This isn't a homework question. Just something I've been thinking about, so I figured I'd ask it here. Mainly, I'm trying to find proof or disproof of whether multiplication outside of a parentheses has higher precedence than regular multiplication. Such as the difference between 3*3 and 3(3). Some people I've talked to seem to think the latter has higher precedences. This may seem unrelated to the title, so hear me out more. I always thought that both forms of multiplication were the same, as well as thinking that parentheses disappear after distribution. So I when I tried to prove to someone that both forms of multiplications were the same, I explained by solving: - 2*5(x+1)=20 - 10(x+1)=20 - 10x+10=20 - 10x = 10 - x = 1 Then when plugging in. - 2*5(1+1)=20 - 2*5(2)=20 - 10(2)=20 - 20 = 20 Then I try to show them what happens if I assume multiplication with * has lower precedence, and this was before I had any second thought about parentheses disappearing: - 2*5(x+1)=20 - 2*5(x+1)=20 - 2*5x+5=20 - 10x+5=20 - 10x=15 - x = 1.5 Plugging in. - 2*5(1.5+1)=20 Skipping all the steps, solving it my way, you get 25 = 20, which is just plain wrong. But THEN.... when they solved it their way, they didn't take away parentheses after distribution. So they did: - 2*5(x+1)=20 - 2(5x+5)=20 - 10x+10=20 - 10x = 10 - x = 1 Which is basically doing the same thing as what I was doing. But I still couldn't prove that parentheses disappear after distribution. So THEN I tried putting subtraction outside of the parentheses instead of multiplication in an equation: Their way: - 2-5(-2x+1)=40 - 2(10x-5)=40 - 20x-10=40 - 20x=50 - x = 2.5 Plugging in: - 2-5(-2(2.5)+1)=40 - 2-5(-5+1)=40 - 2-5(-4)=40 - 2(20)=40 - 40=40 At this point, I'm thinking "okay, maybe they're right", but I try it the other way anyway. - 2-5(-2x+1)=40 - 2+10x-5=40 - 10x=40+5-2 - 10x=43 - x=4.3 Plugging in: - 2-5(-2(4.3)+1)=40 - 2-5(-8.6+1)=40 - 2+43-5=40 - 45-5=40 - 40=40 At this point, I just lose it. After all that, both methods seem to work. But there's a consequence! I have two equations saying: 2-5(-2(2.5)+1)=40 and 2-5(-2(4.3)+1)=40 Which is nonsensical! I tried some online calculators which ultimately said the way I initially thought was correct, but that doesn't actually tell me anything. I actually want to see a demonstration of why either one is correct or wrong. If one method is wrong, I want to see a mathematical consequence that shows it. Basically, I want to see one of the methods fail.