Expanded Form
Expanded form is a way of writing a number as the sum of the values of each of its digits according to place value.
Definition
Expanded form means breaking a number apart to show how much each digit is worth, writing each part as its own value and then adding them all together. It expresses a numeral as a sum of products, each product being a digit multiplied by its place value: for a $4$-digit number $abcd$, this is $a \times 1000 + b \times 100 + c \times 10 + d \times 1$. More generally, expanded form is the explicit polynomial evaluation of a numeral in base $b$, $n = \sum_{k=0}^{m} d_k \cdot b^k$, which generalizes to any base and to fractional digits (positions with $k < 0$).
Example
$4{,}352$ in expanded form is $4{,}000 + 300 + 50 + 2$. $70{,}405 = 7 \times 10{,}000 + 0 \times 1{,}000 + 4 \times 100 + 0 \times 10 + 5 \times 1 = 70{,}000 + 400 + 5$, where zero-valued places are often omitted. The binary number $1011$ in expanded form is $1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 8 + 0 + 2 + 1 = 11$ in decimal, the same polynomial structure working regardless of base.
Key Insight
Expanded form is like unpacking a suitcase: you take each digit out and show exactly what value it carries, and it reveals why addition with carrying works, since a "carry" is just regrouping $10$ ones into $1$ ten. In algebra, this structure motivates the definition of polynomial rings and the evaluation homomorphism: understanding $n = 10a + b$ (a two-digit number) makes divisibility proofs, digital root calculations, and carrying algorithms all provable from first principles.