Mills

[KWF]

1 mill = $(1/1000) or $0.001. This can be expressed as $(1/1000)/1 mill. I think that both sides of the equal sign were divided by 1 mill to equal $(1/1000)/1 mill.

Is there another calculation that can be used to change 1 mill = $(1/1000) into $(1/1000)/1 mill other than by dividing both sides by 1 mill?

 

[mmm4444bot]

1 mill = $(1/1000) or $0.001. This can be expressed as $(1/1000)/1 mill. No, it cannot.

The number 1 can be expressed as the ratio [$(1/1000)]/[1 mill] because the numerator and denominator are equal.

Is there another calculation that can be used to change 1 mill = $(1/1000) into $(1/1000)/1 mill NO!

There are not any ways to change an equation into a fraction. We can only change an equation into another equation.

Given the equation 1 mill = 1/1000 dollars, we can write two different conversion factors (one for converting dollars into mills and another for converting mills into dollars).

[1 mill]/[1/1000 dollars] is the conversion factor for converting dollars into mills.

[1/1000 dollars]/[1 mill] is the conversion factor for converting mills into dollars.

Do you have a specific exercise on which you are working?

 

[KWF]

Thanks mmm4444bot for the reply!

"Do you have a specific exercise on which you are working?" No
1 mill = $(1/1000) or $0.001. This can be expressed as $(1/1000)/1 mill. No, it cannot.

Why do you think that 1 mill = $(1/1000) cannot be expressed as $(1/1000)/1 mill?

I still need some clarification.

I thank you for your reply. Is there a way to rate your answers?

 

[mmm4444bot]

Why do you think that 1 mill = $(1/1000) cannot be expressed as $(1/1000)/1 mill?

Because these are two different types of statements. One is an equation and the other is not.

It's like comparing apples to oranges. They both equal fruit, but they do not equal each other.

Is there a way to rate your answers? Thankfully, no.

\(\displaystyle 1 \ \text{mill} \;=\; \frac{1 \ \text{dollar}}{1000}\)

This is called an "equation". It is a mathematical statement that the quantity to the left of the equals sign has the same value as the quantity to the right of the equals sign.

If we divide both sides of this equation by 1 mill, then we get another equation. We can't just throw away half the equation, including the equals sign!

\(\displaystyle \frac{1 \ \text{mill}}{1 \ \text{mill}} \;=\; \frac{\frac{1 \ \text{dollar}}{1000}}{1 \ \text{mill}}\)

The lefthand side of this new equation simplifies.

\(\displaystyle 1 \;=\; \frac{\frac{1 \ \text{dollar}}{1000}}{1 \ \text{mill}}\)

This resulting equation is a statement that the ratio on the righthand side has a value of 1.

If we now write only the righthand side (i.e., we erase the "1=" part), then we no longer have an equation. All we have is a ratio; which is a new type of statement. The ratio itself and the equation from which it comes are not the same thing.

\(\displaystyle \frac{\frac{1 \ \text{dollar}}{1000}}{1 \ \text{mill}}\)

We can say that this ratio equals 1, but it does not make sense to say that it equals some equation.

Again, an equation is not a number. An equation is simply a statement that relates two numbers.

Here's an analagous scenario.

If I write 1 = 1, that's an equation, yes?

Now, if I divide both sides by 2, I get the following.

1/2 = 1/2

You're trying to claim that the number 1/2 has the same value as "1 = 1".

That claim makes no sense.

I mean, if somebody asks you, "How much is one-half?", and you answer, "One equals one", they will look at you funny because your answer does not make sense.

You're trying to do the same thing with 1 mill.

If somebody asks you, "How much is a thousandth of a dollar divided by one mill?", and you answer, "one mill equals a thousandth of a dollar", they will say, "That's not what I asked".

Your response gives them sufficient information to determine the answer, but your response is not the answer itself (which is "1").

Again, I ask you: What motivated your question? I mean, what led you to think about this stuff?

 

[KWF]

Hello mmm4444bot:

I want to thank you for your reply and great answer.

If I understand correctly 1 mill does not equal $(1/1000)/1 mill because the two terms separated by the equal sign are not equal. Is this correct.

I saw this in a textbook: "1 mill = $1.00 divided by 1000."

After reading what you wrote in your answer. This equation cannot be true either because $1.00 divided by 1000 equals $0.001 not 1 mill. (The units do not match. Where does "mills" come from?)
I can indicate $1.00 divided by 1000 as 1/1000 X $1.00. This equals $0.001. I do not know why the author represented 1 mill as $1.00 divided by 1000. Some things author omit mathematical steps and go directly to the final answer.

In regards to the following: "What motivated your question? I mean, what led you to think about this stuff?"

Good question! I hope that I can provide a good answer. I am not a student in a class nor am I trying to determine my property taxes. I am just a student of business mathematics, so to speak. I read from an old textbook that I used in high school. There is a section on property taxes and "mills" are used, but sometimes the authors do not fully explain the calculations. I guess they think people can read between the lines.

I am just curious about mathematics that I can use in my daily life and the meanings behind the calculations. I can better understand the reasoning for the solution if I can understand the reasoning of the calculations.

If this does not answer your question, let me know, and I will try again.

Thanks again for your help!

 

[mmm4444bot]

If I understand correctly 1 mill does not equal $(1/1000)/1 mill because the two terms separated by the equal sign are not equal. Is this correct.

Yes.

\(\displaystyle 1 \ \text{mill} \ = \ \$0.001\)

\(\displaystyle \frac{\$0.001}{1 \ \text{mill}} \ = \ 1\)

0.001 does not equal 1

I saw this in a textbook: "1 mill = $1.00 divided by 1000."

After reading what you wrote in your answer. This equation cannot be true either because $1.00 divided by 1000 equals $0.001 not 1 mill.

No, this equation is true. In fact, this equation gives us a definition for the noun "mill".

$(1/1000) is just another way of writing $0.001, so both of these expressions mean the same thing: one-thousandth of a dollar.

1 mill = $0.001

$(1/1000) = $0.001

Therefore, 1 mill = $(1/1000).

We can think of the word "mill" as an abbreviation for saying "one thousandth of a dollar".

Where does "mills" come from? According to the Wikipedia article HERE: The term comes from the Latin "millesimum", meaning "thousandth part".

I do not know why the author represented 1 mill as $1.00 divided by 1000.

That's the definition for the noun "mill" !

The author simply used an equation to state this definition.

The author could just as easily have written an English sentence, instead:

\(\displaystyle 1 \ \text{mill is one-thousandth of a dollar.}\)

(In this sentence, the verb "is" represents the equal sign.)

Do you now have a clear understanding of why the following ratio equals 1 versus $0.001 ?

\(\displaystyle \frac{\$0.001}{1 \text{mill}} \ = \ 1\)

In any ratio, if the numerator (the value on top) and the denominator (the number on bottom) are EQUAL values, then the value of the ratio itself is 1. (0/0 is an exception because division by zero is not defined.)

\(\displaystyle \frac{0.001}{0.001} \ = \ 1\)

\(\displaystyle \frac{0.001}{1/1000} \ = \ 1\)

\(\displaystyle \frac{4}{4} \ = \ 1\)

\(\displaystyle \frac{6 - 2}{4} \ = \ 1\)

\(\displaystyle \frac{x}{x} \ = \ 1\)

\(\displaystyle \frac{\clubsuit\clubsuit}{\clubsuit\clubsuit} \ = \ 1\)

\(\displaystyle \frac{\$0.001}{1 \text{mill}} \ = \ 1\)

\(\displaystyle \frac{1 \text{mill}}{\$0.001} \ = \ 1\)

Okay now?

Previous edit: Deleted implication that 0/0 = 1 :roll:

 

[Denis]

Re:

In any ratio, if the numerator (the value on top) and the denominator (the number on bottom) are EQUAL values, then the value of the ratio itself is 1. Always!

0 / 0 ; ahem :roll:

 

[mmm4444bot]

0 / 0 ; ahem Yes, this is an exception. (I will correct my post.) Thank you, Denis.

 

[KWF]

Hello mmm4444bot:

I cannot thank you enough for your help. I understand most of your comments except the fact that 1 mill = $0.001.

I understand that $(1/1000) = $0.001. For one reason the units are the same on both sides of the equal sign.

It would be more logical to me if the mills in "1 mill = $0.001" should be converted to dollars as in ($0.001/1 mill) X 1 mill = $0.001. The result would be $0.001 = $0.001.

Although this equation 1 mill = $0.001 is the same as 100 cents = $1.00. Mathematics, to me, isn't always logical in my mind!

Thanks again for your help.

 

[Mrspi]

You started this whole discussion with

1 mill = $(1/1000)

$(1/1000) = $0.001

What part of

1 mill = $0.001

do you not understand?

No, the units are not the same on both sides. You've seen dozens (or perhaps hundreds) of statements like this:

1 foot = 12 inches

16 ounces = 1 pound

100 cents = 1 dollar

2.54 cm = 1 inch

In each case, we could make a "unit fraction" from the given statement.

1 foot / 12 inches = 1 (or, 12 inches / 1 foot = 1)
16 ounces / 1 lb = 1 (or 1 lb / 16 ounces = 1)
100 cents / 1 dollar = 1 (or 1 dollar / 100 cents = 1)
2.54 cm / 1 inch = 1 (or 1 inch / 2.54 cm = 1)

Each of these is called a "unit fraction" because the numerator and denominator represent equivalent quantities.

Unit fractions are used in "conversion" problems where one needs to convert from one unit to another.

For example, if you are asked "How many centimeters are there in 6 inches?" you could use the unit fraction 2.54 cm / 1 inch ....since the value of this fraction is 1, multiplying 6 inches by 2.54 cm / 1 inch will NOT change the 6 inches, but it WILL change the form.

6 inches * (2.54 cm / 1 inch)
The unit "inch" cancels, leaving us with 6 * 2.54 cm, or 15.24 cm

I don't know what more you need.....you've gotten HOURS of response time from one of our finest volunteers.

If there is something more specific that needs to be answered, please repost.

 

[mmm4444bot]

I understand most of your comments except the fact that 1 mill = $0.001.



Do you have any issue with these equations?

1 penny = $0.01

1 cent = $0.01

1 penny = 1 cent

1 dime = $0.10



Although this equation 1 mill = $0.001 is the same as 100 cents = $1.00.

I would not say that those two are "the same" equation, but rather "equivalent".

If doing arithmetic with mills is perplexing, then maybe you're just not used to thinking abstractly, or symbolically.

1 mill x 1,000 = $1

This operation might seem strange, to you, because you're not making the mental substitution of $0.001 for the unit "mill".

This substitution is as simple as thinking:

$0.001 x 1,000 = $1

If you're okay with 1 penny = $0.01, then you should be just fine with 1 mill = $0.001.

I'll leave you with another abstraction:

1 penny = $0.01

$0.001 = $0.01/10

1 mill = penny/10

One mill is a tenth of a penny, too. Which means, if you bought 10 candies for a penny in 1945, you paid 1 mill for each of them.

 

[KWF]

Hello mmm4444bot:

I understand. Thanks so much for your time and efforts in answering my questions!

 

[WilliamLKerr]

Thanks for the tips and clear examples folks.