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# 3.4: Exercises


Exercise $$\PageIndex{1}$$

Traditionally, computers keep track of the time/date using a format known as Unix time, which counts the number of seconds that have elapsed since 00:00:00 UTC on Thursday, 1 January 1970. But there's a problem if we track Unix time using a fixed-width integer, since that has a maximum value. Beyond this date, the Unix time counter will roll-over, wreaking havoc on computer systems. Calculate the roll-over date for:

1. Ordinary (signed) 32-bit integers
2. Unsigned 32-bit integers, which do not reserve a bit for the sign (and thus store only non-negative numbers).
3. Signed 64-bit integers
4. Unsigned 64-bit integers

Exercise $$\PageIndex{2}$$

Find the runtime of each of the following Python code samples (e.g. $$O(1)$$ or $$O(N)$$). Assume that the arrays x and y are of size $$N$$:

1. z = x + y
2. x[5] = x[4]
3. z = conj(x)
4. z = angle(x)
5. x = x[::-1] (this reverses the order of elements).

Exercise $$\PageIndex{3}$$

Write a Python function uniquify_floats(x, epsilon), which accepts a list (or array) of floats x, and deletes all "duplicate" elements that are separated from another element by a distance of less than epsilon. The return value should be a list (or array) of floats that differ from each other by at least eps.

Exercise $$\PageIndex{4}$$

(Hard) Suppose a floating-point representation uses one sign bit, $$N$$ fraction bits, and $$M$$ exponent bits. Find the density of real numbers which can be represented exactly by a floating-point number. Hence, show that floating-point precision decreases exponentially with the magnitude of the number.

3.4: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.