# Homotopy Type Theory decimal numeral representations of the natural numbers > history (changes)

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## Idea

The decimal numeral representations of the natural numbers as familiar from school mathematics.

## Definition

Let the type of decimal digits, which we shall denote as $D$, be the type inductively generated by the terms $0:D$, $1:D$, $2:D$, $3:D$, $4:D$, $5:D$, $6:D$, $7:D$, $8:D$, $9:D$

The type of decimal numeral representations of the natural numbers, which we shall denote as $N_D$, is a higher inductive type generated by:

• A term $\epsilon: N_D$ representing the empty string of digits

• A function: $(-)(-): D \times N_D \to N_D$ representing concatenation of a digit and a string of digits.

• A dependent product of identities

$\zeta: \prod_{a:N_D} 0a = a$

representing that adding 0 to the left of any string of digits does not change the underlying value of the decimal numeral representation.

## Properties

TODO: Define a successor function $s:N_D \to N_D$ and show that $(N_D, \epsilon, s)$ is equivalent to the natural numbers. Then prove the addition and long multiplication algorithms for decimal numeral representations.