My eighth grade math class got to the mathematics of scientific notation, especially how huge multiplication problems can turn into simpler additions of exponents. When the curriculum shifted over to addition with scientific notation, a deviation from the students’ expectations led them, in response to a problem involving adding planetary distances, to an answer larger than the size of the universe without them even wondering if their answer made sense. This led me off into the relative scale of things. I used this link (http://htwins.net/scale2/?bordercolor=white) and started moving outward. We discovered that the entire observable universe was three orders of magnitude smaller than the answer they had put down This led me into a series of thought problems each evening that I shared with them the following mornings. The final thought problem was this. So, a hundred years ago, the best astronomers in the world thought that our galaxy was the entire universe. Everything observable was within the “confines” of our galaxy. Our galaxy is 10 x 10^{21 }meters in diameter so that is the dimension within which the best astronomers of that time thought. Now, we are thinking the observable universe is 10 x 10^{27} meters across. How much bigger is this universe than the one a century ago? This turns into: how many times bigger is 10^{27} than 10^{21}? We find this by dividing the first number by the second. Because of scientific notation, this turns into 10^{(27-21)} which is 10^{6}. That is 1 with 6 zeroes after it. It’s 1,000,000 times further to the edge of the universe now than astronomers thought a hundred years ago. However, space is a volume and since we measure space in three dimensions, the width must be cubed to give us the volume. The kids had learned that when a power is raised to a power, you multiply the exponents so one million cubed is (10^{6})^{3} which is 1 x 10^{(6×3)} which is 1 x 10^{18}. That is 1 with 18 zeroes after it. So we have found the answer to our question in a couple of minutes. The universe that the astronomers of our present time think about is 1,000,000,000,000,000,000 – one quintillion – times larger than the universe that the best astronomers of a century ago thought about.

# Scientific Notation and the Size of the Universe – Cairns #76

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