What Is Infinity Plus One?
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- Crossing the Street This photo is of Wikis Take Manhattan goal code R9, Children/Families, crossing street. |} (Photo credit: Wikipedia)
One-to-one correspondence
Materials
- Children
- Hands
- Street to cross
Instructions
- Ask each of your grandchildren to take your hand when you cross the street
- Ask them if you have enough hands for each child
- If you do, ask them to name other families who have enough hands to walk their children across the street
- Ask them to name other families who do not have enough hands to walk their children across the street
- Ask them to name other families who have more hands than they need to walk a child across the street.
What Should Happen?
Your grandchildren are identifying sets – parents who have two hands and two children, parents who have two hands and one child, parents who have two hands and more than two children.
Why Is This Important?
A set, in math, is a well-defined group. You know what is in the group and what is outside.
Parents with two hands and three children do not belong in the group of parents with two hands and two children. But, all parents with two hands and two children, whether they are tall or short, American or not, live nearby or not, fit in the set of parents with two hands and two children.
Georg Cantor, a mathematician in the 1800s, invented set theory. He discovered that you can find out whether one set is bigger or smaller than another set by applying one-to-one correspondence. That is, the set of parents with two hands and two children is the same size as the set of parents with two hands and three children as long as there is one parent in each set that corresponds to a parent in the other set. If you run out in either set, the other set is larger.
He applied this insight to infinity, adding and multiplying an infinite number of things.
So, the answer to what is infinity plus one is infinity because if you compare an infinite set of things, like real numbers (1,2,3,4…) to another set of infinite things, like fractions (1/2, 1/3, ¼, 1/5, 1/6…) you will never run out of either set.
Similarly, the answer to what is infinity times infinity is infinity.
In answer to the question are there any infinite sets of numbers that do not correspond one-to-one, that is, are different sizes, and if so, how do you tell which is larger?
Yes. The set of real numbers, which includes all the rational numbers, fractions and irrational numbers, square roots, and pi, along a continuous line (1, 1 ½, 2, 2 ¼…) out to infinity, is larger than the infinite set of natural numbers (for counting, 1,2, 3, 4…) and ordering (first, second, third…) because you cannot pair up numbers in the two sets.
The set of real numbers is larger because it has numbers left over when you match them to natural numbers.
Thanks to talkingmathwithkids.com for this activity.
Carol Covin, Granny-Guru
Author, “Who Gets to Name Grandma? The Wisdom of Mothers and Grandmothers”
http://newgrandmas.com
Filed in: education • family activities
I'm not gonna lie… I just got a rush of story problem math anxiety 😉
I love the picture used to illustrate this concept. You can see the little boy holding his hands behind his back so his Mom can't take his hand across the street. Since infinity has no end, you can add something to it and multiply it and it will still have no end.
Very nice. It's perfect just to use when grandchildren take on that attitude, "I don't need to hold your hand." Love how the learning is like a game. You're brilliant Cheryl!
This one made my brain hurt.
Sorry, Jon. The basic concept is that since infinity has no end, if you add something to it, it's still infinity. It's a little harder to think about whether one set of infinite objects is larger or smaller than another set of infinite objects. That's where one-to-one correspondence comes in.
This is a cool way to engage you kids/grandkids in what the concept is about and to have some fun too. Great job Carol. 🙂
Thanks, Susan. Starting with just the simple game of seeing if Mom has enough hands for each child, and who, among their friends, has the same number of children, introduces a pretty sophisticated concept that they'll understand later.
I don't believe it… we played many games where infinity plus one trumped infinity!
I yield to the logic of children!
Never thought of using infinity plus one. My children and grandchildren didn't ever consider not holding my hand when I said take my hand.