Using the Distributive PropertyDate: 03/06/2002 at 12:23:14 From: Melissa Subject: Integers This question is actually from my chemistry class: -2.4222 x 10^-19 - (-6.0556 x 10^-20) I tried typing it in as is on the calculator. As well I have tried -2.18 x 10^-18 + .06556 x 10^-18 The teacher gave the answer as +1.81664 x 10^-19. I keep getting the opposite "sign" on my answers. Date: 03/06/2002 at 12:44:45 From: Doctor Ian Subject: Re: Integers Hi Melissa, The idea behind adding numbers like this is that if you have the same exponent, you can use the distributive property: 2.13 * 10^-5 + 3.46 * 10^-5 = (2.13 + 3.46) * 10^-5 = 5.59 * 10^-5 It's the same as if you had a variable instead of an exponent: 2.13x + 3.46x = (2.13 + 3.46)x = 5.59x When you start with different exponents, you have to pick a common exponent - usually one of the exponents you already have - and convert both numbers to it: 2.13 * 10^-5 + 3.46 * 10^-6 = 21.3 * 10^-6 + 3.46 * 10^-6 = (21.3 + 3.46) * 10^-6 = 24.76 * 10^-6 Changing the exponent doesn't change the sign: -2.13 * 10^-5 + 3.46 * 10^-6 = -21.3 * 10^-6 + 3.46 * 10^-6 = (-21.3 + 3.46) * 10^-6 = -(21.3 - 3.46) * 10^-6 = -17.84 * 10^-6 When you use the distributive property explicitly, as I have here, it can help you avoid making sign errors. (And you don't even have to enter the exponents in the calculator.) For your first problem, I get -2.4222 * 10^-19 - (-6.0556 * 10^-20) = -2.4222 * 10^-19 - (-0.60556 * 10^-19) = (-2.4222 - -0.60556) * 10^-19 = (-2.4222 + 0.60556) * 10^-19 = -(2.4222 - 0.60556) * 10^-19 = 1.81664 * 10^-19 This is the answer your teacher got. Can you see now why it's correct? Note that writing (or typing) numbers with lots of decimals is a good way to make careless transcription errors. This is one of the reasons that variables were invented. I would do the problem this way: -2.4222 * 10^-19 - (-6.0556 * 10^-20) = -a * 10^-19 - (-b * 10^-20) = -a * 10^-19 - (-b/10 * 10^-19) = (-a - -b/10) * 10^-19 = (-a + b/10) * 10^-19 = -(a - b/10) * 10^-19 = -(2.4222 - 0.60556) * 10^-19 = 1.81664 * 10^-19 This makes it much easier to check that you haven't messed up a sign, and it's a lot easier on your fingers. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 03/06/2002 at 18:08:13 From: Melissa Subject: Integers Thanks for getting back to me. Still having difficulty. I understand everything up until here: = -(2.4222 - 0.60556) * 10^-19 = 1.81664 * 10^-19 Where did that negative sign in front of the brackets come from? Please refresh my mind on this procedure as it has really been about 10 years since I learned all this the first time! Date: 03/06/2002 at 18:37:33 From: Doctor Ian Subject: Re: Integers Hi Melissa, That's a shortcut for a particular use of the distributive property, a(b + c) = ab + ac The longer version looks like this: -2.422 + 0.60556 = (-1 * 2.422) + (-1 * -0.60556) Each term is -1 times something. = -1 * (2.422 + -0.60556) Distributive property. = -1 * (2.422 - 0.60556) To add a negative, subtract a positive. = -(2.422 - 0.60556) Keep the sign. In practice, when you have something like (a - b + c - d + e ...) changing the sign of the whole thing also changes the signs of the individual variables: (a - b + c - d + e ...) = -(-a + b - c + d - e ...) Again, the distributive property explains why this works; but in practice, you just use the shortcut. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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