Number Families and Base Conversion

In AQA’s convention, natural numbers include zero and support counting. Integers also include negative whole numbers. A rational number can be written as a ratio of integers with a non-zero denominator; integers are rational because n=n/1. Irrational numbers, such as √2, cannot be expressed in that form.

The real numbers include rational and irrational values and model continuous measurements. A finite computer representation generally approximates rather than stores every real number exactly. Ordinal numbers describe positions in an ordering, such as first or third, rather than simply the quantity of objects.

In base b, successive integer positions have weights 1,b,b² and so on. Binary uses digits 0 and 1; hexadecimal uses 0–9 and A–F for values 10–15. A subscript or another explicit label distinguishes a base-16 representation from a decimal number.

Convert to decimal by summing digit×place-value products. Convert a non-negative integer to another base by repeated division, recording remainders and reading them in reverse order. Verify by expanding the result back into decimal rather than trusting a remembered visual pattern.

One hexadecimal digit represents exactly four binary bits because 16=2⁴. Group binary digits into fours from the binary point or integer right-hand end, adding harmless leading zeroes when needed. A hexadecimal string is therefore a compact readable representation of the same bit pattern.

The grouping does not perform encryption or change the stored information. Leading zeroes can matter when a fixed-width identifier or bit pattern is required, even though they do not change an unsigned integer’s value. Keep value, width and representation distinct throughout the conversion.

Attempt the task before opening its worked response. Record each change of state, explain the rule used, and check the result against the original requirements. Code examples labelled Python 3 use that language’s conventions; pseudocode states its own assumptions.