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-X x -X=+!!!!!!?
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Can someone explain in layman's terms why a minus number multiplied by a minus number makes a plus?
Any use of concrete items like bananas, pears, apples etc would be appreciated.
How does it go to the right of the decimal point for heavens sake?!
Any use of concrete items like bananas, pears, apples etc would be appreciated.
How does it go to the right of the decimal point for heavens sake?!
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one explantion
Multiplication by a negative number can be regarded as
repeated addition as well. For instance, 3 � -2 can be thought of as 3 groups, with -2 in each group. 3 � −2 = (-2) + (−2) + (-2) = −6. Notice that this keeps multiplication commutative: 3 � −2 = −2 � 3 = −6.
Applying the same interpretation of "multiplication by a negative number" for a value that is also negative, we have:
−4 � −3 = − (−4) − (−4) − (−4)
= 4 + 4 + 4
= 12
Multiplication by a negative number can be regarded as
repeated addition as well. For instance, 3 � -2 can be thought of as 3 groups, with -2 in each group. 3 � −2 = (-2) + (−2) + (-2) = −6. Notice that this keeps multiplication commutative: 3 � −2 = −2 � 3 = −6.
Applying the same interpretation of "multiplication by a negative number" for a value that is also negative, we have:
−4 � −3 = − (−4) − (−4) − (−4)
= 4 + 4 + 4
= 12
what we mean by negative numbers is simply working below zero.
In other words anything below zero is imaginary or abstract if you like.Multiplication of a negative by a negative means we are working out numbers of the same nature therefore its answer will be positive.For example,I want three mangoes of which each costs 4pence,however I have nothing in the pocket to buy the 3 mangoes.This means I need 12pence for me to get the 3 mangoes.
In other words anything below zero is imaginary or abstract if you like.Multiplication of a negative by a negative means we are working out numbers of the same nature therefore its answer will be positive.For example,I want three mangoes of which each costs 4pence,however I have nothing in the pocket to buy the 3 mangoes.This means I need 12pence for me to get the 3 mangoes.
Fibonacci and Erick84.. Thank you both. I understand the defecit of 12p for the mangoes I can't buy.
If I have an overdraught of -4p for one mango in the bank and I want to multiply this by 3, then I will have -12p in the bank. The - 4p in the bank is not imaginary but if the bank were to multiply it by -3p, how would I do this? It would walk right toward the decimal point and go to 8? I know 12 is the answer but I still don't understand. Sorry I am such a pain.
If I have an overdraught of -4p for one mango in the bank and I want to multiply this by 3, then I will have -12p in the bank. The - 4p in the bank is not imaginary but if the bank were to multiply it by -3p, how would I do this? It would walk right toward the decimal point and go to 8? I know 12 is the answer but I still don't understand. Sorry I am such a pain.
Firstly, it is nothing to do with Decimal Point. What we use to illustrate positive and negative numbers is a straight horizontal line, from minus infinity on the left, through zero in the middle, to plus infinity on the right.
Fibonacci (nice name btw) explained how to multiply +ve and -ve numbers.Actions like commutative and distributive are actually part of the rulebook in Maths, building blocks without which we couldn't make the next step. It may be easier for you just to accept that multiplying 2 negatives makes a positive - not everything can be explained in apples and pears.
How do you feel about i the square root of -1 ? :)
Fibonacci (nice name btw) explained how to multiply +ve and -ve numbers.Actions like commutative and distributive are actually part of the rulebook in Maths, building blocks without which we couldn't make the next step. It may be easier for you just to accept that multiplying 2 negatives makes a positive - not everything can be explained in apples and pears.
How do you feel about i the square root of -1 ? :)
Let a and b be any two real numbers. Consider the number x defined by
x = ab + (-a)(b) + (-a)(-b).
We can write
x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0 = ab
Also,
x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).
So we have
x = ab and x = (-a)(-b)
abis positive so (-a)*(-b) must also be positive
x = ab + (-a)(b) + (-a)(-b).
We can write
x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0 = ab
Also,
x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).
So we have
x = ab and x = (-a)(-b)
abis positive so (-a)*(-b) must also be positive
A better way- think of temperature, a glass of water and some plastic cubes (full of other plastic cubes containing ice (temp below zero)
� plus times plus is plus (adding many hot cubes raises the temperature)
� minus times plus is minus (removing many hot cubes reduces the temperature)
� plus times minus is minus (adding many cold cubes reduces the temperature)
� minus times minus is plus (removing many cold cubes raises the temperature)
� plus times plus is plus (adding many hot cubes raises the temperature)
� minus times plus is minus (removing many hot cubes reduces the temperature)
� plus times minus is minus (adding many cold cubes reduces the temperature)
� minus times minus is plus (removing many cold cubes raises the temperature)
Oops- gremlins got in to my keyboard. Should read:
think of temperature, a glass of water and some plastic cubes (some full of hot water and somecubes containing ice at below zero degrees C)
� plus times plus is plus (adding many hot cubes raises the temperature)
� minus times plus is minus (removing many hot cubes reduces the temperature)
� plus times minus is minus (adding many cold cubes reduces the temperature)
� minus times minus is plus (removing many cold cubes raises the temperature)
think of temperature, a glass of water and some plastic cubes (some full of hot water and somecubes containing ice at below zero degrees C)
� plus times plus is plus (adding many hot cubes raises the temperature)
� minus times plus is minus (removing many hot cubes reduces the temperature)
� plus times minus is minus (adding many cold cubes reduces the temperature)
� minus times minus is plus (removing many cold cubes raises the temperature)
Square root of -1 is i (a complex number).
In terms of the overdraft, why would you multiply -�5 multiplied by -�5? That would be 25 square pounds! Do you mean multiply -�5 by -5? The answer would be �25. But why would you do that- the overdraft may be -�5 but where does the other -5 figure come from?
Oh yes, and quare root of -1 is i (a complex number) but I doubt you'll need to worry about that.
In terms of the overdraft, why would you multiply -�5 multiplied by -�5? That would be 25 square pounds! Do you mean multiply -�5 by -5? The answer would be �25. But why would you do that- the overdraft may be -�5 but where does the other -5 figure come from?
Oh yes, and quare root of -1 is i (a complex number) but I doubt you'll need to worry about that.
Factor, thank you for your reply.It comes from trying to understand why the bank cannot x -5x--5! I have always had a blockage with Maths as if I don't understand and just have to follow rules, I am not happy.
What is i? . There cannot be a square root of -1 because everyone says so....ha! Please reply.
What is i? . There cannot be a square root of -1 because everyone says so....ha! Please reply.
Sorry but I'm not going to spend more time on this unless you can explain why you think the banks should ever need to multiply -5 by -5. Sorry but I think you are talking nonsense! Can you convince me I'm wrong and that you are talking sense. Can you explain why you think the banks should multiply minus �5 by minus five.
As for square root of -1, here is some info.
http://people.bath.ac.uk/acpw20/Complexhistory .html
However, I get the feeling that by inviting further discussion you may be taking the mickey and may actually be an expert on complex numbers. What do you think of Cauchy's theorem?
As for square root of -1, here is some info.
http://people.bath.ac.uk/acpw20/Complexhistory .html
However, I get the feeling that by inviting further discussion you may be taking the mickey and may actually be an expert on complex numbers. What do you think of Cauchy's theorem?
Good Evening Factor. I am far from an expert on Complex numbers. It would not even occur to me to take the mickey. I apologise if I have offended you. I really don't know who Cauchy is. I attained a Grade B in GCE Maths many years ago with great difficulty. The rest were all A's. I have a double Hrs degree. I am not stupid., just have a BLOCKAGE . I just can't see the light........Rosaxx
I'm afraid I just can't work out what you want to know!
What do you mean about multiplying an overdraft of minus five pounds by minus five pounds. What are you wanting to show/happen?
What is the problem with (-a x -b) being equal to (a x b)? There are lots of good explanations here and on other sites. Why does it trouble you?
And what's all this stuff about " will now rule out the decimal point in my thinking" and "walking right past the decimal point". Can't see what you're getting at there.
I will try to help if you want but I don't know what you are getting at.
What do you mean about multiplying an overdraft of minus five pounds by minus five pounds. What are you wanting to show/happen?
What is the problem with (-a x -b) being equal to (a x b)? There are lots of good explanations here and on other sites. Why does it trouble you?
And what's all this stuff about " will now rule out the decimal point in my thinking" and "walking right past the decimal point". Can't see what you're getting at there.
I will try to help if you want but I don't know what you are getting at.
Hi
I've read through the whole thread and would like to add a few points to try to clarify.
Negative numbers do not need to be considered as abstract - you are correct in thinking about it in terms of debt or the other suggestion of being temperature related.
(I would forget about the root of negative one - this is a whole different area of maths which is pretty abstract, especially if maths is not your thing!)
I guess you are happy with the idea of multiplication being repeated addition... so if I give you �5 three times, that is +5+5+5 or 5x3, making �15.
If I take �5 from you three times, then it is -5-5-5 or -5x3, making -�15.
If you then consider that you owe someone �5, it can be considered to be -�5.
If I takeaway that debt, I takeaway your negative amount, which is increasing your money. So if I - -�5 that is like +�5.
If you owe two people �5 and I takeaway your debt, then I
- -5 and - - 5, which will 'gain' you �10.
You could link this back to the earlier multiplications and see it as -5 x -2, the result of this being a gain of �10, so +10. You don't have �10, but you no longer owe �10.
So if you have two overdrafts of �100 each and the bank TAKES AWAY both of these debts, they do -100 x -2, making you �200 better off. You don't have �200, but that is the increase in your monetary situation, each account will have a balance of 0.
Not sure whether this will help, but as a maths teacher I thought it was worth a try!!
I've read through the whole thread and would like to add a few points to try to clarify.
Negative numbers do not need to be considered as abstract - you are correct in thinking about it in terms of debt or the other suggestion of being temperature related.
(I would forget about the root of negative one - this is a whole different area of maths which is pretty abstract, especially if maths is not your thing!)
I guess you are happy with the idea of multiplication being repeated addition... so if I give you �5 three times, that is +5+5+5 or 5x3, making �15.
If I take �5 from you three times, then it is -5-5-5 or -5x3, making -�15.
If you then consider that you owe someone �5, it can be considered to be -�5.
If I takeaway that debt, I takeaway your negative amount, which is increasing your money. So if I - -�5 that is like +�5.
If you owe two people �5 and I takeaway your debt, then I
- -5 and - - 5, which will 'gain' you �10.
You could link this back to the earlier multiplications and see it as -5 x -2, the result of this being a gain of �10, so +10. You don't have �10, but you no longer owe �10.
So if you have two overdrafts of �100 each and the bank TAKES AWAY both of these debts, they do -100 x -2, making you �200 better off. You don't have �200, but that is the increase in your monetary situation, each account will have a balance of 0.
Not sure whether this will help, but as a maths teacher I thought it was worth a try!!
Bloss. Thank you.
I now understand the following:
I have 2 x �100 overdrafts.
State of accounts: 2 x -�100. ie: - �200
So -100 x 'take this away twice' =0
So -100 x -2 0r 'take it away twice' = back to the decimal point! + 200 or 0 in the bank!!
If the above is correct, then you have resolved my problem.
I now understand the following:
I have 2 x �100 overdrafts.
State of accounts: 2 x -�100. ie: - �200
So -100 x 'take this away twice' =0
So -100 x -2 0r 'take it away twice' = back to the decimal point! + 200 or 0 in the bank!!
If the above is correct, then you have resolved my problem.
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