The influence of the place-value structure of the Arabic number ...

the output of the right node would be 1, because 0 .67 − 1 < 0. 5, respectively. Generally, the
output is calculated as follows:
⎧ t
⎫
j
if t
j
− a
j
< d
⎪
⎪
o
j
= ⎨a
j
if t
j
− a
j
≥ d ⎬
(2)
⎪ a if rt = rt ⎪
max
⎩
j
⎭
Where t j denotes the correct output and a j reflects the actual activation of node j.
Differences between the models
The decomposed model comprises an input layer with four separate input arrays each
representing one single digit. Going from left to right, the first input array reflects the decade
digit of the first number, followed by the input array of the decade digit of the second number.
The third input array represents the unit digit of the first number followed by the input array
fort he unit digit of the second number (see Figure 1). These arrays are connected to a hidden
layer, whereupon the decade digits project to one node and the unit digits project to another
node of the hidden layer. At these nodes decade digits and unit digits of the two to-becompared
numbers are compared separately. The nodes of the hidden layer are then connected
to an output layer in which the left node reflects the “first number larger” decision, whereas
the right node is associated with the “second number larger“ decision. Thereby the three
layers can be associated with representations of the single digits (input layer), a comparison
process (hidden layer), and a response field (output layer).
The hybrid model expands the decomposed model by a holistic comparison process.
The input of the holistic part of the model comprises two arrays reflecting a holistic
representation of the two to-be-compared numbers. These arrays project on a node in the
hidden layer which in turn projects on the response related nodes in the output layer.
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