Facts about floating-point number systems

• To ensure full 6-bit precision, a number should be normalized--that is, adjusted so that its leftmost bit is 1. The more modern term for such a number is normal. Because it is known to be 1, the leading significant bit of a normal number need not be represented explicitly in an encoding of the number for storage. This use of an implicit leading bit saves a bit in the encoding of the number.
• Any finite, nonzero value in a floating-point number system has a unique representation as a normal number. By contrast, and are just two unnormalized representations of the value 1.
• A large floating-point number--in this system any number whose exponent is at least 5--is an integral value, with no fractional part at all.
• Numbers just larger than 1 are spaced apart, while numbers just smaller than 1 are spaced half as far apart. In fact, the normal powers of 2 divide the number system into binades, the binary analog of decades. Numbers in the binade below a power of two are spaced half as far apart as numbers in the binade above. Figure 54 is a picture of the situation near 1.

• Each binade has members, including the leftmost power of 2.
• Almost half of the positive floating-point numbers are between 0 and 1. Each binade stretches half the remaining distance to zero, but because the system is finite, the numbers (like Zeno's paradoxical arrow) never reach
  their target.

• The smallest normal number is . It is possible to represent even smaller values by admitting unnormalized values with the same exponent, emin. The IEEE binary floating-point arithmetic standard introduced these values as denormalized; the more modern term is subnormal.
• With subnormal numbers, the computed difference between any two different numbers is nonzero. For example, the difference between the two smallest normal numbers, , is the smallest subnormal number. (On most pre-IEEE platforms, this teeny difference would underflow to zero.)
• Subnormal numbers populate the interval between zero and the smallest normal number, with the same spacing as the normal numbers in the smallest normal binade ( in this case). Subnormal numbers reach all the way to zero because they have the same spacing as the normal numbers in the smallest binade. Figure 55 illustrates subnormal numbers.

Units in the last place

units in the last place of a specific value). For example, one ulp of , where abcde is any 5-bit sequence,
is . One ulp of is . One ulp of the subnormal value is .