Distributive law and negative numbers

[conwy]

Hi all,

I'm trying to understand how the distributive law applies to expressions involving negative numbers and how, when reading an expression, to distinguish between a negative number and a subtraction operation.

In this Khan Academy video - Difference of squares intro - the following expression is given:

[math](x + a)(x - a)[/math]
The narrator then goes on to apply [imath](x + a)[/imath] to first [imath]x[/imath], then [imath]-a[/imath].

For some reason I find this counter-intuitive.

My default way of thinking was to treat [imath]x[/imath] and [imath]a[/imath] as terms, but [imath]-[/imath] as an operator between those terms. So my first instinct was to apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]a[/imath], and then to subtract the the latter from the former.

I guess when I see a [imath]-[/imath] sign, I assume it is a subtraction, because a lot of the time it actually is. But it seems my default way is wrong and instead I should have been treating [imath]-a[/imath] as a unit, meaning "negative a".

But now I feel confused because I don't know how to differentiate between the use of the minus sign as part of a term vs. as an operator, when applying the distributive law.

Suppose the expression was instead:

[math](x + a)(x \div a)[/math]
Then should I apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]\div a[/imath]? But how could I multiply by [imath]\div a[/imath]? How can such a term as "divided-by-a" even exist? That's nonsensical!

So evidently the subtraction symbol, [imath]-[/imath], is a special case, as, unlike most other operators, it can mean either subtraction or a prefix which makes the number following it a negative number, and so its meaning has to be partly determined based on context and it is not guaranteed that in all cases it is an operator.

So I guess it's probably simple enough to determine which meaning is intended in most cases, but still I find this very confusing.

Does my reasoning make sense, or can you see a more fundamental flaw in my thinking?

Thanks in advance.

 

[conwy]

Screenshot from the Khan video, for reference:

 

[topsquark]

Hi all,

I'm trying to understand how the distributive law applies to expressions involving negative numbers and how, when reading an expression, to distinguish between a negative number and a subtraction operation.

In this Khan Academy video - Difference of squares intro - the following expression is given:

[math](x + a)(x - a)[/math]
The narrator then goes on to apply [imath](x + a)[/imath] to first [imath]x[/imath], then [imath]-a[/imath].

For some reason I find this counter-intuitive.

My default way of thinking was to treat [imath]x[/imath] and [imath]a[/imath] as terms, but [imath]-[/imath] as an operator between those terms. So my first instinct was to apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]a[/imath], and then to subtract the the latter from the former.

I guess when I see a [imath]-[/imath] sign, I assume it is a subtraction, because a lot of the time it actually is. But it seems my default way is wrong and instead I should have been treating [imath]-a[/imath] as a unit, meaning "negative a".

But now I feel confused because I don't know how to differentiate between the use of the minus sign as part of a term vs. as an operator, when applying the distributive law.

Suppose the expression was instead:

[math](x + a)(x \div a)[/math]
Then should I apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]\div a[/imath]? But how could I multiply by [imath]\div a[/imath]? How can such a term as "divided-by-a" even exist? That's nonsensical!

So evidently the subtraction symbol, [imath]-[/imath], is a special case, as, unlike most other operators, it can mean either subtraction or a prefix which makes the number following it a negative number, and so its meaning has to be partly determined based on context and it is not guaranteed that in all cases it is an operator.

So I guess it's probably simple enough to determine which meaning is intended in most cases, but still I find this very confusing.

Does my reasoning make sense, or can you see a more fundamental flaw in my thinking?

Thanks in advance.

[imath](20 + 3)(10 + 7) = 20(10 + 7) + 3(10 + 7)[/imath]. This is just the standard table form for multiplication:
[imath]\begin{array}{lll} \\ ~ & 1 & 7 \\ \times & 2 & 3 \\ \hline \\ ~ & 5 & 1 \\ 3 & 4 & 0 \\ \hline \\ 3 & 9 & 1 \\ \end{array}[/imath]

For [imath](x + a) \div a[/imath], note that (a + b)c = ac + bc. So:
[imath](x + a) \div a = x \div a + a \div a[/imath]

-Dan

 

[Cubist]

Screenshot from the Khan video, for reference:

View attachment 34585

The post#2 image is very misleading in my opinion. The minus "-" is definitely intended to be the "binary" form, it accepts two operands "x" and "a". But the underlining in blue and purple imply a totally different meaning (x)(-a) which is multiplication ??‍

To properly convert that binary minus operation to a binary plus AND a "unary minus" then you'd have to write the following...



Take a look at these unary minus to binary minus conversions...
[math]x + (-a) = x - a\\ -a = 0 - a[/math]

 

[JeffM]

I understand your confusion.

[math]\alpha \beta \pm \alpha \gamma \equiv \alpha ( \beta \pm \gamma).[/math]
That is the way that the distributive law should, in my opinion, be Introduced to students. As a matter of pure theory, we can dispense with the operations of subtraction and division. We effect the same result as subtraction by adding the additive inverse and the same result as division by multiplying by the multiplicative inverse. In other words

[math]\theta - ( + \phi ) \equiv \theta + (- \phi).[/math]
As cubist points out, the minus symbol (and, to a lesser extent, the plus symbol) are used in three different senses in mathematics.

A plus symbol preceding a numeral indicates that the specified number is [imath]\ge[/imath] 0 whereas a minus symbol preceding a numeral indicates that the specified number is [imath]\le[/imath] 0. I like to call this usage the sign symbol.

A plus or minus sign following a number or pronumeral and also preceding a number or pronumeral represents the binary operation of addition or subtraction respectively and is called a binary operator.

A minus symbol preceding a pronumeral, a letter representing a number that has not been specified and whose sign is not known, indicates the additive inverse of the indicated number, does not specify which, if either, is positive or negative, and is called a unary operator.

This notation is admittedly confusing and takes a while to get used to, but you can avoid all confusion if you remember two things:

[math]\alpha \beta \pm \alpha \gamma \equiv \alpha ( \beta \pm \gamma).\\ \theta - ( + \phi) \equiv \theta - \phi \equiv \theta + (- \phi).[/math]
Multiplication distributes over addition and subtraction.

a - b means both a - (+b) and a + (-b); they are just different ways to say the same thing so use the one that appeals to your intuition.

 

[Dr.Peterson]

I'm trying to understand how the distributive law applies to expressions involving negative numbers and how, when reading an expression, to distinguish between a negative number and a subtraction operation.

In this Khan Academy video - Difference of squares intro - the following expression is given:
[math](x + a)(x - a)[/math]The narrator then goes on to apply [imath](x + a)[/imath] to first [imath]x[/imath], then [imath]-a[/imath].

For some reason I find this counter-intuitive.

My default way of thinking was to treat [imath]x[/imath] and [imath]a[/imath] as terms, but [imath]-[/imath] as an operator between those terms. So my first instinct was to apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]a[/imath], and then to subtract the the latter from the former.

I guess when I see a [imath]-[/imath] sign, I assume it is a subtraction, because a lot of the time it actually is. But it seems my default way is wrong and instead I should have been treating [imath]-a[/imath] as a unit, meaning "negative a".

But now I feel confused because I don't know how to differentiate between the use of the minus sign as part of a term vs. as an operator, when applying the distributive law.

You're right that the "-" in [imath]x-a[/imath] means subtraction.

What's happening here is that they are assuming that you have enough experience to see [imath](x-a)[/imath] as [imath](x+\ -a)[/imath], knowing the equivalence and automatically making the substitution in your mind. Presumably they have explained this somewhere (such as when they first explain distribution), but you arrived at this video without passing through that explanation.

Suppose the expression was instead:
[math](x + a)(x \div a)[/math]Then should I apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]\div a[/imath]? But how could I multiply by [imath]\div a[/imath]? How can such a term as "divided-by-a" even exist? That's nonsensical!

So evidently the subtraction symbol, [imath]-[/imath], is a special case, as, unlike most other operators, it can mean either subtraction or a prefix which makes the number following it a negative number, and so its meaning has to be partly determined based on context and it is not guaranteed that in all cases it is an operator.

I've occasionally had a related (but opposite) thought: There is no reason we couldn't define a unary [imath]\div[/imath] sign, so that [imath]\div a=1\div a=\frac{1}{a}[/imath]. That is, we would have a unary reciprocal operation corresponding to the unary negation operation. Then we could write \(x\div y=x\times\div y/) to say that division means multiplication by the reciprocal.

In fact, I said that here:

If we wanted, we could use the division symbol "/" to indicate the multiplicative inverse (also called the reciprocal):​

​

a * /a = 1​

​

We define division as multiplication by this inverse:​

​

a / b = a * /b​

​

in much the same way as for subtraction. For some reason this has never caught on, as far as symbols are concerned. Similarly, although we can use "+" as a unary operator (which has no effect on a number, as +a = a, meaning 0+a), we don't happen to use multiplication, "*", in the same way, so that *a = a, meaning 1*a. It wouldn't hurt to do so, but has not been found useful!​

[But don't miss the fact that in your example, since you aren't adding terms in the second factor, you wouldn't actually distribute.]

 

[The Highlander]

Hi all,

I'm trying to understand how the distributive law applies to expressions involving negative numbers and how, when reading an expression, to distinguish between a negative number and a subtraction operation.

In this Khan Academy video - Difference of squares intro - the following expression is given:

[math](x + a)(x - a)[/math]
The narrator then goes on to apply [imath](x + a)[/imath] to first [imath]x[/imath], then [imath]-a[/imath].

For some reason I find this counter-intuitive.

My default way of thinking was to treat [imath]x[/imath] and [imath]a[/imath] as terms, but [imath]-[/imath] as an operator between those terms. So my first instinct was to apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]a[/imath], and then to subtract the the latter from the former.

I guess when I see a [imath]-[/imath] sign, I assume it is a subtraction, because a lot of the time it actually is. But it seems my default way is wrong and instead I should have been treating [imath]-a[/imath] as a unit, meaning "negative a".

But now I feel confused because I don't know how to differentiate between the use of the minus sign as part of a term vs. as an operator, when applying the distributive law.

Suppose the expression was instead:

[math](x + a)(x \div a)[/math]
Then should I apply [imath](x + a)[/imath] to first [imath]x[/imath], then to [imath]\div a[/imath]? But how could I multiply by [imath]\div a[/imath]? How can such a term as "divided-by-a" even exist? That's nonsensical!

So evidently the subtraction symbol, [imath]-[/imath], is a special case, as, unlike most other operators, it can mean either subtraction or a prefix which makes the number following it a negative number, and so its meaning has to be partly determined based on context and it is not guaranteed that in all cases it is an operator.

So I guess it's probably simple enough to determine which meaning is intended in most cases, but still I find this very confusing.

Does my reasoning make sense, or can you see a more fundamental flaw in my thinking?

Thanks in advance.

If it's of any help I always teach my pupils to regard an algebraic expression as a SUM of terms, eg:-

I would encourage them to treat \(\displaystyle (x - 4 - y + 7 + z - 2) \text{ as } (x + ˉ 4 + ˉ y + 7 + z + ˉ 2)\)​

 

[Steven G]

Here is my take on this (I hope I am understanding your question)

a(b - c) = a( b + (-c)) = ab +a(-c) = ab + (-ac) = ab-ac.

a(b-c) = ab - ac or if you prefer, a(b-c) = (+ab)-(ac) = ab-ac

 

[Otis]

My default way of thinking was to treat x and a as terms, but – as an operator between those terms. So my first instinct was to apply (x+a) to first x, then to a, and then to subtract the the latter from the former

Hi conwy. There's nothing wrong with that way of thinking.

(x + a)(x) – (x + a)(a)

If you write out the multiplications, don't forget to enclose within grouping symbols the part that's being subtracted.

x^2 + ax – (ax + a^2)

x^2 + ax – ax – a^2

when I see a − sign, I assume it is a subtraction

It's unfortunate that a lot of materials make no visual distinction between a subtraction sign and a negative sign. Even worse, many instructors seem to speak the words 'negative', 'minus' and 'subtract' as though they're synonyms. (Such sloppiness!) Once you've written out enough practice exercises, you'll be able to mentally parse algebraic expressions involving negation symbols. Until you're comfortable doing that, use the suggestion in post#7 to avoid sign errors, and always check results.

I should have been treating -a as a unit, meaning "negative a"

That's your choice! I often view subtraction as "adding the opposite" and vice-versa, but not always.

And, when negated terms stand alone, like -a, I often think "the opposite of a" (particularly when I don't know the sign of a itself). That is, I view it as (-1)(a):

[x + a]*[x + (-1)(a)]

BTW, when somebody says, "you must do it this way", you're free to hear that as, "you must do it this way when you're in my presence".

I'm glad to see you question these things; it shows that you're engaged.
[imath]\;[/imath]

 

[Otis]

(x+a)(x÷a)
Then should I apply (x+a) to first x, then to ÷a?

Perhaps, you were joshing, but I'll mention for other readers as well that the distribution patterns discussed in this thread (one of which is named with the acronym FOIL) don't apply to that example because x÷a is not a binomial expression.

This is one way to go:

(x + a)[imath]\big(\frac{x}{a}\big)[/imath]

[imath]\big(\frac{x}{1}\big)\big(\frac{x}{a}\big) + \big(\frac{a}{1}\big)\big(\frac{x}{a}\big)[/imath]

Cheers
[imath]\;[/imath]