Which Place?Date: 07/28/2003 at 03:20:15 From: Kelly Mirelle Subject: Place value In which place is the digit 6 in the number 3164297 ? In my book there is an answer that says it is in the 100000, and when I asked my mom she said 10000. I dont know which one is correct! Date: 07/28/2003 at 09:25:47 From: Doctor Ian Subject: Re: Place value Hi Kelly, One way to solve a problem like this is to write down all the possible place values. We do that by starting with 1, and multiplying by 10. 10,000 1000 100 1 Now, these quickly get pretty big! So to make things a little more compact, we use exponents. If you're not familiar with exponents, the basic idea is to use one number to represent a bunch of multiplications: 1 2 = 1 * 2 2 2 = 1 * 2 * 2 3 2 = 1 * 2 * 2 * 2 4 2 = 1 * 2 * 2 * 2 * 2 Do you see the pattern? When we write something like 9 10 we mean start with 1, and multiply it by 10, 9 times that is, 9 10 = 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 1,000,000,000 Now, when we use exponents with 10's, there is a nice pattern: 1 10 = 10 2 10 = 100 3 10 = 1000 Do you see the pattern? The exponent is the same as the number of zeros. So for a number like 24 10 we don't have to do 24 multiplications. We can just write a 1 with 24 zeros after it: 24 10 = 1,000,000,000,000,000,000,000,000 There is one other thing you need to know, one that will eventually seem pretty natural. Anything raised to the 0th power is 1. That is, 0 2 = 1 0 10 = 1 and so on. One way to think of it is from the definition we started with. To compute 2^0 (which is another way to write the exponent, when you want to fit everything on one line), we start with 1, and multiply it by 2 zero times. That is, we start with 1, and do nothing to it, so we end up with 1. Why am I telling you all this? Because now we can find the place value of each digit in a number like 3164297 by using exponents: 6 5 4 3 2 1 0 10 10 10 10 10 10 10 <- place values 3 1 6 4 2 9 7 <- digits So to find the place value of a digit, we count over from the right, starting at 0: 3164297 <- digits ^ | 0 | 1 | 2 |3 4 <- counting from zero When we get to the digit 6, we're at place 4; so the place value of 6 in this number is 10^4, which is 1 with 4 zeros, or 10,000. So you and your mom are correct. By the way, here's a less complicated way to do the same thing. For a number like 32571 we can write it as a sum: 30000 2000 500 70 + 1 ------- 32571 Now we can see immediately what the place values are. For your number, it would look like this: 3000000 100000 60000 <-- 60,000 is 6 times 10,000 4000 so 10,000 is the place value 200 for the digit 6 in this number 90 + 7 -------- 3164297 Note that each of the items in the sum corresponds to one of the exponents that we saw earlier: 3000000 3 times 10^6 100000 1 times 10^5 60000 6 times 10^4 4000 4 times 10^3 200 2 times 10^2 90 9 times 10^1 + 7 7 times 10^0 -------- 3164297 Does this make sense? Write back if any of it wasn't clear, or if you have any questions about it. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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