Pluses and minuses

[errequeerre]

Hello,

I am getting confused with something which is extremely simple, and perhaps somebody can help me.

I have this operation:
4 – 3

I can do it like I’ve done all my life:
4 – 3 = 1 (directly)

or indirectly by adding the opposite:
4 + (-3)
which I understand is particularly useful when dealing with negative numbers, not so much for a simple example like that.

Now, if we say that “adding the opposite” is + (-3), we are accepting that in the original operation (4 – 3) we have two positive numbers:
(+4) – (+3)

The problem I have is when I see the explanations for beginner’s algebra and expanding brackets.

If I have an algebraic expression like:
4 – 3(2x – 5y)

I can think of two possible ways of tackling it.

One of them is:
4 – 3*2x – 3*5y
or 4 – 6x – 15y
In this case, I have taken “3” to be a positive number. As agreed before.

However, I see in my book and in some online pages that in fact “3” is negative, because it is preceded by a minus sign. And they proceed like this:
4 – 3(2x – 5y)
expands to:
4 – (-3)*2x – (-3)*5y
or 4 – (-6)x – (-15)y = 4 + 6x + 15y

Both results are different.
What am I getting wrong?

 

[HallsofIvy]

Hello,

I am getting confused with something which is extremely simple, and perhaps somebody can help me.

I have this operation:
4 – 3

I can do it like I’ve done all my life:
4 – 3 = 1 (directly)

or indirectly by adding the opposite:
4 + (-3)
which I understand is particularly useful when dealing with negative numbers, not so much for a simple example like that.

Now, if we say that “adding the opposite” is + (-3), we are accepting that in the original operation (4 – 3) we have two positive numbers:
(+4) – (+3)

The problem I have is when I see the explanations for beginner’s algebra and expanding brackets.

If I have an algebraic expression like:
4 – 3(2x – 5y)

I can think of two possible ways of tackling it.

One of them is:
4 – 3*2x – 3*5y
or 4 – 6x – 15y
In this case, I have taken “3” to be a positive number. As agreed before.

Unfortunately, both of those are wrong. Note, for example, that, since 3- 1= 2, -(3- 1)= -2. That would be -3+ 2= -1. It is an application of the "distributive law": a(b+ c)= ab+ ac. Here, -3(2x- 5y)= (-3)(2x)- (-3)(5y)= -6 - (-15y)= -6+ 15y.

However, I see in my book and in some online pages that in fact “3” is negative, because it is preceded by a minus sign. And they proceed like this:
4 – 3(2x – 5y)
expands to:
4 – (-3)*2x – (-3)*5y
or 4 – (-6)x – (-15)y = 4 + 6x + 15y

Both results are different.
What am I getting wrong?

Not using the distributive law.

 

[JeffM]

Hello,

I am getting confused with something which is extremely simple, and perhaps somebody can help me.

I have this operation:
4 – 3

I can do it like I’ve done all my life:
4 – 3 = 1 (directly)

or indirectly by adding the opposite:
4 + (-3)
which I understand is particularly useful when dealing with negative numbers, not so much for a simple example like that.

Now, if we say that “adding the opposite” is + (-3), we are accepting that in the original operation (4 – 3) we have two positive numbers:
(+4) – (+3)

The problem I have is when I see the explanations for beginner’s algebra and expanding brackets.

If I have an algebraic expression like:
4 – 3(2x – 5y)

I can think of two possible ways of tackling it.

One of them is:
4 – 3*2x – 3*5y

If, AS WE MAY, we treat the first minus sign as an operator on a positive number 3, then to be consistent we must also treat the second minus sign as an operator preceding a positive 5. So

\(\displaystyle 4 - 3(2x - 5y) = 4 - \{3[2x - (5y)]\} = 4 - [(3 * 2x) - 3(5y)] = 4 - [(3 * 2x) - (3 * 5y)] = 4 - [6x - (15y)] = 4 - 6x + 15y.\)

or 4 – 6x – 15y You were not consistent and so you got a wrong answer
In this case, I have taken “3” to be a positive number. As agreed before.


However, I see in my book and in some online pages that in fact “3” is negative, because it is preceded by a minus sign. And they proceed like this:
4 – 3(2x – 5y)
expands to:
4 – (-3)*2x – (-3)*5y No it does not. If - 3 is considered to be a negative number then it is being added, not subtracted.

\(\displaystyle 4 - 3(2x - 5y) = 4 + (-3)(2x - 5y) = 4 + (-3)\{2x + [(-5)y]\} = 4 + \{[(- 3)(2x)] + [(- 3)(- 5y)]\} = 4 + (-6x + 15y) = 4 - 6x + 15y.\)


or 4 – (-6)x – (-15)y = 4 + 6x + 15y You were not consistent and got a different wrong answer

Both results are different.
What am I getting wrong?

Let me see if I can unconfuse you.

a + (- b) = a - (+ b).

a + (- b) means to add the additive inverse of b to a. Here - is being used as an additive-inverse indicator and + is being used as an operator.

a - (+ b) means to subtract b from a. Here - is being used as an operator and + is redundant.

The fact is that - is used as a sign indicator, an additive-inverse indicator, and an operator. It is admittedly confusing. The way to avoid confusion is to be consistent. Normally that means treating the minus sign as an additive-inverse indicator and not as an operator in algebra. Of course you still need subtraction as an operation in arithmetic: 5 + (-3) = 5 - 3 = 2.

 

[errequeerre]

Thank you both!

HallsofIvy, I have checked the link to the distributive law and it was very helpful.

JeffM, I think you could have given me the key on how to tackle this kind of problems.
If you don't mind I'll spell it out and if you were so kind me to tell me whether my summary is correct I'd be really thankful.

I can deal with the minus sign in two ways (I have subsumed the additive-inverse function):
(1) as an operator
(2) as an indicator of a negative number

(1) If it is an operator, when applied to another minus operator it "multiplies" and turns positive:
4 - (6x - 15y) ends up in ... + 15y because the first minus by the second minus turns it to a + sign.

(2) If it indicates a negative number I separate the whole thing from the 4 preceding it and then "hang" the end result to the 4 that I've left behind.
4 - 3(2x - 5y)
I "isolate" the 4 and concentrate on the parenthesis:
-3(2x - 5y)
2x * -3 - 5y * -3
-12x - (-15y)
-12x + 15y
and then:
4 - 12x + 15y

I hope that's right. I'm sorry to bother you with such a simple point, but I last did maths 30 years ago and basic things sound very distant.

 

[JeffM]

Thank you both!

HallsofIvy, I have checked the link to the distributive law and it was very helpful.

JeffM, I think you could have given me the key on how to tackle this kind of problems.
If you don't mind I'll spell it out and if you were so kind me to tell me whether my summary is correct I'd be really thankful.

I can deal with the minus sign in two ways (I have subsumed the additive-inverse function):
(1) as an operator
(2) as an indicator of a negative number

(1) If it is an operator, when applied to another minus operator it "multiplies" and turns positive:
4 - (6x - 15y) ends up in ... + 15y because the first minus by the second minus turns it to a + sign.

(2) If it indicates a negative number I separate the whole thing from the 4 preceding it and then "hang" the end result to the 4 that I've left behind.
4 - 3(2x - 5y)
I "isolate" the 4 and concentrate on the parenthesis:
-3(2x - 5y)
2x * -3 - 5y * -3
-12x - (-15y)
-12x + 15y
and then:
4 - 12x + 15y

I hope that's right. I'm sorry to bother you with such a simple point, but I last did maths 30 years ago and basic things sound very distant.

I cannot advise the road you are going down at all. It is bad enough that plus and minus signs are used in multiple senses, an accident of history that is too late to fix now, but it just makes it worse to discard the primary sense. Let's try it a different way.

In a purely formal sense, algebra requires only two operations, addition and multiplication. There is an additive identity element, namely zero. There is a multiplicative identity element, namely 1. a + 0 = a. a * 1 = a. Notice why 0 and 1 are called identity elements; when 0 is added to a number, the answer is identical to the number added to. When a number is multiplied by 1, the answer is identical to the number multiplied.

With respect to addition, every number has an additive inverse. The primary meaning of the minus sign in algebra is to identify an additive inverse. The additive inverse of the number a is - a and has the following property, a number plus its additive inverse equals 0, the additive identity element.

Notice that - a does NOT mean that a is less than 0. It does NOT mean that - a is a negative number. Let's consider a specific example. Let's say that when it is getting warmer, the change in temperature is plus. So if it was - 3 degrees at dawn and it has got 3 degrees warmer by noon, what's the temperature at noon? Why clearly zero. So what is the negative inverse of - 3? By definition it is + 3, which is a positive number.

Now what is the negative inverse of the negative inverse of - 5. Well the negative inverse of - 5 is 5. And the negative inverse of 5 is - 5. So the negative inverse of the negative inverse of - 5 is - 5.

With respect to a numeral such as 3 or 8, - 3 means 3 less than zero and + 8 means 8 more than zero. These are sign indicators.

With respect to a pronumeral (a letter used as the name of a number), - a means the additive inverse of the number named a. This is using the minus sign as an additive-inverse indicator. But we don't really need a not-additive-inverse indicator. So we can leave + out of account when thinking about additive inverses.

With respect to an expression (meaning a combination of numbers, pronumerals, and operation indicators), we have to have a sign for the operation of addition so we use +. But we do not need the operation of subtraction at all. So I think it best when thinking carefully about math to eschew the whole concept of subtraction and therefore not use the minus sign to indicate an operation.

How does that work.

13 - 5, where the minus sign is intended to indicate the operation of subtraction, can be viewed as shorthand for

13 + (- 5), where the operation of subtract has been replaced by the operation of addition and the number being subtracted has been replaced with its additive inverse.

Why does it work?

13 + (- 5) = 8 + 5 + (- 5) = 8 + 0 = 8. I just never think about subtraction at all.

Now suppose we are given with a mess like this

13 - (-3). Someone is using the first minus sign to indicate the operation of subtraction and the second sign to indicate a negative number. Confusing.

So I can get rid of that subtraction operation. I replace it with the addition of the additive inverse.

So 13 + the negative inverse of - 3. Well the negative inverse of - 3 must equal zero when added to - 3. So it must be + 3.

13 - (-3) = 13 + 3.

Now you have a reason for your rule.

Furthermore we have eliminated one meaning of the minus sign.

The minus sign in algebra means formally either a negativity indicator or an additive-inverse indicator.

The plus sign in algebra means formally either a positivity indicator or an addition operator.

Now obviously people do talk of subtraction in algebra and use the minus sign in all three senses. But when you are trying to get your head around algebra, forget about subtraction and use the minus sign in only two senses. Algebra will not be quite as confusing.

I hope this helps. If not keep asking questions.

 

[errequeerre]

Thanks!

Hi JeffM,

Not only does it help, but it was quite a pleasure to read it.

Thanks very much indeed!

It was my first post here and I am really grateful for the help I've received. As a self-learner, I will keep visiting and asking for help.

Best regards

 

[errequeerre]

Remember that there is an implied -1 before the brackets;
really means -1(6x - 15y), or -1 times (6x - 15y);
-1 times 6x = -6x
-1 times -15y = 15y

Thanks Denis!