A rational number is a number that can be written as the quotient, or fraction, of two integers. An irrational number, on the other hand, is a number that has an infinite number of decimal digits and cannot be written as a ratio of two integers.
Note that there is the exception of repeating decimals. Even though they have an infinite sequence of decimal digits, they are still rational numbers because they can be expressed as a fraction. The picture at right shows examples of rational numbers.
The diagram below shows the subsets of numbers under real numbers. We can see that rational numbers also include all integers, whole number, and natural numbers.
Example 1. Which of the following is a rational number?
Choice c) is a rational number. 3.14 is a terminating decimal, so raising 3.14 to the third power will also result in a terminating decimal, which can then be converted to a ratio of integers.
Choice a) results in an irrational number, namely 0.63246... since 2/5 is not a perfect square. Choice b) is irrational because its digits run on forever. Choice d) is a ratio, but a rational number must be the ratio of two integers, an pi is an irrational number.
When a number is raised to a negative exponent, this indicates the number of times to divide by the base of the exponent, instead of multiply, like we would do with a positive exponent.
Example 2. Simplify the fraction such that the result contains only positive exponents.
The fraction can be simplified by combining exponents. Let's combine exponents by subtracting the denominator exponents from the respective numerator exponents.
Scientific notation is used to represent a number that is too large or small to be written in a convenient decimal form. We can round a number when writing in scientific notation, as long as the decimal itself is between 1 and 10.
Example 3. Convert the following into scientific notation.
We take the first non-zero digit, add a decimal point, and count the places of 10 up to the ones place.
Absolute value describes the magnitude of a real number without its sign. In other words, it is the distance of any number to zero on the number line. Absolute value is represented by two bars ||.
Example 4. Solve the following absolute value equation for x.
We divide the problem into two equations:
x - 5 = 7, so x = 7 + 5 = 12
x - 5 = -7, so x = -7 + 5 = -2
The two solutions to the example above gives us a sense of what the graph for an absolute value equation may be. y = |x| is the parent function. The function can be transformed into y = a|x-b|+c. The vertex, or pivot, is at (b, c).
When c > 0, the graph shifts up by c units, and down when c < 0. When b > 0, the graph shifts to the right by b units, and vice versa. When |a| > 1, the graph is stretched vertically, since a functions as the absolute value of the slope of the rays stemming from the vertex. Thus, when |a| < 1, the graph will look flatter. A negative a flips the graph across the x-axis, so the graph will look like an upside-down "v".