# Real Numbers

The set of real numbers contains all the numbers we normally deal with in real world applications. It consists of the rational numbers and the irrational numbers.

Rational numbers are all the fractions – that is, all the numbers that can be expressed as a ratio of two integers. All repeating decimals are rational, because they can be expressed as the ratio of two integers.

Irrational numbers are all the other real numbers. They are only called irrational because none of them can be expressed as a ratio of two integers. Examples of irrational numbers are most of the roots – for example, most of the square roots, cubed roots, 4th roots, etc. The square root of 2 is an irrational number. Indeed, the square root of any integer or rational number that is not a perfect square of some other integer or rational number is irrational. \begin{align}\pi\end{align} is also irrational. Although many think it’s equal to 3.14 (“pi day”), or to 22/7, both of these are rational numbers, and are only approximations to \begin{align}\pi\end{align}. \begin{align}\pi\end{align} is actually an infinite non-repeating decimal, and is irrational.

We normally express real numbers as integers, ratios of integers, or decimals. They can be positive or negative. They can also be very small. For example, .0000000000000000000000001 is a real number which is positive but not much bigger than 0.

Example 1)

If the range of the five numbers 1, 2, 3, 4, and x is 10, then the minimum possible value for x is

A. 6
B. 11
C. – 11
D. – 10
E. – 6

Explanation: The range of a set of numbers is the difference between the smallest and the largest. In this case, we want to know the minimum possible value for x to make the range 10. We are tempted to say 11, because all the numbers listed are positive, and we’re thinking in terms of positive numbers, and 11 – 1 = 10. But negative numbers are also possible, and in this case, since 4 is the largest number listed, and 4- (-6) = 4 + 6 = 10, then -6, which is less than 11, is the correct answer. This is answer E.

Let’s try a similar problem:

If the range of the five numbers -1, -2, -3, -4, and x is 10, then the minimum possible value for x is what?

A. 11
B. -14
C. -11
D. -12
E. 6

Explanation: The range of a set of numbers is the difference between the smallest and largest. In this case, we want to know the minimum or smallest possible value for x to make the range 10. Since -1 is the largest number listed, and -1 – (-11) = -1 + 11 = 10, then -11, answer C, is the correct answer.

Here’s one more similar problem:

If the range of the five numbers -1, -2, -3, -4, and x is 10, then the maximum possible value for x is what?

A. 10
B. -10
C. -6
D. 6
E. 12

Explanation: The range of a set of numbers is the difference between the smallest and largest. In this case, we want to know the maximum possible value for x to make the range 10. Since -4 is the smallest number listed, and 6 – (-4) = 10, then 6, answer D, is the correct answer.

Example 2)

If x, y, and z are positive numbers, such that x + y + z = 1, then the possible range of

S = 20x + 30y + 40z

is defined by:

A. 10 ≤ S ≤ 90
B. 10 < S < 90
C. 20 < S < 30
D. 20 < S < 40
E. 20 ≤ S ≤ 40

Explanation: Because x, y, and z are all positive, they must all be greater than 0 and, because their sum is 1, they must all be less than 1. But that’s all we know about them.

However, if we modify the expression

S = 20x + 30y + 40z

to look like

20x + 20y + 20z = 20(x + y + z) = 20

or to look like

40x + 40y + 40z = 40(x + y + z) = 40

it’s easy to see that S must be greater than 20 and less than 40. It cannot be equal to either 20 or 40, because, for example, the value 30y will make it a little more than 20 and a little less than 40. So

20 < S < 40

So D is the correct answer.

Example 3)

Define an operator □ as follows: For all non-zero real numbers, a and b,

a □ b = (a + 2b)/ab

If 2 □ 3 = 4/3, what is 3 □ 2?

A. 3/4
B. 4/3
C. 5/6
D. 1
E. 7/6

Explanation: Problems like these can be fun, because they generalize the ideas of addition and multiplication, both of which are themselves operations. So, let’s plug in the numbers and see what happens with this operation:

2 □ 3 = (2 + 2∙3)/2∙3 = (2 + 6)/6 = 8/6 = 4/3

So we can see that the first part is true. Now let’s reverse the roles of 2 and 3:

3 □ 2 = (3 + 2∙2)/2∙3 = (3 + 4)/6 = 7/6

Therefore, the correct answer is E.

Example 4)

How many leading zeroes to the right of the decimal point are in the expansion of

\begin{align}(\frac{4}{100})^5\end{align}

A. 8
B. 6
C. 4
D. 9
E. 10

Explanation: \begin{align}4^5 = 2^{10}\end{align}, and \begin{align}100^5 = 10^{10}\end{align}, so \begin{align}(\frac{4}{100})^5 = \frac{2^{10}}{10^{10}}\end{align}.

Now, \begin{align}\frac{1}{10^{10}} = 10^{-10} = .000\,000\,000\,1\end{align}.

We’ve put spaces between every 3 zeroes to make them easier to count. As you can see, there are 9 leading zeroes, which is 10 minus the number of digits in the numerator of \begin{align}\frac{1}{10^{10}} \end{align}. There is just 1 digit in this numerator, so the number of leading zeroes is \begin{align}10\,-\, 1 = 9.\end{align}

Therefore, the number of leading zeroes in \begin{align}\frac{2^{10}}{10^{10}}\end{align} depends on the value of \begin{align}2^{10}\end{align}, which is\begin{align}1024.\end{align} Since there are 4 digits in\begin{align}1024,\end{align}then the number of leading zeroes in \begin{align}\frac{2^{10}}{10^{10}}\end{align} is \begin{align}10\,-\, 4 = 6.\end{align}

So B is the correct answer.