log 2(10)⌉ Maximum Number of Bits in a d-Digit Integer.Here’s the equivalent formula:ī min = ⌈log 2(10 d-1)⌉ = ⌈(d-1) Since we are dealing with powers of ten we can use the ceiling function here (as long as d > 1) there is no positive power of ten that is also a power of two. In this form, we take the logarithm of a small constant instead of a large variable. We can make this a more efficient computation by using the logarithmic identity log a(x y) = y The minimum number of bits required for a d-digit integer is computed simply by using the specific number formula on the minimum d-digit value:
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This diagram shows the ranges: Number of Bits In Four-Digit Decimal Integers Minimum Number of Bits in a d-Digit Integer Here’s how the examples look from that viewpoint: Why does this occur? Because that single power of ten range spans all or part of five consecutive power-of-two ranges.
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For example, 1344 requires 11 bits, 2527 requires 12 bits, and 5019 requires 13 bits. The number of bits varies between those extremes. Using the above formula you’ll see that the smallest four-digit number, 1000, requires 10 bits, and the largest four-digit number, 9999, requires 14 bits. For example, consider four-digit decimal integers. How many bits do numbers in this range require? It varies. Number of Bits in a d-Digit Decimal IntegerĪ positive integer n has d decimal digits when 10 d-1 ≤ n ≤ 10 d – 1. However, this fails when n is a power of two. You might be tempted to use the ceiling function - ⌈ x⌉, which is the smallest integer greater than or equal to x - to compute the number of bits as such: You can think of this step as accounting for the 2 0th place of your binary number, which then gives you its total number of bits.
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All of the discussion assumes positive integers, although it applies to negative integers if you temporarily ignore their minus signs. I will be discussing pure binary and decimal, not computer encodings like two’s complement, fixed-point, floating-point, or BCD. In this article, I will show you those calculations. Those values can be computed directly as well. For any d-digit range, you might want to know its minimum, maximum, or average number of bits. For example, four-digit decimal integers require between 10 and 14 bits. A range of integers has a range of bit counts. Sometimes you want to know, not how many bits are required for a specific integer, but how many are required for a d-digit integer - a range of integers. But there’s a way to compute the number of bits directly, without the conversion. For example, the two-digit decimal integer 29 converts to the five-digit binary integer 11101. To find the number of binary digits (bits) corresponding to any given decimal integer, you could convert the decimal number to binary and count the bits. Except for 0 and 1, the binary representation of an integer has more digits than its decimal counterpart.
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Every integer has an equivalent representation in decimal and binary.