Decimals as factors

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Cambridge101

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If I have 2(0.5+0.7)

Technically you can read this a 2 being factored out of each term in the bracket.
That said, it can only be a factor if you can divide 1 and 1.4 by 2 and get a whole answer, which you don't.
Is 2 therefore factored out or not?
 
Cambridge101 said:
If I have 2(0.5+0.7)

Technically you can read this a 2 being factored out of each term in the bracket.
That said, it can only be a factor if you can divide 1 and 1.4 by 2 and get a whole answer, which you don't.
Is 2 therefore factored out or not?
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No, no, no.

There is no restriction of the distributive property to whole numbers.

[math]a(b + c) \equiv ab + ac \text { for all real numbers.}[/math]
It is true that many of us like to work with whole numbers, but there is no requirement that you can only "factor out" whole numbers.
 
Cambridge101 said:
If I have 2(0.5+0.7)

Technically you can read this a 2 being factored out of each term in the bracket.
That said, it can only be a factor if you can divide 1 and 1.4 by 2 and get a whole answer, which you don't.
Is 2 therefore factored out or not?
Click to expand...
Why does the common factor have to be a whole number?
 
JeffM said:
No, no, no.

There is no restriction of the distributive property to whole numbers.

[math]a(b + c) \equiv ab + ac \text { for all real numbers.}[/math]
It is true that many of us like to work with whole numbers, but there is no requirement that you can only "factor out" whole numbers.
Click to expand...
I am talking about the language specifically, I am aware that the distributive property holds.
I am saying, to say 2 is being factored out is incorrect language, as 2 is not a factor of 1 or 1.4.
Obviously you can have 2 x (0.5+0.7).
 
BigBeachBananas said:
Why does the common factor have to be a whole number?
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The definition of a factor, is a number which divides into some x and gives a whole answer. So I was curious to see if you could technically say 2 has been factored out in that expression.
 
Cambridge101 said:
Factors are always whole numbers or integers and never decimals or fractions. From the first page on google.
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I believe that you're referring to this:
1641420507630.png
The word "factor" was used in the context of the divisor. It's different when talking about the distributive property.
 
BigBeachBananas said:
I believe that you're referring to this:
View attachment 30480
The word "factor" was used in the context of the divisor. It's different when talking about the distributive property.
Click to expand...
What do you mean by 'its different'. So in this case, when you 'factor out' the 2, you're not really taking out a common factor of both terms are you. You're re-writing 1+1.4 as 2 lots of 0.5+0.7 which obviously works. But is it factoring out? surely not
 
Cambridge101 said:
What do you mean by 'its different'. So in this case, when you 'factor out' the 2, you're not really taking out a common factor of both terms are you. You're re-writing 1+1.4 as 2 lots of 0.5+0.7 which obviously works. But is it factoring out? surely not
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English is hard. The difference is between a verb and a noun, i.e. factor vs factoring. When you take 5/2, you get a divisor of 2 (factor) and the remainder of 1.
Where as what you're doing is factoring, which refers to the distributive property. As JeffM stated, there's no requirement that you can only "factoring" out whole numbers.
 
BigBeachBananas said:
English is hard. The difference is between a verb and a noun, i.e. factor vs factoring. When you take 5/2, you get a divisor of 2 (factor) and the remainder of 1.
Where as what you're doing is factoring, which refers to the distributive property. As JeffM stated, there's no requirement that you can only "factoring" out whole numbers.
Click to expand...
I see, so factoring/factorising refers to the process of diving by a common amount if you like and putting that amount outside the bracket. Whereas a factor is a integer m which can divide into an integer n and give a whole answer.

Thats cool, I just wanted to clarify that indeed factoring can take place even if the amount your taking out of each term isn't a factor itself of the term you're taking it out off.
 
BigBeachBananas said:
English is hard. The difference is between a verb and a noun, i.e. factor vs factoring. When you take 5/2, you get a divisor of 2 (factor) and the remainder of 1.
Where as what you're doing is factoring, which refers to the distributive property. As JeffM stated, there's no requirement that you can only "factoring" out whole numbers.
Click to expand...
In conclusion, factorising and taking out a common factor are different things.
 
I would agree that "factoring/factorising refers to the process of diving by a common amount if you like and putting that amount outside the bracket. Whereas a factor is an integer m which can divide into an integer n and give a whole answer."

About your conclusion, "factorising and taking out a common factor are different things". Define "common factor".
 
Cambridge101 said:
Do we agree about the above statement
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Actually, I do not.

In the expression xy, x is a factor of y and y is a factor of x.

If we are talking about vectors or matrices, a factor may not even be a number at all.

When, however, we are talking about number theory, which is about integers, then a factor is a whole number because whole numbers are what we are talking about. When we say that a prime number like 7 has factors that are equal to itself and to 1, we are not denying that
3.5 is called a factor in the expression 3.5 * 2.

As often happens, the meaning of a word may depend on context.
 
Cambridge101 said:
I see, so factoring/factorising refers to the process of diving by a common amount if you like and putting that amount outside the bracket. Whereas a factor is a integer m which can divide into an integer n and give a whole answer.

Thats cool, I just wanted to clarify that indeed factoring can take place even if the amount your taking out of each term isn't a factor itself of the term you're taking it out off.
Click to expand...
The difference is the context. When you are talking about factorizing in arithmetic, you are working only with integers, and the factor will always be an integer. You are asking about factorizing in algebra, where the objects you are working with are polynomials (or expressions, more generally), and your 2 is not being thought of as an integer, but just as a term.

We can still talk about common factors in algebra, but we are taking "factor" in this different sense, as one of two or more polynomials whose product is a given polynomial.
 
BigBeachBananas said:
I would agree that "factoring/factorising refers to the process of diving by a common amount if you like and putting that amount outside the bracket. Whereas a factor is an integer m which can divide into an integer n and give a whole answer."

About your conclusion, "factorising and taking out a common factor are different things". Define "common factor".
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Common factor. Well, a factor being defined as an integer m whose product with some other integer p has a product of some other integer n. Thus, both p and m are factors of their multiple n, as you can divide n by either p or m and get a whole answer being p or m.

A common factor specifically being a factor which is common for separate integers or more than one entity. Where you can divide two integers by some integer n (factor) and get a whole number, thus n is a common factor of both integers.

Something like this.... my language may not be perfect.
 
Cambridge101 said:
Yeah exactly. So if you take the 2 out of the 1 and 1.4, and re-writing as 2(0.5+0.7) you're not taking out a common factor as defined in arithmetic context?
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If you wish to make the distinction, I would recommend defining factoring as the process of diving by a common amount if you like and putting that amount outside the bracket. The common factor is a special case of a common amount, where the amount is a common divisor.
 
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