Explain to me why the simple subtraction problem of 1-(-1) = 2 using just marbles?
You can think of the negative sign as debt. So if you HAVE a marble that means you have positive one (+1), and if you OWE a debt of one marble that's a negative one (-1).
When you subtract a negative, it's like taking away a debt.
So 1-(-1) = 2 is as if I give you a marble (the positive 1) and I also take away your one marble debt (subtract the negative 1), which combined is the same impact as if I had given you two marbles.
5-(-2)=.....? Just like as if I gave you five marbles (the 5) and also took away your 2 marble debt (the -2), so your net gain is 7 marbles.
Explain to me why 1x1x1x1x1 = 1 using your cups and marbles analogy?
So imagine you have a certain # of marbles in each cup, and a certain number of cups in each box, and a certain number of boxes in each truck.
1x1x1x1 is saying that you have one marble in each cup, one cup in each box, one box in each truck, and just one truck. So how many marbles do you have total? Only one.
If I insert the symbols, it's basically saying one (1) marble in each (x) cup, one (1) cup in each (x) box, one (1) box in each (x) truck, and one (1) truck = one (1) total marble.
You can change any of the #'s and the equation works perfectly. If you have 2 cups in each box but everything else in the same, your equation becomes 1x2x1x1 = 2 marbles. If you have two marbles in each cup, two cups in each box, two boxes in each truck, the equation becomes 2x2x2x1 = 8 marbles. If you get a second truck with the same setup, now you have 2x2x2x2 = 16 marbles.
Explain to me why -1x-1 = 1 using marbles?
So imagine the first -1 is your debt of one marble.
If I had written -1 x 5, that's basically like applying a debt of one marble five times. So now you have a debt of five marbles (-5). But if you do -1 x 1, then you're only applying that debt once, so your debt is just 1 marble (-1).
When you choose to multiply by a negative instead of a positive, you're inverting the action. So -1 x -1 is the opposite of applying a 1 marble debt once. What's the inverse of applying a debt? Just like in the addition problem, you're now removing the debt! So -1 x -1 means that you removed a 1 marble debt once. If I remove a one marble debt from you, that's the equivalent of giving you 1 marble.
Why is 100^0 = 1... but 100^1 = 100 using cups and marbles?
I think the easiest way to understand this is to work backwards from larger exponents first.
So you know that 100^2 is the same as 100 x 100 = 10,000
And you know that 100^3 is the same as 100 x 100 x 100 = 1,000,000
How do we get from 100^2 to 100^3? We just multiplied by 100, right? 100^3 is one more multiplication of 100 than 100^2 was.
Okay, so let's go backwards. How do you get from 100^3 to 100^2? You DIVIDE by 100, because division is the opposite of multiplication.
So 100^3 divided by 100 will give you 100^2. Just like 1,000,000 divided by 100 gives you 10,000.
Now let's keep going. How do you get from 100^2 to 100^1? You have to divide by 100 one more time. So what's 10,000 divided by 100? The answer is 100.
This makes sense, because if 100^3=100x100x100 (three multiplied 100s), and 100^2 = 100x100 (two multiplied 100s), then 100^1=100 (just one 100).
Now you can see how we get 100^0. To get from 100^1 to 100^0, you just divide 100^1 by 100. What is 100 divided by 100? It's one!
So 100^0 = 100^1 divided by 100 = 1
Why is 1^∞ undetermined....but 1^1,000,000 = 1 using cups and marbles?
1^∞ is only undetermined when "1" can possibly be part of an equation with variables, which would then have to be solved by limits. In THAT specific case, which is the sort of thing you'd run into in Algebra II, then you have to consider 1^∞ an indeterminate form because it doesn't find its limit at 1, since a number even infinitesimally larger than 1 will go to infinity, not 1, when multiplied by itself infinite times. I could give a longer, better explanation, but I don't feel like reteaching Algebra II from scratch right now.
The important part is that IF you assume that 1 is a constant, not part of a variable equation, then you CAN say that 1^∞ = 1.
So just be comforted in the fact that 1^∞ = 1 if 1 is assumed to be a constant. And don't worry about the indeterminate answer unless you're actually solving limit equations.
Thanks Professor