Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?
[begin{array}{ll} text{Counting numbers} & 1,2,3,4,….. text{Whole numbers} & 0,1,2,3,4,…. text{Integers} & ….−3,−2,−1,0,1,2,3,…. end{array}]
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What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for (frac{1}{3}) is the number (0.overline{3}). The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.
Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.
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Every rational number can be written both as a ratio of integers (frac{p}{q}), where p and q are integers and (q≠0), and as a decimal that stops or repeats.
Are there any decimals that do not stop or repeat? Yes! The number ππ (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.
[π=3.141592654…]
The square root of a number that is not a perfect square is a decimal that does not stop or repeat.
A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.
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Let’s summarize a method we can use to determine whether a number is rational or irrational.
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
Later in this course we will introduce numbers beyond the real numbers. Figure illustrates how the number sets we’ve used so far fit together.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify (−sqrt{25})? Is there a number whose square is (−25)?
[()^2=−25?]
None of the numbers that we have dealt with so far has a square that is (−25). Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to (sqrt{−25}). The square root of a negative number is not a real number.
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Category: WHICH
