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Chapter 5: Problem 143

Why is \(\frac{0}{4}\) equal to 0 , but \(\frac{4}{0}\) undefined?

Short Answer

Expert verified
\(\frac{0}{4}\) equals 0 because 4 fits into 0 zero times. On the other hand, \(\frac{4}{0}\) is undefined because there is no number that can be multiplied by 0 to result in 4 and also, zero could theoretically fit into any number an infinite number of times.

Step by step solution

01

Understand Division

In the context of this problem, it's helpful to consider division as an operation that describes how many times one quantity is contained in another. For example, if you have 10 apples and 2 baskets, you could divide the number of apples by the number of baskets to find out how many apples could go in each basket. This would give you \(\frac{10}{2} = 5\), meaning you could put 5 apples in each basket.
02

Analyze \(\frac{0}{4}\)

When we have zero divided by any number, as \(\frac{0}{4}\), we are asking 'how many times does 4 fit into 0'? Since 0 has nothing in it, 4 can fit into 0 zero times. Therefore, \(\frac{0}{4}\) is equal to 0.
03

Analyze \(\frac{4}{0}\)

However, when we try to divide a nonzero number by zero, as in \(\frac{4}{0}\), we are essentially asking 'how many times does 0 fit into 4'? There is an issue here because 0 can technically fit into 4 an infinite number of times. On the other hand, we could also say that it doesn't fit at all because zero times anything always results in zero, not four in this case. Such a contradiction leads us to say that \(\frac{4}{0}\) is undefined.

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Most popular questions from this chapter

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 20.85 & 21.27 & 21.70 & 22.13 & 22.57 & 23.02 \\ \hline \end{array} $$ $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project Texas's population, in millions, for the year 2020 . Round to two decimal places.

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=40,000, r=0.1\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(a_{1}=-20, d=-4\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2000, r=-1\)
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