The Zero Power of Two
Date: 12/10/98 at 10:51:56 From: David Burns Subject: Exponents/powers of two Dear Dr. Math, In fifth grade we've learned that 2 to the third power = 8, two squared = 4, 2 to the first power = 2, and 2 to the zero power = 1. Could you please explain how 2 the zero power = 1 because I'm having trouble understanding this. For example, 2 cubed means that you multiply 2 by itself 3 times. How do you multiply 2 by itself 0 times in 2 to the zero power? I understand the pattern of 2 cubed, squared, to the first power, and to the zero power (8, 4, 2, 1), but I'm still having trouble with this idea. Could you help? I looked through your elementary archives and found nothing on this subject. David Burns
Date: 12/10/98 at 13:01:51 From: Doctor Rick Subject: Re: Exponents/powers of two Hi, David. Good question! Actually we do have material on why a number to the zero power is 1, but I'm not surprised that it isn't in the Elementary Archives. Questions about why numbers behave as they do are best answered when you get to study algebra. Here is our FAQ (Frequently Asked Questions) page about this question: http://mathforum.org/dr.math/faq/faq.number.to.0power.html You will see some things there that you won't understand, but some of it may help you convince yourself. You know, there was a time when the only numbers people knew were the counting numbers 1, 2, 3, .... Zero hadn't been invented yet, so nobody could ask your question. Then zero and negative numbers were invented, and fractions and decimals, and even more that you probably haven't heard of yet. Each time new numbers were invented, mathematicians had to figure out how those numbers behave. You don't want to have a whole new set of rules for the new numbers - you want them to follow the same old rules, but to take them where no number has gone before. This is what happened with powers. When zero is added to the counting numbers, you need to figure out what 2^0 (2 to the 0 power) is. The old definition doesn't help you, because as you say, multiplying zero 2's together doesn't make sense. But you want powers to keep working the same way they always did, and one rule is this: if you divide a number to a power by the same number to a different power, the answer is the same number raised to the difference of the first two powers. For example, 3 2 (3-2) 1 ---- = 2 = 2 2 2 What happens when the powers in the numerator and denominator are the same? 3 2 (3-3) 0 ---- = 2 = 2 3 2 But you know that 8/8 = 1. So 2^0 must equal 1. You can do the same sort of thing to figure out what 2^(-1) should be, or what 2^(1/2) should be. I hope this helps you. Keep asking those "why" questions, and you will be all set for algebra, and more! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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