• If an integer is four bytes, how to we number the bytes?
• Two normal answers, both with extreme proponents.

The Problem of Real Numbers

Representing Reals

• We've found ways to represent integers. Will that help with real numbers?
• One approach, similar to that for characters: Map between real numbers and integers.
  • Unfortunately, there isn't a natural mapping of reals onto the integers.
  • See your Math professor for more details.
• If we're going to used a fixed number of bits, we're unlikely to be able to represent many irrational numbers or particularly small numbers, so we will never be able to get all the reals in a particular range. We must approximate them.
• We'll design a few different representations and I'll challenge you to critique them.
  • You are allowed to reflect on the reading in your criticisms.

One approach: Rational numbers

• Since we have no irrational reals, we can represent many real numbers fairly easily as rational numbers: ratios of two integers.
• Reserve some number of bits for the numerator, represented as a signed integer in the notation of your choice.
• Reserve some number of bits for the denominator.

Another approach: Fixed precision

• A common early representation of reals was the fixed-point strategy.
• Given a sequence of bits to use to represent a real number, select a position for the decimal point (except that it should be called the binary point).
• The column to its left is the 1's column, the next column to the left is the 2's, then the 4's, and so on and so foth.
• The column to the right of the point is the 1/2's column. The next column is the 1/4's, and so on and so forth.
• Note that this is effectively a representation of rational numbers with a fixed denominator.

Detour: Scientific Notation

• Rather than trying to design a representation from scratch, perhaps we should reflect on common practices in other disciplines.
• Many folks who work with real numbers (including approximations of real numbers) use scientific notation, as in 1.231 x 10⁵.
• What is involved in scientific notation?
  • A sign, which indicates whether the number is positive or negative,
  • A mantissa, which gives the primary digits of the number.
  • A base of exponentiation (in this case, 10).
  • An exponent.
• Might we not use the same general technique in designing a representation for real numbers?
• Such notations are called floating point because the point moves depending on the exponent.

IEE Single-Precision Floating-Point Numbers

• Rather than having each computer manufacturer decide on a particular floating point representation, standards groups worked do design a few standard representations.
• Two of the most popular representations are IEEE Single-Precision and Double-Precision floating-point numbers.
  • We'll concentrate on single-precision numbers. Double-precision numbers follow similar conventions.
• Single-precision numbers use 32 bits.
  • One bit gives the sign.
  • Eight bits give the exponent (represented in bias-127 notation).
  • The remaining 23 bits give the mantissa, using fixed-point notation.
• Note that the base of exponentiation is not represented. It is instead fixed at 2.
• There are some important restrictions on the various parts.
  • The exponent must be between -126 and 127
  • The mantissa must be no smaller than 1.0 and cannot be as large as 2.0.
• Note the clever trick: Given that restriction on the mantissa, you don't need to represent the first bit of the mantissa (since it's always 1).

Special Cases

• There are some parts the previous discussion ignores.
• The range of exponents only gives 254 different values. We have two additional unused values, corresponding to bit sequences 00000000 and 11111111.
• If the mantissa can be no smaller than 1.0, how do we represent 0?

Special Case: Representing Small Values

• If the exponent consists of only 0's, we use -126 as the exponent and restrict the mantissa to values no less than 0 and less than 1.
• Hence, 0 has an exponent of all 0's and a mantissa of all 0's.

Special Case: Error Values

• If the exponent consists of only 1's, the float represents some special value.
• Positive infinity uses all 0's for the mantissa, as does negative infinity (sign bit gives the sign).
• Anything else is NaN.

Lab

• Lab.
• This is an optional lab. We probably won't have time for it.