Nines of nines

In the operations business we like to talk about nines of things, especially regarding service levels.


  • "one nine of availability" = available 0.9 of the time,
  • "two nines of availability" = available 0.99 of the time,
  • and so on...

then generally,

  • "n nines of availability" = available (1 - 10^{-n}) of the time,


This works for any whole number n: e.g. 5 nines is 1 - 10^{-5} = 1 - 0.00001 = 0.99999.

There's a problem with this simple generalisation, and that is, when people say "three and a half nines" the number they actually mean doesn't fit the pattern. "Three and a half nines" means 0.9995, but

  • 1 - 10^{-3.5} \approx 0.9996838, and going the other way,
  • 0.9995 \approx 1 - 10^{-3.30103}.

We could resolve this difficulty by saying "3.3ish nines" when we mean 0.9995, or by meaning ~0.9996838 when we say "three and a half nines." But there's at least one function that fits the half-nines points as well!

Let's start with the function above: f(n) = 1 - 10^{-n}. For every odd integer, it just has to be lower by a small, correspondingly decreasing amount. We can do this by increasing the exponent of 10 by k = 0.5 + \log_{10}(0.5) \approx 0.19897.

One function for introducing a perturbation for halfodd integers is p(n) = \sin^2(\pi n). When n is a whole integer, p(n) = 0, and when n is half an odd integer, p(n) = 1.  Multiply this function by some constant and you're in business.

Thus, define a new function g(n) for all n:

 g(n) := 1 - 10^{-n + k p(n)} = 1 - 10^{-n + (0.5 + \log_{10}(0.5))\sin^2(\pi n)}

which, when plotted, looks like this:

Screen Shot 2015-04-23 at 5.21.45 PM

a negative exponential curve with a negative exponential wiggle.  And it has the desired property that at every integer and half-integer it has a value with the traditional number of nines and trailing five (or not).

French brioche toast

Makes 3 slices.


  • 3 slices of brioche loaf;
  • 2 eggs;
  • Dash of milk (soy works) or if you're feeling really naughty, cream;
  • Pinch of salt;
  • A cap-ful of vanilla;
  • Sugar to taste (I usually put 1tsp);
  • Butter (salted/unsalted/whatevs) for the pan.
  1. Mix the eggs, milk, salt, sugar,  and vanilla together in a bowl until the sugar is dissolved.
  2. Put the frypan on lowish heat, melt a curl or two of butter.
  3. Dip a brioche slice into the egg mix. Don't worry about soaking it through but be careful that the slice doesn't fall apart. Brioche is weaker than regular bread.
  4. Fry until brown on both sides and puffed up.
  5. Repeat with more slices until the mixture is used up.

Enjoy on their own (they're pretty rich) or with fresh sliced strawberries, blueberries, or fried banana and lightly dusted with icing sugar or drizzled with maple syrup.

Electric bike!

Last week, Bob's old mountain bike (which he gave me when he stopped riding, and which I didn't ride all that much in Hobart but made my regular commute after moving to Sydney) had died. I say "died," but in fairness, it's just a bit wonky and in need of repair.

Having fallen in lesbians with some of the loaner electric bikes at work (BH E-motion, very fun premium electric bikes), I figured it was time to get something similar. On the weekend I walked over to Sydney Electric Bikes and test-rode a few, before finally choosing the model I wanted. Having parted with a chunk of money via bank transfer on Monday, it arrived and was ready on Tuesday afternoon. I excitedly hurried  to the store to get my paws on it because it's awesome.

The model I got is the SmartMotion e20 folding electric bike. It's made by a New Zealand company (going to visit the country soon!) who devised the electric system they use on the postie bicycles there. This bike is fun. It has five levels of pedal assist, throttle, integrated lights, clicky-shifter derailleur sprockets, and a freaking USB port on the 12Ah battery for charging my other devices. It's all mine and I'm going to ride it all over the place—the electric system is apparently good for 40-60km on a charge.

The folding part is interesting. I've not owned a folding bike before. Benefit, the first: I can take it travelling far more easily. It also has 20-inch diameter wheels. These are ostensibly to take up less space, but mechanically the bike has better torque than standard size bikes, which is good for hill-climbing. (Sydney's no Hobart, but it's no Melbourne either.) The only downside seems to be the increased bumpiness, but I'd be a whinger to complain about that seriously.

Oh, and it's red. 😀


Leaves your eyeballs feeling minty-fresh.