first non-zero digit) is easily visible when you open the cabinet. Just like ensuring your primary detail is showcased helps in quick retrieval of information, normalizing helps in maximizing numerical precision.

So, if we increase the number of bits, the range will increase as well as accuracy will also increase.

The accuracy of a floating point number representation is determined by the number of bits allocated to the significand. In standard 32-bit floating point representation, 23 bits are used for the significand, resulting in an accuracy of about 2^-23. If we want to improve this accuracy or allow for larger ranges of numbers, we can utilize more bits (for example in a 64-bit representation). The more bits available, the more precise the number can be, and the larger numbers can be represented without overflow.

Consider a ruler: a short ruler gives you limited measurement options, and it is difficult to measure large objects accurately. If you switch to a longer ruler (which corresponds to more bits), you can see much further and measure more precisely, effectively increasing your measurement range and accuracy.

In floating point representation also that IEEE has given a format 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 for floating point representation defines how real numbers should be represented in a computer. It includes formats for single precision (32-bit) and double precision (64-bit). The primary difference lies in the number of bits allocated to the exponent and significand, which also indicates the range of values and accuracy that can be achieved.

Think of a language with different dialects. The 32-bit format is for simpler communications (like having a brief chat), while the 64-bit format enables more detailed conversations (like writing a novel). The structure provided by IEEE 754 helps ensure everyone understands the format regardless of their computing environment, just as a standard language helps facilitate communication between people from different regions.

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