I also put out a container of Sour Cream for anyone (such as myself) who wants to "dollop" it. While my personal preference is Parmesan, some of my family members prefer American while others prefer Swiss. I serve it with a "side salad and either a loaf of "really heavy" and "crusted" bread (an Italian loaf or a French "bagette" is good, but my family favors either Sour Dough or Pumpernickel) on the side, or refrigerated biscuits or homemade dumplings "on top". Prior to serving, I emulsify at least PART of my "batch" (perhaps 1/3rd) in my blender and return it to "the pot".
I rarely include potatoes, but if/when I do, they are "pre-cooked" and added along with the carrots. I usually cook the soup in my crock pot - to which I add either (fresh or dried) parsley, oregano, basil, coriander you know, whatever I like/whatever I feel like - on "low", so that it REALLY slow cooks, adding carrots during the last hour (when I raise the temperature to "high"). I generally use dried beans (usually 1 lb.), chicken bullion cubes (6-8) and just about any kind of pork, bacon, sausage or frankfurter (which I sautee in butter and with onions first) I have on hand. Please make a donation to keep TheMathPage online.Split pea soup is a staple in our home (at least once in any 2 week cycle, especially during the winter) and this is a good "core" recipe for it. That will be Problem 13 of the Lesson on Common Factor.
To prove that (− a) b is the negative of ab in what some call a rigorous manner, we would have to apply the definition of the negative of a number. This same logical principle will apply to division and fractions.
That is, "Unlike signs produce a negative number."Īnd upon introducing another negative factor, the sign changes back: It must be negative - it must be the negative of ab. Thus if ab is positive, then (− a) b cannot also be positive. If we now apply this principle to multiplication:Ī negative factor changes the sign of a product. (As for 0, it is best to say that it has both signs: −0 = +0 = 0. Thus if the value of x is positive, then the value of − x must be negative, and vice-versa.Īnd so since we call the positive or negative value of a number its sign, then we can state the following principle:Ī minus sign changes the sign of a number.Ī minus sign is the logical equivalent of "not." Geometrically, a minus sign reflects a number symmetrically about 0. Now in algebra we do not have true or false, but we do have the logical equivalent: positive or negative. If the statement was true, "not" makes it false, and vice-versa. Rather, we have to respect the either-or, yes-or-no nature of logic.įor example, the introduction of the word "not" into a statement changes its truth value. To decide how negative numbers should behave, we are not able to copyĪrithmetic. To multiply fractions, multiply the numeratorsĪnd multiply the denominators, as in arithmetic. In summary, here again is the Rule of Signs. Evaluate each of the following as a positive or negative fraction in lowest terms, or as an integer. Multiplication is always simpler if factors will produce 10, or 100, or any power of 10. ( Lesson 1.) The multiplication therefore will be simpler if we first multiply 2 We have now only to multiply those numbers.īut the order of factors does not matter. Before even multiplying, we can see that because there are an odd number of negative factors, the sign will be negative. While an odd number of negative factors produces a According to the previous problem:Īn even number of negative factors produces a Let the value of y depend on the valueĬalculate the value of y that corresponds to each value of x: In other words, any problem that looks like this. And so even though the problem means to subtract (5 times −2), we may interpret it to mean: −5 times −2 = +10.