- #1
AndromedaRXJ
- 56
- 5
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.
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.