So the sign in front of a term(to the left of it) always belongs to that term except-

[The_Bad_Guy]

......When your doing a basic math problem/expression such as -6-8÷(-4)? Why is that the only(?) instance where "the sign in front of a term" doesn't belong to that term?

 

[mmm4444bot]

So the sign in front of a term(to the left of it) always belongs to that term except-......When your doing a basic math problem/expression such as -6-8÷(-4)? Why is that the only(?) instance where "the sign in front of a term" doesn't belong to that term?

-6 - 8 ÷ (-4)

The red symbols are negative signs.

The green symbol is an arithmetic operator (subtraction).

I'm not sure that I understand your question about the "only instance". Can you rephrase it?

 

[The_Bad_Guy]

-6 - 8 ÷ (-4)

The red symbols are negative signs.

The green symbol is an arithmetic operator (subtraction).

I'm not sure that I understand your question about the "only instance". Can you rephrase it?

Absolutely. I'm reviewing math and making my way along. My book mentioned that "The sign in front of a term belongs to that term" always. When you combine like terms, use the distributive property, etc. it certainly applies.

In the combining like terms section, it gives this example:
7x^2-2x-5x
And it mentions that the terms are: +7x^2, -2x, and -5x

But with problems like:
6-2·(-2)+8, -6-8÷(-4), or (-1)^3-4·(-2) the negative sign in front of the term(excluding terms in parenthesis) doesn't render it negative and does not belong to that particular term. But in the example above I gave you from the combining like terms section, the term with a negative sign in front of it and outside parenthesis does make the term negative in that instace. Why is that?

 

[mmm4444bot]

6-2·(-2)+8, -6-8÷(-4), or (-1)^3-4·(-2)

... the term with a negative sign in front of it and outside parenthesis [makes] the term negative in that instance. Why is that?

Those characters are not negative signs; they are subtraction operators. What "makes" the term negative is the switch from subtraction to addition.

6 - 2·(-2)

When I look at this expression, I could see a positive two and a negative two multiplied together, and the result subtracted from 6:

6 - (-4)

But, since algebra defines the operation of subtraction as "adding the opposite", I will most likely view that expression as a negative two and a negative two multiplied together, and the result added to 6:

6 + (-2)·(-2)

Either way you view it, the result is 10. (With practice, you'll be thinking in terms of addition more and more often.)

So, if we're given 6 - 2·(-2) we may view the first factor of 2 as either positive or negative, depending upon whether we view the operator in front of it as subtraction or addition, respectively.

If we view that first two as negative, we've changed the operation from subtraction to addition. If we view it as positive, we have not changed the subtraction operation. The following expressions are equivalent.

6 + (-2)·(-2)

6 - (2)·(-2)

You know, in grade school, the standard model for understanding subtraction is that it "takes something away". For example, if you have five apples and you give two of them to your friend you have three apples remaining.

5 - 2 = 3

We "took away" two apples from a bag of five apples, and 3 apples are what's left over.

This model served us well in grade school because we always worked with positive numbers (and zero). With algebra, the number system increases to include this "new" kind of number: negative numbers. We need to increase our viewpoint, accordingly, in order to work with signed numbers.

Here's a thought experiment. Suppose there were such a thing as a new kind of apple: negative apples. If you were to place a negative apple near a regular apple, then poof, they both disappear (kinda like equal amounts of matter and anti-matter annihilating one another when they come into contact). Let's call the regular apples: positive apples. Negative apples are the "opposite" of positive apples.

In such a scenario, if you were to add two negative apples to a bag of five positive apples, poof, three apples would remain. In other words, instead of reaching into a bag and pulling out two apples (subtraction), you could achieve the same result by adding something, instead.

5 + (-2) = 3 instead of 5 - 2 = 3

-2 is the opposite of 2

So, I think the issue for you is to learn from practice the ability to distinguish which characters (they're called hyphens) represent subtraction operators and which represent negative signs, and to remember that the operation of subtraction may be viewed in terms of "adding the opposite".

---

-6 - 8 ÷ (-4)

The hyphen in front of the 8 is a subtraction operator, so the 8 is positive.

If we view the 8 as negative, then we've switched the operation from subtraction to addition.

-6 + (-8) ÷ (-4)

Either way, the end result is -4.

---

(-1)^3 - 4·(-2)

(-1)^3 + (-4)·(-2)

Both the same. Given the first line, people may call that positive four "negative", AND, if they do, they've already switched to the second line!

Either way, the end result is 7.

"The sign in front of a term belongs to that term" always.

That's correct, but in the expression

7x^2 - 2x - 5x

there is no sign in front of 2x or 5x, IF you're thinking in terms of subtraction operators.

In the combining like terms section, it gives this example:
7x^2 - 2x - 5x
And it mentions that the terms are: +7x^2, -2x, and -5x

That's correct because they are now thinking in terms of addition, instead of subtraction.

7x^2 + (-2x) + (-5x)

After you switch your viewpoint from subtraction to addition, there are signs in front of the 2x and 5x.

It is very regrettable that the very first keyboards were designed with only a hyphen key. Obviously, the inventors of the manual typewriters were not mathematicians. Otherwise, we would have two different characters, to distinguish between negative signs and subtraction operators. (My Texas Instruments calculator does; not only does it have two different buttons for these, but the respective characters which appear on the screen look distinctly different). We all have to learn to deal with this "foolish" situation.

Keep practicing; it gets better. As your signed-number-sense starts to coalesce, you'll start to parse expressions more easily into what's positive and what's negative when multiplying/dividing, and then what gets added/subtracted to what. :cool:

 

[The_Bad_Guy]

Those characters are not negative signs; they are subtraction operators. What "makes" the term negative is the switch from subtraction to addition.

6 - 2·(-2)

When I look at this expression, I could see a positive two and a negative two multiplied together, and the result subtracted from 6:

6 - (-4)

But, since algebra defines the operation of subtraction as "adding the opposite", I will most likely view that expression as a negative two and a negative two multiplied together, and the result added to 6:

6 + (-2)·(-2)

Either way you view it, the result is 10. (With practice, you'll be thinking in terms of addition more and more often.)

So, if we're given 6 - 2·(-2) we may view the first factor of 2 as either positive or negative, depending upon whether we view the operator in front of it as subtraction or addition, respectively.

If we view that first two as negative, we've changed the operation from subtraction to addition. If we view it as positive, we have not changed the subtraction operation. The following expressions are equivalent.

6 + (-2)·(-2)

6 - (2)·(-2)

You know, in grade school, the standard model for understanding subtraction is that it "takes something away". For example, if you have five apples and you give two of them to your friend you have three apples remaining.

5 - 2 = 3

We "took away" two apples from a bag of five apples, and 3 apples are what's left over.

This model served us well in grade school because we always worked with positive numbers (and zero). With algebra, the number system increases to include this "new" kind of number: negative numbers. We need to increase our viewpoint, accordingly, in order to work with signed numbers.

Here's a thought experiment. Suppose there were such a thing as a new kind of apple: negative apples. If you were to place a negative apple near a regular apple, then poof, they both disappear (kinda like equal amounts of matter and anti-matter annihilating one another when they come into contact). Let's call the regular apples: positive apples. Negative apples are the "opposite" of positive apples.

In such a scenario, if you were to add two negative apples to a bag of five positive apples, poof, three apples would remain. In other words, instead of reaching into a bag and pulling out two apples (subtraction), you could achieve the same result by adding something, instead.

5 + (-2) = 3 instead of 5 - 2 = 3

-2 is the opposite of 2

So, I think the issue for you is to learn from practice the ability to distinguish which characters (they're called hyphens) represent subtraction operators and which represent negative signs, and to remember that the operation of subtraction may be viewed in terms of "adding the opposite".

---

-6 - 8 ÷ (-4)

The hyphen in front of the 8 is a subtraction operator, so the 8 is positive.

If we view the 8 as negative, then we've switched the operation from subtraction to addition.

-6 + (-8) ÷ (-4)

Either way, the end result is -4.

---

(-1)^3 - 4·(-2)

(-1)^3 + (-4)·(-2)

Both the same. Given the first line, people may call that positive four "negative", AND, if they do, they've already switched to the second line!

Either way, the end result is 7.


That's correct, but in the expression

7x^2 - 2x - 5x

there is no sign in front of 2x or 5x, IF you're thinking in terms of subtraction operators.

That's correct because they are now thinking in terms of addition, instead of subtraction.

7x^2 + (-2x) + (-5x)

After you switch your viewpoint from subtraction to addition, there are signs in front of the 2x and 5x.

It is very regrettable that the very first keyboards were designed with only a hyphen key. Obviously, the inventors of the manual typewriters were not mathematicians. Otherwise, we would have two different characters, to distinguish between negative signs and subtraction operators. (My Texas Instruments calculator does; not only does it have two different buttons for these, but the respective characters which appear on the screen look distinctly different). We all have to learn to deal with this "foolish" situation.

Keep practicing; it gets better. As your signed-number-sense starts to coalesce, you'll start to parse expressions more easily into what's positive and what's negative when multiplying/dividing, and then what gets added/subtracted to what. :cool:

Wow ok it makes more sense. Thank you. I'll definitely keep doing more problems and get the hang of it. Thank you!