Last time we showed how you can take any two coprime positive integers `x`

and `m`

and compute a third positive integer `y`

with the property that `(x * y) % m == 1`

, and therefore that `(x * z * y) % m == z % m `

for any positive integer `z`

. That is, there always exists a “multiplicative inverse”, that “undoes” the results of multiplying by `x`

modulo `m`

. That’s pretty neat, but of what use is it?

Continue reading

# Category Archives: Integer arithmetic

# Math from scratch, part thirteen: multiplicative inverses

Here’s an interesting question: what is the behavior of this infinite sequence? WOLOG let’s make the simplifying assumption that `x`

and `m`

are greater than zero and that `x`

is smaller than `m`

. Continue reading

# Math from scratch, part twelve: Euclid and Bézout

Now that we have an integer library, let’s see if we can do some 2300-year-old mathematics. The Euclidean Algorithm as you probably recall dates from at least 300 BC and determines the greatest common divisor (GCD) of two natural numbers. The Extended Euclidean Algorithm solves Bézout’s Identity: given non-negative integers `a`

and `b`

, it finds integers `x`

and `y`

such that:

a * x + b * y == d

where `d`

is the greatest common divisor of `a`

and `b`

. Obviously if we can write an algorithm that finds `x`

and `y`

then we can get `d`

, and `x`

and `y`

have many useful properties themselves that we’ll explore in future episodes.

# Math from scratch, part eleven: integer division

Now we come to the controversial one. As I pointed out back in 2011, the `%`

operator is the *remainder *operator. When extending the division and remainder operators from naturals to integers, I require that the following conditions be true for dividend `x`

, non-zero divisor `y`

, quotient `q`

and remainder `r`

:

`x == y * q + r`

- if
`x`

and`y`

are non-negative then`0 <= r < y`

`(-x) / y == x / (-y) == -(x / y)`

Let’s take a look at the consequences of these three facts. Continue reading

# Math from scratch, part ten: integer comparisons

I’m going to use the same technique I used for Natural comparisons to do Integer comparisons: make a helper function that returns -1 if `x < y`

, 0 if `x == y`

and 1 if `x > y`

, and then have every entry point simply call that helper function. This way we know that all the operators are consistent. The big difference here of course is that I do not need to have a special case that says that null sorts before any value. Continue reading

# Math from scratch, part nine: integer arithmetic

Almost all the `Integer `

operations are simply wrappers around the `Natural `

operations, plus some gear to deal with the signs. Continue reading

# Math from scratch, part eight: integers

The integers are an extension to the naturals that closes them over subtraction. Rather than do anything fancy, I’m simply going to say that an integer is a natural with an associated sign, and that zero is always “positive”. We don’t want to end up in a situation where there are two forms of zero, since that is confusing. (We could of course have three signs, positive, negative and zero, but, meh. We don’t gain any great benefits from having a third sign.)

This time however I’m going to make a struct. C# requires that `default(SomeStruct)`

, where all the fields are their default values, be a valid value of the type. The default should logically be zero, as it is with all the built-in number types. So rather than checking for null, as we did with the reference type `Natural`

, I’m going to check the `sign`

field for null, which will only be possible if we’ve got a default struct. We’ll simply replace that with zero. (I am in general opposed to modifying the formal parameters of a method; I prefer to treat them as read-only variables. I’ll make an exception in this case because it keeps the code short and clear.) Continue reading