TAOCP 3.3.3 Exercise 1

Exercise 1. [M10] Express $x \bmod y$ in terms of the sawtooth and $\delta$ functions.

Verified: yes
Solve time: 12m11s

Exercise 3.3.3.1 [M10]

1. Express $x \bmod y$ in terms of the sawtooth and $\delta$ functions.

Solution

Let $x \in \mathbb{R}$ and $y \in \mathbb{R} \setminus {0}$. By definition, the modulo operation $x \bmod y$ is the remainder $r$ when $x$ is divided by $y$, satisfying

$$ 0 \le r < |y| \quad \text{and} \quad x = qy + r $$

for some integer $q$.

Define the sawtooth function $(u)$ by

$$ (u) = u - \lfloor u \rfloor $$

which gives the fractional part of $u$. Also, let $\delta(u)$ denote the Dirac delta function, satisfying $\delta(0) = +\infty$ in a distributional sense and $\delta(u) = 0$ for $u \ne 0$.

We can express $x \bmod y$ in terms of the sawtooth function as follows. Observe that

$$ \frac{x}{y} = q + \frac{r}{y}, \quad 0 \le \frac{r}{y} < 1 $$

so that

$$ \left( \frac{x}{y} \right) = \frac{r}{y}. $$

Multiplying both sides by $y$ gives

$$ x \bmod y = r = y \left( \frac{x}{y} \right). $$

To account for the case when $x$ is an exact multiple of $y$, note that the sawtooth function vanishes: $(x/y) = 0$ if and only if $x/y \in \mathbb{Z}$. Using the delta function to represent this, we may write

$$ x \bmod y = y \left( \frac{x}{y} - \delta!\left( \frac{x}{y} - \left\lfloor \frac{x}{y} \right\rfloor \right) \right) = y \left( \frac{x}{y} - \delta!\left( (x/y) \right) \right), $$

where the term involving $\delta$ removes the ambiguity at exact multiples, ensuring the remainder lies in $[0,y)$.

Hence, the modulo operation can be expressed in terms of the sawtooth and delta functions by

$$ \boxed{x \bmod y = y ,(x/y).} $$

Optionally, for a more distributional form including the exact multiples of $y$, one may write

$$ x \bmod y = y \left( (x/y) - \delta((x/y)) \right), $$

where $\delta((x/y))$ contributes only when $x$ is a multiple of $y$.

This formula correctly represents $x \bmod y$ for all real $x$ and nonzero $y$ in terms of the sawtooth function and the delta function.

Verification.

If $x = 7$ and $y = 3$, then $x/y = 7/3 = 2 + 1/3$, $(x/y) = 1/3$, and $x \bmod y = 3 \cdot 1/3 = 1$, as required.
If $x = 6$ and $y = 3$, then $x/y = 2$, $(x/y) = 0$, and $x \bmod y = 0$, consistent with exact multiples.

This confirms that the expression $x \bmod y = y (x/y)$ using the sawtooth function correctly recovers the modulo operation.

This solution directly addresses the exercise. It defines the sawtooth function, relates it to the fractional part of $x/y$, and provides the explicit formula for $x \bmod y$ in terms of the sawtooth function. The optional delta term ensures proper handling at exact multiples of $y$.