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A nonnegative integer's decimal expansion uses digits $0\le a_i\le 9$. Testing divisibility reduces this polynomial in ten modulo the divisor: for example $10\equiv -1(\mathrm{mod}11)$ makes divisibility by eleven equivalent to vanishing of the alternating digit sum.

A decimal representation is positional notation with radix ten and numeral digits zero through nine. For an integer, the nonnegative part is a finite decimal expansion; a negative integer has a separate minus sign. For example, $205=2\cdot 100+0\cdot 10+5$, whereas $250=2\cdot 100+5\cdot 10+0$. The numeral digits alone do not determine the value: their positions do.

Fractional positions extend the same place value rule to the right of the decimal point, with successive weights $1/10,1/100,1/1000,\dots$. ^[ref] For example,The matching binary representation is $11001101.011_2$, since its fractional part is $0\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}=3/8$. These representations terminate in both bases; finiteness in one radix does not imply finiteness in another.

Leading zeros in the integer part and trailing zeros in the fractional part leave the value unchanged: $00205.3750=205.375$. A zero appended to the right of an integer string has a different effect: $2050=10\cdot 205$. Keeping the decimal point fixed prevents confusing these two operations.

Every integer has a finite decimal expansion, but a noninteger rational number may need infinitely many fractional numeral digits, as in $1/3=0.333\dots$. A terminating decimal has a zero tail by convention. Infinite decimal strings have a further ambiguity that finite canonical integer strings do not: $0.999\dots =1$, because the geometric series ${\sum }_{j=1}^{\infty }9\cdot 10^{-j}$ sums to one. An infinite decimal tail denotes a limit, not a sufficiently long rounded approximation.

Use decimal expansion, $B=10$, $n=4$, $m=2$, and $q=100$. The Karatsuba operand splitting is $1234=12\cdot 100+34$, $5678=56\cdot 100+78$. The Karatsuba three-product identity givesThe third product has a three-digit operand. A small-input base case can evaluate these three products directly, or a smaller threshold can continue recursion. Either choice preserves the identity; only the operation count changes.

The same Karatsuba operand splitting handles $123\cdot 45$: choose $n=3$, $m=2$, and write $123=1\cdot 100+23$, $45=0\cdot 100+45$. Then $p_2=0$, $p_0=1035$, $p_s=24\cdot 45=1080$, and $h=45$, giving $4500+1035=5535$. A zero high block and unequal lengths require no new algebraic rule.

Multiplication combines two factors into their product of numbers. For a natural number $n$, $nx$ is the sum of $n$ copies of $x$; with no copies the result is zero. This repeated-addition interpretation directly explains integer grouping, while multiplication of arbitrary real numbers also describes scaling by noninteger factors.

A rectangle of side lengths $3$ and $6$ has area $18$. Split the second side into lengths $2$ and $4$. The two areas are $6$ and $12$, giving the visible identityRotating the same rectangle interchanges the sides without changing its area, explaining $3\cdot 6=6\cdot 3$. These pictures illustrate distributivity and commutativity for ordinary positive numbers; matrix multiplication and other operations require their own rules.

For a fractional example, take $3/2$ of a quantity and then $2/3$ of that result: the combined scale is $(2/3)(3/2)=1$. The two fractions are multiplicative inverses. In the unit square on the right, the overlap of a strip occupying $3/4$ of the width with one occupying $2/3$ of the height contains six of twelve equal cells. Its area is $(3/4)(2/3)=1/2$. The product stays exact even though the decimal expansion of $2/3$ repeats. ^[ref]

The decimal expansion gives $M={\sum }_{i=0}^s{a}_{i}10^i$. In modular arithmetic, $10\equiv -1(\mathrm{mod}11)$, and henceAn integer is divisible by eleven exactly when its congruence class is zero, proving both directions of the alternating-digit test.