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OpenStax-CNX module: m38348 1 Rational Numbers - Grade 10 [CAPS] * Free High School Science Texts Project Based on Rational Numbers by Rory Adams Free High School Science Texts Project Mark Horner Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 1 Introduction As described in Review of past work, a number is a way of representing quantity. The numbers that will be used in high school are all real numbers, but there are many dierent ways of writing any single real number. This chapter describes rational numbers. Khan Academy video on Integers and Rational Numbers This media object is a Flash object. Please view or download it at <http://www.youtube.com/v/kyu-IQ-gBIg&arel=0&hl=en_US&feature=player_embedded&version=3> Figure 1 * Version 1.4: Jun 13, 2011 9:25 pm -0500 http://cnx.org/content/m31331/1.5/ http://creativecommons.org/licenses/by/3.0/ http://cnx.org/content/m38348/1.4/ OpenStax-CNX module: m38348 2 2 The Big Picture of Numbers Figure 2 The term whole number does not have a consistent denition. Various authors use it in many dierent ways. We use the following denitions: • natural numbers are (1, 2, 3, ...) • whole numbers are (0, 1, 2, 3, ...) • integers are (... -3, -2, -1, 0, 1, 2, 3, ....) 3 Denition The following numbers are all rational numbers. 10 , 1 −1 , −3 21 , 7 10 , 20 −3 6 (2) You can see that all denominators and all numerators are integers. Denition 2: Rational Number A rational number is any number which can be written as: a b (2) where a and b are integers and b 6= 0. a as −a Note that because we can write −b b (in other words, one can always nd an equivalent rational expression where b > 0) mathematicians typically dene rational numbers not as both a and b being integers, but rather that a is an integer and b is a natural number. This avoids having to worry about zero in the denominator. tip: Only fractions which have a numerator and a denominator (that is not 0) that are integers are rational numbers. This means that all integers are rational numbers, because they can be written with a denominator of 1. Therefore √ 2 , 7 π 20 (2) are not examples of rational numbers, because in each case, either the numerator or the denominator is not an integer. A number may not be written as an integer divided by another integer, but may still be a rational number. This is because the results may be expressed as an integer divided by an integer. The rule is, if a number can be written as a fraction of integers, it is rational even if it can also be written in another way as well. Here are two examples that might not look like rational numbers at rst glance but are because there are equivalent forms that are expressed as an integer divided by another integer: −1, 33 133 = , −3 300 http://cnx.org/content/m38348/1.4/ −3 −300 −100 = = 6, 39 639 213 (2) OpenStax-CNX module: m38348 3 3.1 Rational Numbers 1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers? a. 56 b. a 3 c. b 2 d. 1c Click here for the solution1 2. If a1 is a rational number, which of the following are valid values for a? a. 1 b. −10 √ c. 2 d. 2, 1 Click here for the solution2 4 Forms of Rational Numbers All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers. 4.1 Investigation : Decimal Numbers You can write the rational number 1. 2. 3. 4. 5. 1 2 as the decimal number 0,5. Write the following numbers as decimals: 1 4 1 10 2 5 1 100 2 3 Do the numbers after the decimal comma end or do they continue? If they continue, is there a repeating pattern to the numbers? You can write a rational number as a decimal number. Two types of decimal numbers can be written as rational numbers: 4 1. decimal numbers that end or terminate, for example the fraction 10 can be written as 0,4. 2. decimal numbers that have a repeating pattern of numbers, for example the fraction 31 can be written as 0, 3̇. The dot represents recurring 3's i.e., 0, 333... = 0, 3̇. For example, the rational number 56 can be written in decimal notation as 0, 83̇ and similarly, the decimal number 0,25 can be written as a rational number as 14 . You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal. tip: 1 http://www.fhsst.org/l35 2 http://www.fhsst.org/l3N http://cnx.org/content/m38348/1.4/ OpenStax-CNX module: m38348 4 5 Converting Terminating Decimals into Rational Numbers A decimal number has an integer part and a fractional part. For example 10, 589 has an integer part of 10 and a fractional part of 0, 589 because 10 + 0, 589 = 10, 589. The fractional part can be written as a rational number, i.e. with a numerator and a denominator that are integers. Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For example: • • 1 10 1 100 is 0, 1 is 0, 01 This means that: 10, 589 = = = 10 + 8 100 589 10 1000 10589 1000 5 10 + + 9 1000 (2) 5.1 Fractions 1. Write the following as fractions: a. 0, 1 b. 0, 12 c. 0, 58 d. 0, 2589 Click here for the solution3 6 Converting Repeating Decimals into Rational Numbers When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example. If we wish to write 0, 3̇ in the form ab (where a and b are integers) then we would proceed as follows x = 0, 33333... 10x = 3, 33333... 9x = 3 3 9 x = = multiply by 10 on both sides (subtracting the second equation from the rst equation) (2) 1 3 And another example would be to write 5, 4̇3̇2̇ as a rational fraction. x = 1000x = 5, 432432432... 5432, 432432432... multiply by 1000 on both sides (2) 999x = x = 5427 5427 999 3 http://www.fhsst.org/l3R http://cnx.org/content/m38348/1.4/ = 201 37 (subtracting the second equation from the rst equation) OpenStax-CNX module: m38348 5 For the rst example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the rst example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring. In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet? The number of zeros is the same as the number of recurring digits. √ Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like 2 = 1, 4142135... cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals. 6.1 Repeated Decimal Notation 1. Write the following using the repeated decimal notation: a. 0, 11111111... b. 0, 1212121212... c. 0, 123123123123... d. 0, 11414541454145... Click here for the solution4 2. Write the following in decimal form, using the repeated decimal notation: a. 23 3 b. 1 11 5 c. 4 6 d. 2 19 Click here for the solution5 3. Write the following decimals in fractional form: a. 0, 6333̇ b. 5, 313131 c. 0, 999999̇ Click here for the solution6 7 Summary Real numbers can be either rational or irrational. A rational number is any number which can be written as The following are rational numbers: a. b. c. d. a b where a and b are integers and b 6= 0 Fractions with both denominator and numerator as integers. Integers. Decimal numbers that end. Decimal numbers that repeat. 4 http://www.fhsst.org/l3U 5 http://www.fhsst.org/l3n 6 http://www.fhsst.org/l3Q http://cnx.org/content/m38348/1.4/ OpenStax-CNX module: m38348 8 End of Chapter Exercises 1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers? a. 56 b. a3 c. 2b d. 1c Click here for the solution7 2. Write each decimal as a simple fraction: a. 0, 5 b. 0, 12 c. 0, 6 d. 1, 59 e. 12, 277̇ Click here for the solution8 3. Show that the decimal 3, 211̇8̇ is a rational number. Click here for the solution9 4. Express 0, 78̇ as a fraction ab where a, b ∈ Z (show all working). Click here for the solution10 7 http://www.fhsst.org/l3v 8 http://www.fhsst.org/l3f 9 http://www.fhsst.org/l3G 10 http://www.fhsst.org/lOf http://cnx.org/content/m38348/1.4/ 6