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The blocks in the Karatsuba three-product identity are integers, not single digits: $p_0$ can need $2m$ positions and $h$ can exceed $q-1$. Concatenating the displayed blocks therefore fails. Accumulate their shifted values with ordinary addition, normalizing each digit position by Euclidean division: a coefficient $v$ becomes the digit $v\mathrm{mod}B$, with carry $\lfloor v/B\rfloor$ passed to the next position. Subtraction likewise propagates borrows. Although $h\ge 0$ for these nonnegative integers, intermediate digitwise subtraction can still borrow.

In the four-digit Karatsuba worked multiplication, grouping by base $q=100$ gives unnormalized coefficients $[672,2840,2652]$, written from high to low. Normalize from the right: $2652=26\cdot 100+52$; $2840+26=2866=28\cdot 100+66$; $672+28=700=7\cdot 100+0$. The final groups are $[7,00,66,52]$, yielding $7,006,652$. The zero group must retain its two positions.

Since $a,b,c,d<B^m$, their sums are less than $2B^m\le B^{m+1}$ and can need $m+1$ digits. Allocate that extra digit before the third recursive product; discarding it can change the answer. The GMP manual describes this sum growth and uses a difference-based form to avoid it.^[ref]

Karatsuba multiplication is a divide-and-conquer algorithm for multiplication of integers. Splitting each operand into two blocks seems to require four smaller products; the Karatsuba three-product identity recovers the two cross terms together from just three products. Repeating that saving through recursion changes the time complexity from quadratic to $O(n^{{\log }_{2}3})$, where $n$ counts digits in a fixed positional notation. The exponent is about $1.585$: the benefit comes from the algorithm, even though each splitting step needs more addition and subtraction.

The Karatsuba worked multiplication below computes $1234\cdot 5678=7,006,652$ using $12\cdot 56$, $34\cdot 78$, and $46\cdot 134$. The last operand already shows why “half-size” needs care: Karatsuba carry handling and Karatsuba recursion termination account for the extra digit. Karatsuba digit-operation cost states the operation model; Karatsuba practical crossover explains why the asymptotic saving alone does not select a useful threshold.

The original 1962 paper presents a binary numeral system automaton construction: its Theorem 2 and proof reduce multiplication to taking squares of numbers, then reduce a larger square of a number to three smaller squares of numbers. Its footnote explicitly addresses the extra digit in a sum of halves. The sequential time complexity analysis below uses a different operation model from that paper's automaton size and depth bounds, including its Boolean circuit realization without feedback.^[ref]

Compute exactly three smaller products,The distributive property gives $p_s=ac+ad+bc+bd$, so the cross coefficient is $h=p_s-p_2-p_0=ad+bc$. Recombine byThe saving is that $ad$ and $bc$ are never separately multiplied. Multiplying a digit block by $q$ or $q^2$ shifts its position in positional notation; it does not require another general integer multiplication. Addition of the shifted results still requires Karatsuba carry handling.