Monthly Archives: April 2013


Buttermilk Biscuits

Last week, Alivia and I decided to stick to a firm budget for food. We had devised such a plan before, but fell off the budgeting bandwagon once our schedules became sufficiently hectic. The exciting part of finding yourself on Friday night with only one dollar remaining in the weekly food budget is that it forces you to get creative in the kitchen. For whatever reason, some of the best meals I’ve ever cooked have come from days when I’ve needed to work with a severely limited list of ingredients. Last night was one of those meals.

We had a ton of greens (kale and beet greens), a pint of buttermilk, a can of tomatoes, and the usual cooking staples (rice, flour, butter, etc.) at our disposal. So I decided to make sautéed greens, Spanish rice and buttermilk biscuits. The greens and rice turned out well, but the biscuits were the stars of the show. I’d never made buttermilk biscuits before, but it turns out they are relatively easy to make and ridiculously delicious. Here is a recipe for about 12 biscuits:


  • 2 cups white flour
  • 1 tsp baking powder
  • 1/4 tsp baking soda
  • 1/2 tsp salt
  • 6 tbsp butter (very cold)
  • 1 cup buttermilk


Preheat the oven to 450 degrees. Combine dry ingredients in a large bowl and mix thoroughly. Cut the butter into 1/4 inch cubes and add to the flower. Mix around and mash with a pasty blender or fork until the butter is in pieces a bit bigger than grains of rice and is evenly distributed. Add the buttermilk and stir until just combined. Roll the (sticky) dough onto a well-floured surface and spread the dough out by hand until it is about 3/4 inch thick. Fold it in half, pat down and repeat four or five times. Work quickly so the butter doesn’t soften. Once the dough is rolled out for a final time, cut the biscuits into circles (I used a small glass as a cookie cutter). Put the biscuits on a lightly greased cookie sheet and bake for 8 – 12 minutes until lightly browned on top.


I was surprised at how decadent these biscuits turned out to be. I suppose any small starchy morsel with a half tablespoon of butter per serving is bound to taste alright, but I was still blown away.

This morning, I made a variation of eggs Florentine with the leftovers: two halves of a biscuit topped with the sautéed greens and a fried egg on top of each (who has the patience to poach eggs?). It was easily the best breakfast I’ve had this year. Highly recommended.


Math 32A Midterm 1 Review Questions

I have compiled some review materials for the first midterm for Math 32A this quarter. Here are links to the questions and solutions. As always, feel free to let me know in the comments below if you have any questions or notice any errors.


The Storytelling of Science

Some of the biggest names in popular science came together for a panel as part of the Origins Project at ASU. Bill Nye, Neil deGrasse Tyson, Richard Dawkins, Brian Greene, Ira Flatow, Neil Stephenson, Tracy Day and Lawrence Krauss shared some of their favorite stories about science. I’ve always enjoyed how the folklore of science can grant humanity and pathos to what might otherwise be viewed as an austere subject. The stories shared by the participants are at once entertaining and inspiring.

Following the presentations by the panelists, the floor was opened up for questions. The subsequent discussion among the panelists ran the gamut of topics from the politics of science to philosophy. They hit upon some of my favorite topics such as the role of governmental funding in basic research, the unreasonable effectiveness of mathematics in the sciences, and the perils of watering down science for popular consumption.

Part I of the video shows the prepared presentations, while Part II includes the lively and informal responses to audience questions.


art musings

Yet another reason I like living in Los Feliz

This showed up on an electrical box at the end of my street overnight:

Larry David Bowie


Triangle Inequality and the Like

I didn’t finish a careful proof of the triangle inequality in class, so I am presenting a more polished argument here. I abandoned the text’s argument in favor of what I hope is a more intuitive (albeit longer) proof.

I’ve also had a number of requests for hints on a couple homework problems, specifically numbers 3.7 and 3.8. So here are my comments on those problems:

Problem 3.7 This problem asks us to prove that \(|b| < a\) if and only if \(- a < b < a\), and some variations on this result. Most often, it is easiest to prove "if and only if" statements in two parts: first prove that if \(|b| < a\) then \(-a < b < a\); then prove that if \(-a < b < a\) then \(|b| < a\). Notice that the statement \( a < b \) is equivalent to \(a \leq b\) and \(a \neq b\). Therefore, you can use all of the order axioms for the strict inequality (\(<\)) except that O2 doesn't apply. You can prove the first implication by breaking the problem up into the cases where \(b\) is positive and negative. Also, you can assume that \(a > 0\), for otherwise the assumption \(|b| < a\) is impossible. If \(b \geq 0\) then \(|b| = b\) so that \(|b| < a \Rightarrow b < a\). On the other hand, we get \(-a < 0 < b\). Combining this with the previous inequality gives \(-a < b < a\), as desired. The case where \(b\) is negative is similar. For the opposite implication (assuming \(-a < b < a\)), it is again helpful to break the proof up into cases depending on the sign of \(b\). For example, if \(b \geq 0\), then \(|b| = b < a\), which does it. If \(b \leq 0\), you'll have to appeal to the assumed inequality \(-a < b\). Part (b) asks you to prove that \(|a - b| < c\) if and only if \(b - c < a < b + c\), and (c) asks you to prove the same thing replacing each \(<\) with \(\leq\). You can prove (b) by appealing to (a): just use \(a - b\) for \(b\) and \(c\) for \(a\) in part (a) and see what you get. For part (c), you can just argue that the same inequality as (a) holds with \(\leq\), so the proof of (c) is essentially the same as the proof of (b). Don't worry about going through all the details again -- just convince me that you understand what is going on in you write-ups. Problem 3.8 In this problem, you are asked to prove that for \(a, b \in \mathbf{R}\) if \(a \leq b_1\) for every \(b_1 > b\), then \(a \leq b\).

I think the easiest way to approach this problem is to do a proof by contradiction. The idea is that in order to prove that a statement is true, you prove derive a contradiction from its negation. The negation of the statement we are trying to prove is, “for all \(b_1 > b\), \(a \leq b_1\) and \(a > b\).”

If you think about what this statement is saying, it should be clear that it cannot possibly be true. For example, if \(a > b\), then we can choose \(b_1 = (a + b) / 2\) so that \(b < b_1 < a\). But this choice of \(b_1\) contradicts the assumption that for all \(b_1 > b\) \(a \leq b_1\). Since the negation of the statement we are trying to prove leads to a contradiction, the statement must be true.


Binomial Theorem

I just finished writing up a proof of the Binomial Theorem that I wasn’t able to complete carefully in class. The write-up is available here. Let me know in the comments if anything us unclear or incorrect.