Binary number system — conversions from - to decimal, counting in binary: Foundations

Before you can convert, count, or even read a binary number, you must own every symbol the parent note quietly used. This page builds them one at a time, from absolute zero. Nothing here assumes you have seen the notation before.

0. What is a "digit"? What is a "number"?

Look at the figure: the same pile of five dots can wear decimal clothing ("5") or binary clothing ("101"). Same pile, different marks.

Figure — Binary number system — conversions from - to decimal, counting in binary

1. The base $b$ — how many marks are allowed

Plain-words picture: the base is the size of your box of stamps. Decimal gives you a box of 10 stamps; binary hands you a box of only 2. You must build every number using only stamps from your box.

2. The subscript notation $(\dots)_{2}$ and $(\dots)_{10}$

3. Exponents: $b^{i}$ — repeated multiplication

The parent note is full of things like $2^{3}$ , $1 0^{2}$ , $2^{i}$ . You cannot read the place-value formula without owning this symbol first.

4. The position index $i$ and the "rightmost is 0" convention

Picture: the slots are houses on a street. The house nearest the "units end" is house 0. Its weight is $b^{i} = b^{0} = 1$ ; the next house is worth $b^{1}$ , then $b^{2}$ …

5. Weight — what one slot is worth

Plain-words picture: think of coins. In binary the coins are worth $1, 2, 4, 8, 16, \dots$ each. A digit "1" in a slot means "I have one coin of that value"; a digit "0" means "I have none of that value".

6. The summation symbol $\sum$ — "add all of these up"

The capital Greek "S", $\sum$ , scares people. It is just an instruction to add.

7. The floor bracket $⌊ ⌋$ and remainders

The decimal→binary algorithm uses $⌊ N /2 ⌋$ and "remainder". Both must be built.

8. Putting it together — the picture of a whole conversion

Now every symbol in $\sum_{i = 0}^{n} d_{i} \cdot 2^{i}$ is owned:

$d_{i}$ — the digit (0 or 1) sitting in house $i$ ,
$2^{i}$ — the weight of house $i$ (a coin value),
$\sum$ — add up the coins you actually have,
the subscript $_{2}$ — reminder we are in the base-2 box.

With that, "convert binary to decimal" reads in plain English as: add the coin-values of every slot showing a 1. Nothing mystical remains.

Prerequisite map

Equipment checklist

A number is really a

a count of things; the digits are just clothing for that count.

The base

b

means

how many different digits (stamps) the system allows.

Allowed digits in base

b

satisfy

0 \leq d < b

; so binary uses only 0 and 1, never a "2".

The subscript in

(101)_{2}

tells you

which base to read the digits in (here base 2).

b^{0}

equals

1, for every base — this is why the rightmost slot is worth 1.

2^{3}

equals

2 \times 2 \times 2 = 8

The rightmost slot is position

0, not 1; positions count from the right starting at 0.

The weight of the digit at position

i

b^{i}

The symbol

\sum_{i = 0}^{n}

tells you to

plug in

i = 0, 1, \dots, n

and add all the results.

⌊ 6.5 ⌋

equals

6 (throw away the fraction, keep the whole part).

The remainder of

N \div 2

0 if

N

is even, 1 if

N

is odd — it is the units bit.

Why "odd/even" fixes the last bit

only the units weight

2^{0} = 1

is odd, so it alone decides oddness.

Connections

Parent: Binary number system — these foundations feed directly into every conversion there.
Place-value and number bases — the general $\sum d_{i} b^{i}$ idea for any base $b$ .
Hexadecimal and Octal — the same foundations with $b = 16$ and $b = 8$ .
Bits and Bytes — once you own "digit" and " $2^{n}$ values", a byte is just 8 of these slots.
Logic gates and switches — the physical reason the digit box has exactly two stamps.
Binary addition and overflow — needs carrying, built from the "no digit 2" rule here.
Two's complement — reuses weights and position index to encode negatives.