Monday, September 27th, 2010...10:06 pm

Millions (and trillions) of cats

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In the classic children’s book, Millions of Cats written by Wanda Gág in 1928, a lonely, elderly couple enter a delightful tale about caring and self-esteem that also lends itself to a discussion of number sense. Toward the beginning of the story, the woman tells her husband that she thinks a cat would make them happy. The comments set the husband off on a journey to find his love’s desired feline. His travels lead him to a hillside covered with hundreds, thousands, millions, billions, and trillions of cats! He tries to pick and, in a sense, does–them all!

Let’s look at the story from the perspective of series, the current topic in my Calculus class. For instance, the old man encountered c_1sum_{k=1}^{10^2} 1, c_2sum_{k=1}^{10^3} 1, c_3sum_{k=1}^{10^6} 1, c_4sum_{k=1}^{10^9} 1 and c_5sum_{k=1}^{10^{12}} 1 of cats, where c_1, c_2, c_3, c_4 and c_5 are some constants since he met more than 1 trillion cats! While a big number, a trillion is less than a drop in the ocean of infinity. In fact, even writing 1 and then zeros for the next two days produces a number that would be lost in an infinite landscape. This should help us begin to sense why concepts, such as sum_{i=0}^infty u_k, that involve the infinite can be nonintuitive and should be approached carefully. Let’s more clearly see this by gaining a better sense of the size of a trillion.

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Returning to the story, the man travels home with the cats, indeed “trillions” of them. We’ll take the plural of trillion to equal 2 trillion. The massive caravan approaches a lake where each kitty takes a sip, leaving only a muddy mass in the place of the previous body of water. How much water did the group discover and consume? Suppose each cat’s sip is a teaspoon. Since 768 teaspoons make a gallon, 2cdot 10^{12} cats would sip 2 cdot 10^{12}/768 = 2.6 cdot 10^9 gallons. Can you visualize 2 billion gallons of water? To help, note that 550,000 gallons of water are estimated to roar over Niagara Falls per second. If the cats continuously took their teaspoon-sized sips, these picturesque falls would be left silent, with not a drop falling, for 2cdot 10^{12}/(768 cdot 550000 cdot 60 cdot 60) = 1.3 hours! Want to change the amount of water each cat might drink? Use WolframAlpha by clicking here. For instance, if each cat drank a cup of water, Niagara would be left without its roaring falls for just over 2 days! Such numbers give a sense as to the magnitude of a trillion. Still, it can be difficult to fully grasp a number of such size which further underscores how difficult it should be for our minds to carefully consider the infinite!

Later in the trip back to the cottage, the elderly man and his cats stop at a hill where each cat takes a mouthful. They leave a barren mound upon their departure. Let’s estimate the size of the hill? Searching the internet, it appears that a healthy lawn contains about 400 blades of grass per square inch. There are 4,014,489,600 square inches per square mile! So, if each cat took only one blade of grass, the hill would be 2 cdot 10^{12}/(400 cdot 4,014,489,600) which equals just over a square mile! But, the text states that each kitty consumes a mouthful. Suppose this correlates to each cat eating a square inch of grass! Then, the man walked up to a “hill” that was about 500 square miles in area! Check the computation or change the kittys’ consumption at WolframAlpha by clicking here.

This helps us see why the woman is so shocked to open the door to her husband and his entourage of millions, billions and trillions of cats. What happens next? What other mathematical questions can you ask? Read the book and ask (and answer) your own questions, further contemplate the size of a trillion and if you are studying infinite series, keep in mind that if a trillion can be tricky to consider then infinity can be even harder to grasp by our finite minds!

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