I was actually just talking with my students the other day about how different kinds of numbers are useful for different purposes, as a lead-in to introducing complex numbers (for @Stoid, complex numbers are just a kind of number that can describe rotations, which it turns out makes a lot of math make more sense). A shepherd has use for positive integers, but has no use for fractions. A merchant or architect has use for fractions, but not for negative numbers. A banker can use integers, fractions, and negative numbers, but has no use for complex numbers. An electrical engineer can use all of those.
They do after a wolf shows up
I suppose I should have tagged @Stoid for that question because their answer is what I want. I’m well aware how negative multiplication works.
I don’t think your debt explanation works for someone that thinks “where did my physical frisbee that’s in my hand physically go when math happens?” Like, who’s the debit to, does it transport into the upside down?
Or, to a pure mathematician, it’s an object worth studying in its own right.
A mathematician would answer the thread title’s question by looking at how “zero” and “multiplication” are defined, and seeing what logically follows from those definitions. It’s nice when the definitions match the way the concepts are usually used by ordinary people in the real world, but mathematical definitions are prescriptive, not descriptive.
I’m pretty sure that’s not the kind of answer the OP was looking for, though.
This resonates for me more than anything else that anybody has said. And it reminds me of how I learned to understand a financial statement.
My brain was screaming when the person who was trying to teach me how to be a bookkeeper was telling me that in order to increase, a section of the statement I needed to “debit” it. My thing is not math my thing absolutely is words, and in English that is as it is Understood and spoken by the vast majority of us to debit a thing means to decrease it so telling me to increase by debiting was giving me heart failure.
Then my teacher came to understand the problem and said OK let’s stop using those words for a while; instead of using debit and credit, let’s say to increase here, you “orange” it. To decrease here you "apple "it.
The sky opened and the Angels sang; I felt like Helen Keller finally understanding that the finger movements represented water. Does that make sense to you?
I think for some people, math just clicks. It’s not something that they learn exactly. That’s how I was. I loved math. It seemed very natural. It didn’t feel like I was learning math. It felt like I was learning how to speak a language that I already knew. It’s like I already had the math concepts swirling in my head and just needed the math structure to make sense of concepts. But not everyone is like that. One of my kids struggled in math. Since I love math, I was eager to help her with math, but it was a struggle. I don’t feel it ever clicked with her. She learned the concepts through brute force in order to get through the class, but it remained sort of abstract to her. It was just an esoteric subject that she learned in order to pass the class.
And in ordinary English, to “multiply,” at least in one sense of the word, means to increase (as in “go forth and multiply”). But mathematically, the result of multiplication need not be larger than the numbers multiplied. (Think of multiplication as a scaling factor. You could scale up or down.)
I have “Touch RPN” installed on my smart phone. It’s a pretty good imitation of an HP-15C. Never tried programming it, though.
Oof. I really don’t think that’s an accurate description. We already have a way to describe “take it away”: that’s subtraction.
Multiplication can mean many things. When dealing with objects like frisbees, “equal groups of” is the most common model. You can do repeated addition with it also. But “take it away” isn’t what multiplying by zero means.
In this particular example it does, though. Multiplying by zero means to turn a number into zero. If that number represents a quantity of something, then that quantity becomes zero.
It is absolutely an oversimplification, but in this case an oversimplification is required.
The term “multiply” is a confusing one in math when a person doesn’t have a strong grasp of the basics. Because what multiplication does is dependent on the values of the numbers you’re dealing with. Multiplying by one changes nothing. Multiplying by a value greater than one resembles addition. Multiplying by a fraction lesser than one resembles subtraction. Multiplying by zero is effectively subtracting everything. (And that’s only talking about positive numbers.)
So while “take it away” is more accurately used to describe subtraction, what is the practical difference between multiplying by zero and subtracting a value by its own value?
Right–but you’re not taking something away when you multiply.
Consider two problems:
1-1=0
Attach units to the numbers, and write a sentence:
One Frisbee, minus one frisbee, equals zero frisbees.
That’s pretty easy to understand, right? You can act it out.
Now:
1x0=0
How would you rewrite that as a sentence, with units?
It’s not necessarily “taking away”, though. It could just be “making irrelevant”.
Let’s say that @Stoid and all of her dog-loving friends each have three Frisbees, and take them with them to the park. When five friends go, there are 15 Frisbees at the park. When none of them go, there are 0 Frisbees at the park. Nobody took away @Stoid 's Frisbees; she still has all three of them. They’re just not at the park.
One more way to think about this that’s used when first introducing these concept that I think might satisfy @Stoid and the more mathematically inclined is the number line.
Think of a number as an arrow pointing from the origin to the magnitude of the number you’re talking about. Adding means the arrow gets longer (pointing further to the right on the number line). Subtracting means the arrow gets shorter (provided it’s positive). Multiplying by 2 means the arrow gets twice as long.
Ok, so your arrow is length 1. Multiply by 0.5, what happened? What about if you multiplied by 0.1? 0.00001? As the number you multiply by approaches zero, does the length of the arrow approach anything?
If you want to talk about frisbees, instead of the length of the arrow talk about the radius of the frisbee or something.
That model works so much better, in part because it works with any numbers. Five friends, three frisbees each? 5x3=15. A million friends, thirteen frisbees each? 1,000,000x13=13,000,000. One friend, two frisbees each? 1x2=2. No friends, three frisbees each? 0x3=0.
To suggest that multiplying by zero means you turn it into zero suggests EITHER that multiplying by zero is handled differently from any other multiplication, OR that 3x4=4, and 3x7=7, and so on.
“\times\ 0” means to reduce the multiplicand to zero.
So …
“All your frisbees will be taken away. There will be zero frisbees left.”
Then what does x2 mean – to reduce the multiplicand to two? Or do we multiply by zero in a completely different way from how we multiply by any other number?
It has the effect of returning a product of zero. But if we say it means that we “take away” or “reduce,” we confuse the issue by incorrectly explaining what it means to multiply in general.
Note that your sentence doesn’t attach units to the numbers. It handles it completely differently from all other multiplication sentences. That’s an incorrect model.
I like to watch quiz shows. Contestants self-select as being more than averagely knowledgeable (and pass some kind of pre-show test).
I have frequently noticed that once a question begins to look as if it is maths-related, many contestants immediately pass. These are not usually hard questions - “What is 30% of 100? How many square feet are there in a square yard?”
This reluctance to take arithmetic questions seems to apply more to females than males.
@Stoid, one reason you might be getting confused it because modeling multiplication IS different from addition or subtraction.
In your classic addition problem (number plus number equals bigger number), all three numbers represent the same kind of thing. “Five hippos plus three hippos equals eight hippos” models 5+3=8. “Ten nerds plus six nerds equals 16 nerds” models 10+6=16.
In subtraction, it’s the same thing. “Eight pizzas minus two pizzas equals six pizzas” is 8-2=6.
One of the things I teach kids is that you HAVE to be talking about the same kind of thing with addition and subtraction. A very common question I ask kids is like, “Eight crayons plus four boxes of crayons equals twelve–twelve what?” They stumble, and I explain if they’re adding eight crayons plus four boxes of crayons, they’re probably not understanding the question adequately.
We call those “kinds of things” “units.” I always always tell kids to label their numbers with units when solving story problems.
In addition and subtraction, the units are all the same.
In multiplication, they’re rarely the same. (Ignore this note if it’s confusing, but it’s arguable that in finding area, the numbers are sometimes the same).
Consider this problem:
Bob eats worms for breakfast every morning. He eats eight worms every day. Over the course of five days, how many worms does Bob eat?
You can model that with 5 x 8 = 40, right? But put units on your numbers:
5 DAYS x 8 WORMS EACH DAY = 40 WORMS
Notice how the first and second unit are different.
When you’re talking about multiplying objects, you’re almost always talking about groups of objects, with the same number of objects in each group. You can model it by making the first number represent the number of groups, and the second number be the number of objects in a group.
This means something really important: THE FIRST NUMBER DOESN’T DO SOMETHING TO THE SECOND NUMBER LIKE IT DOES IN ADDITION OR SUBTRACTION. It doesn’t increase the first number by the second number; it doesn’t decrease the first number by the second number. THE TWO NUMBERS REPRESENT DIFFERENT THINGS.
If this is confusing, the absolute best thing to do is to play around with it. Get out some beans, and put them in equal piles. Write multiplication equations that represent the piles.
Once you’re super comfortable writing those equations, work backwards: write an equation, then build piles of beans that model it.
Only when you’re super comfortable modeling equations should you try this final challenge: model 1x0 using your piles of beans.
Let me ask a different question. Keeping the subject to the context of integers, what does it mean to divide a number? If I have 10 chocolate bars and a group of five kids, how many chocolate bars does each kid get? Now how about if I have 10 chocolate bars and zero kids? What’s the answer there?
Clearly, zero is a special number and it does indeed have to be treated differently, and I suspect that this is the root of the OP’s confusion. Normal integers literally multiply a quantity; if I order a pack of two gadgets and indicate “x5” I will get ten gadgets. Same with negative multipliers in terms of multiplying debt. Zero is neither a positive nor negative number. Zero is special.
Sure it does, just like your sentence. The unit is frisbees.
Again, you’re incorrect. Dividing by zero is done with the same sentences as dividing by any other number. Put X into Y equal groups; how many are in each group?
Put 10 chocolate bars into one equal group; ten in each group. 10/1=10
Put 10 chocolate bars into two equal groups; five in each group. 10/2=5
Put 10 chocolate bars into ten equal groups; one in each group. 10/10=1
Now try that sentence with zero.
Put 10 chocolate bars into zero equal groups.
YOU CAN’T DO IT!
I demonstrate exactly this to kids to show that if you apply exactly the same procedure to zero, you get an impossible task. But you have to understand the procedure in order to understand why it doesn’t work.
Treating it like you’re doing something different is going to mislead people.
I don’t even know what to do with this. Please reread the examples. You’re simply incorrect. You did not attach numbers to units. YOu only used a single number, instead of all three numbers from the equation.