Evaluating Parentheses, Brackets, Braces and Grouping SymbolsDate: 01/20/2007 at 16:58:10 From: Nelson Subject: Pre-algebra math (parentheses, brackets and braces) (18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]} Parentheses, brackets and braces are very confusing to me. For instance, the mathematical sentence above displays more brackets, parentheses and braces than numbers. According to the mathematical rules in place today, what am I supposed to know in order to see this problem and quickly know where to start and where to follow next in order to ensure that I will always get to the right answer? Even though I know that I am supposed to start with the innermost parentheses and then follow with the [ ], and then the { }, I still feel very uncomfortable with these figures mixed with the numbers. (18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]} Here, I start by: 9 x 9 = 81 - 1 = 80. But then what do I do? Keep going through the ( ) and divide 80 by 2 = 40? If that's the right course of action, then I'd follow with: 7 x 9 = 63 - 2 = 61, then I would come back and do: 5 x 20 = 100 - 61 = 39. And then, last, I'd divide 18 by 2 = 9. Then, 40 - 39 = 1; then 9 x 1 = 9? Obviously, I am going more by instincts than by logic. What do I need to know in order to laugh about problems like the one above instead of worrying about them? Date: 01/20/2007 at 23:50:33 From: Doctor Peterson Subject: Re: Pre-algebra math (parentheses, brackets and braces) Hi, Nelson. The first thing to do is to start with something not quite so complicated, so you can get used to the ideas first. But let's go ahead with this one, to check your work. Here's one way I write these to demonstrate how to think it through: (18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]} \____/ \_________/ \_________/ 9 {[ 80 / 2]-[5 x 20 - 61 ]} \______________/ \____________________/ 9 { 40 - 39 } \_______________________________________/ 9 * 1 \____________________/ 9 So you're correct, and your work was fine. The only thing I did differently was to evaluate 18/2 earlier, because nothing stood in its way; I could have waited as you did. What parentheses do is to contain a subexpression that has to be fully evaluated before it can be used in any containing expression. That's why you work from the inside out: you can't use what's inside until you evaluate it all, so you might as well start there. But if you forgot to, you'd still have a reminder. Here's an example: 2[(3 + 7)(3 - 2) - 3(2 + 2)] If I didn't bother with the inside-out "rule", I might just start trying to evaluate at the left (paying attention to the order of operations, of course): 2 times ... what? Well, the second number in that multiplication is the whole thing inside [...], so I have to put it on hold until I do that. So I focus on (3 + 7)(3 - 2) - 3(2 + 2) Now I start that. The first piece is (3 + 7), so I evaluate that whole thing and get 10. Now I have to multiply it by (3 - 2), so I stop and evaluate that, which gives 1. Now I can multiply 10 by 1 and get 10. So I keep going; I have to subtract something from that, but since the next bit is a product, I have to do that first. I'll have 3 times the next parenthesis; that's 3 times 4, so I have 12. The subtraction I put off is 10 - 12 = -2. Now, this is what the whole [...] is, so I go back and do that last multiplication: 2*(-2) = -4 That was pretty ugly, wasn't it? That's how a computer would evaluate it (more or less); it just keeps putting something on hold and coming back to it when it's ready, because it has a really good memory and won't forget to do it! We don't recommend this method for humans, but it can be done. Here's how I'd usually do it: 2[(3 + 7)(3 - 2) - 3(2 + 2)] \_____/\_____/ \_____/ 2[ 10 * 1 - 3* 4 ] \______/ \___/ 2[ 10 - 12 ] \_________________________/ 2* -2 \______________/ -4 Again, I just evaluated ANY parenthesized subexpression that didn't have anything standing in its way (that is, ALL the innermost ones); then I went through it again evaluating everything that was left the same way. In reality, on paper, my work might look like this, just jotting down the value of each parenthesis as I came to it, but doing most of the work in my head: 2[(3 + 7)(3 - 2) - 3(2 + 2)] = 2[10 - 12] = -4 10 1 4 My point is that inside-out is not a magic formula; it's just a natural result of what parentheses do. Wherever you find a parenthesis, you make sure it gets evaluated before you use that particular result in any further operations. So, for example, that (18/2) in your example could be done at any time, because it wasn't needed until the end anyway. I suspect your "instinct" is to do exactly this, and it fits with the logic perfectly (when you let it operate). If you have any further questions, feel free to write back. If I've totally confused you, let me know and I'll try again from a different perspective. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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