Foundations — Data types — char, short, int, long, float, double, size_t

Before you can understand char, int, float, or size_t on the parent note, you need a small toolbox of ideas. This page builds every single one from nothing — no symbol used before it is drawn.

1. The bit — one switch

The picture: a single lightbulb. Dark = 0, lit = 1. That is the entire alphabet of a computer — just two symbols.

Why the topic needs it: every type (char, int, double) is measured in bits and bytes. If "bit" is fuzzy, "8 bits" and "how many patterns fit" stay fuzzy too.

2. The byte — a row of 8 switches, and counting patterns

Now the key question the whole topic leans on: how many different arrangements can 8 switches make?

The picture: a branching tree — one fork per bit. Count the leaves at the bottom: exactly $2^n$ of them for $n$ bits.

Why the topic needs it: the range of every integer type is derived straight from this count. No $2^n$, no ranges.

3. The exponent notation $2^n$

The picture: the same branching tree — $n$ is simply how many rows of forks you drew.

Why the topic needs it: ranges like $[0,2^n-1]$ and $[-2^{n-1},2^{n-1}-1]$ are written with exponents. You must read $2^{n-1}$ as "one row of forks fewer than $2^n$", i.e. half as many patterns.

4. Interval notation $[a,b]$

The picture: a number line with a solid dot on $a$, a solid dot on $b$, and the whole stretch between them coloured in.

Why the topic needs it: every range in the parent — signed char $[-128,127]$, unsigned char $[0,255]$ — is written this way. If you misread the bracket you miscount by one at each end (the classic off-by-one bug).

5. Signed vs unsigned — where the minus sign hides

Here the same 256 patterns get two different meanings. This is the heart of the parent's integer section.

Unsigned: we agree every pattern names a non-negative count. Pattern 00000000 = 0, up to 11111111 = 255. Zero eats one slot, so the top value is $2^n-1$, one less than the count.

Signed (two's complement): we sacrifice the top bit's usual meaning and give it a negative weight, so patterns starting with 1 name negatives.

The picture: a ring (odometer) of the 256 byte patterns. On the unsigned ring the labels run 0…255. On the signed ring the top half is relabelled to negatives: after 127 it flips to −128 and climbs back to −1. Same switches, different labels.

This is the two's-complement idea explored fully in Two's Complement Representation, and the wrap-around at the ring's seam is Integer Overflow and Undefined Behaviour.

Why the topic needs it: the parent's worked example (char c = 200 becoming -56) is exactly walking from the unsigned label to the signed label of the same pattern.

6. Scientific notation — the seed of float

Integers count in equal steps. But to store both an atom's size and a galaxy's size, you can't step evenly — you'd need billions of bits. Instead you store the number the way scientists write it.

The picture: a slider. The digits fix the shape of the number; the exponent slides the decimal point left (small) or right (huge). A few digits + a wide slider covers an enormous range.

Why the topic needs it: IEEE-754 float/double are binary scientific notation — a sign, a mantissa (the digits), and an exponent (the scale). This is the whole of IEEE-754 Floating Point.

7. The pieces of a float's formula

The parent writes $x=(-1{)}^s\times (1.f{)}_2\times 2^{e-\text{bias}}$. Every symbol, plainly:

Why the topic needs it: you cannot read "23 mantissa bits → ~7 digits" without knowing which part is the mantissa.

8. The sizeof question and size_t

The picture: a ruler that only reads $0$ upward (no negative marks) and is long enough to measure the biggest array the machine could ever hold.

Why the topic needs it: the parent's countdown-loop bug (size_t i; i >= 0) only makes sense once you know size_t is unsigned — its odometer can never dip below 0, so it wraps to a giant number instead.

How it all feeds the topic

Equipment checklist

Cover the right side and test yourself. You are ready when every line is instant.