Expanded Form
ArithmeticExpanded 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. You write each part as its own value, then add them all together.
Example
$4{,}352$ in expanded form is $4{,}000 + 300 + 50 + 2$.
Key Insight
Expanded form is like unpacking a suitcase: you take each digit out and show exactly what value it carries.
Definition
Expanded form expresses a numeral as a sum of products, each product being a digit multiplied by its place value. For a $4$-digit number $abcd$: $a \times 1000 + b \times 100 + c \times 10 + d \times 1$. This directly reveals the polynomial structure of base-$10$ representation.
Example
$70{,}405 = 7 \times 10{,}000 + 0 \times 1{,}000 + 4 \times 100 + 0 \times 10 + 5 \times 1 = 70{,}000 + 400 + 5$. Note that zero-valued places are often omitted.
Key Insight
Expanded form shows why addition with carrying works: when a column sum exceeds $9$, the "carry" is just regrouping $10$ ones into $1$ ten, consistent with the expanded representation.
Definition
Expanded form is the explicit polynomial evaluation of a numeral in base $b$: $n = \sum_{k=0}^{m} d_k \cdot b^k$. This form generalizes to any base and to fractional digits (positions with $k < 0$). In algebra, this structure motivates the definition of polynomial rings and the evaluation homomorphism.
Example
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 works regardless of base.
Key Insight
Expanded form directly connects arithmetic to algebra: understanding $n = 10a + b$ (a two-digit number) makes divisibility proofs, digital root calculations, and carrying algorithms all provable from first principles.