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Zero and One We all know that two of the simplest numbers are zero and one, yet something so simple can still be profound. Consider the following: 1. If we didn’t have zero, we couldn’t have base 10! The Roman numeral system was used for quite a while, but they didn't have a way to represent zero. When ten came around, the used the letter "X" and continued on their merry way with other numbers. Have you ever tried to add or subtract with Roman numerals? What about decimals? 2. Zero is the additive identity because when you add it to anything it does not change the number’s identity. Thus, you can add zero to anything and you won't change a thing. 3. One is the multiplicative identity because when you multiply a number by 1 it does not change the number’s identity. Thus, any number multiplied by one will give you the same value. Consider fractions though. 1/3 * 4/4 = 4/12, which changes the look of the fraction but not the value. 4/4 is one. 4. The additive "opposite" of a number is something added to get 0. Like 8 + -8 = 0. Thus the additive inverse is the number that gets you back to zero. 5. The multiplicative "opposite" of a number is something multiplied to get 1. Like 8 * 1/8 = 1. Another term for this is the "reciprocal". 6. One of the oddest uses of zero and one is when any number is raised to the zero power. Any number (except zero) raised to the zero power is 1! Believe it or not! Here's how I explain it: 2 1 0 By the way, ten to the negative one power would continue in this pattern, thus giving 1/10 (not a negative answer). 7. Any number to the first power is that number. One raised to ANY power is always one. Zero to any power (other than zero) is always zero. It is not possible to raise 0 to the 0 power, due to definition of negative exponents and an algebraic proof.. 8. Other number systems, fields, etc have their own mathematical operations but they must operate in certain conditions.. There is an additive inverse and a multiplicative inverse. How do you tell what they are? The additive inverse has to equal the "0". The multiplicative inverse has to make the number equal to the "1". Let's take complex numbers, for instance 5 + 4i. What is it's additive inverse? The answer: whatever it takes make a sum of zero (-5 - 4i). What is it's multiplicative inverse? Whatever 1/ (5 + 4i) turns out to be when you simplify it. 9. All probability is a number between 0 and 1. 10. You can’t divide by zero. 8 divided by 0 cannot be done. Now, 0 divided by 8 is equal to zero, but division is not commutative. If you take 0 objects and split them (divide) 8 ways, every person gets zero. But if you have 8 things and you want to split them up into 0 groups, it's not possible! Everything can divided by 1! Here's a better way to think about it: 6 / 3 is 2 because 3 times 2 gives you an answer of 6. 0 / 8 is 0 because what do you take times 8 to get an answer of 0? Zero! Now, 8 / 0 is not possible because what do you take times 0 to get an answer of 8? Can't be done! A good teaching reminder! N
O O K It's not okay to have a zero in the denominator, but it is ok to have it in the numerator. 11. How many numbers are between 0 and 1? I posed that question to my 7th grade students several years ago and was astounded by the response! Again, I asked, "How many NUMBERS are there between zero and one?". Most of the classes answered that there were none, of course thinking about INTEGERS, but after a little connection to money they soon agreed on decimals and fractions. 12. Zero is different than "none". Zero represents nothing, but zero is a number. It's kind of like banking. When you open an account at a bank, and put no money in it, the balance is 0 (a number). When you don't even have an account, that's NONE! Be careful when your answer is zero versus nothing. 13. What about 0/0 ? There are three possibilities: a. If we concentrate on the numerator, the
answer is 0. If we look at other fractions, 0/3 = 0 , 0/5 = 0 , b. If we concentrate on the denominator, the
answer is not possible. If we look at other fractions, 5/0 is not possible,
10/0 is not possible, 567/0 is not possible (see explanation above). Which way is correct? Mathematicians go with
(b) not possible. It seems dividing by zero takes more importance in the
order of things. Question: What's 1! ? Answer:
1 |