Different-Based Number Systems

We discuss numbers most often in base-10, but other bases can sometimes be useful.

Exploding Dots

Please complete explorations 1-5 on the exploding dots website. Note that there is no interactive web app (right now), but the videos should explain the idea.

Mathematical Definition

A number is expressed in base $b$ if it is written as

$(d_n d_{n-1} d_{n-2} ... d_2 d_1 d_0)_b$

Where $d_i$ are digits from 0 to $b-1$, and the number can be expressed as

$d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + d_{n-1} \cdot b^{n-1} + ... + d_2 \cdot b^2 + d_1 \cdot b^1 + d_0 \cdot b^0$

For example, a base-2 number is always expressed using 1s and 0s, and 17 would be expressed as

$10001_2$ since $17 = 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$

Think about it: How does this all tie into exploding dots?