THE REAL NUMBER SYSTEM

Transcription

1 THE REAL NUMBER SYSTEM Review The real number system is a system that has been developing since the beginning of time. By now you should be very familiar with the following number sets : Natural or counting numbers : 1; ; ; 4; Whole numbers : 0; 1; ; ; 4; Integers : ; ; 1; 0 ; 1; ; ; 1 LESSON Did you know? The three dots ( ) are called ellipses and indicate that there would be more digits to come before or after in the list. And then along came all numbers that could be written as fractions and because fractions are actually ratios the name rational numbers was born. FraCtioNs (Can you see the word ratio hidden inside the word FRACTIONS?) So what exactly then is a rational number? By definition, any number of the form a where a, b are integers and b ¹ 0 is b rational. So let us look at the kind of numbers that fit into this family. 1. All natural, whole and integer numbers are rational. can be written as 1 which fits the definition of a 10 can be written as b 10 which makes it rational 1. All mixed fractions are rational e.g: 1. 1 = = All terminating decimals are rational (they end or have a finite number of decimal places) e.g. : 1. 0,5 = = ,0 = = = Page 1 M911 GRADE 10.indb 1 010/1/1 01:49:0 PM

2 If you type 4,0 into your calculator and hit equals it might change it to 4 4 directly. Ask your teacher to show you if your calculator can 15 change into 504 as well All recurring decimals are rational (there is a repeat pattern identified) e.g.: 1. 0, = 1. 1, = 1 99 = If you type 0, into your calculator and hit equals it converts this recurring decimal into 1 for you. The same holds for 1, being changed to However, you are required to be able to do this conversion manually. So this is how its done. 4.1 Convert 0, to a common fraction. Show all working. Let x = 0, Then 10x =, (Multiply by 10) Now 9x = (Subtract top line from bottom) x = 1 (Divide both sides by 9) 4. Convert 1, to a common fraction. Show all working. Let x = 1, 100 x = 1, (multiply by 100 because we need the part after the decimal comma to be the same) 99x = 11 (subtract top line from bottom x = 11 (divide both sides by 99) Convert 4,04 5 to a common fraction. Show all working. Let x = 4, (multiply by 100 to bring the nonrepeating digits in front of the decimal) 100 x = 40, x = 4045, x = 984 subtract second line from third line x = = the answer you get immediately on your calculator Of course, all rational numbers can be shown on a number line and there are infinitely many of them. What does this mean? Well if we look at the line between integers 1 and 1 a b c Then a = 1 1 = b = 1 4 = 7 4 c = = 15 etc 8 And so we can always place another fraction between any two fractions, and keep on going forever. This means that the rational numbers are densely (tightly) packed on to the number line but there is always room to squeeze in at least one more. A weird thought! Page M911 GRADE 10.indb 010/1/1 01:49:0 PM

3 You would think that we have now mentioned all the numbers that exist, wouldn t you? However, some numbers are not able to be written in the form a b, with a and b integers and b 0 and so they fail the rational test. We call this family of numbers the IRRATIONAL numbers (and yes, they can be thought of as mad!) Example Key the value into your calculator. The screen should show = 1, but in fact = 1, The rest of the digits are not visible/showing on your screen as there is limited space. We have no way of knowing/predicting what the next digits are and so here we have a case of a decimal which does not end (non terminating) and does not re-occur (non recurring) and so this number is not rational, irrational. Here is a quick proof of how we know that is irrational. It is a proof by contradiction. First we suppose that is a rational number. So we can write = a ; a, b are b integers b 0. We also suppose that a is a fraction in simplest form. b Now ( ) = ( a b) square both sides = a b a = b (1) So now a is an even number since it is equal to times something. But = 4; = 9; 4 4 = 16; 5 5 = 5; 7 7 = 49 Can you see that if a is even than a itself has to be even (since (odd) = odd) This means that a = k (can be written as a multiple of ) So now from (1) above = (k) b = 4k b b = 4k b = k This means that b was also an even number. Oops! This is a contradiction! Because we supposed from the start that a is a fraction in simplest form, and b now we get a and b both even. So our original statement that is rational is untrue. It must be irrational. Activity 1 State if these numbers are rational or irrational? Page M911 GRADE 10.indb 010/1/1 01:49:04 PM

4 Another important number which is irrational is p. Remember the ratio of the circumference of a circle to its diameter gives us the quantity p. Circumference = p =, Diameter Rational approximations for p are p = or,14 (rounded to decimals). From 7 now on, all calculations requiring p will need you to use p on your calculator (so hopefully you have a calculator with this button). Now of course, irrational numbers also have to have a position on a number line so we need to find a way to place them. Remember all your working must be shown you can t just type the number into your calculator. Activity In this activity we are going to establish between which two integers any irrational number lies. The first one has been done for you. 1. Now 1 < < 4 ( lies between the square numbers 1 so 1 < < 4 and 4) so 1 < < As before, there are infinitely many irrational numbers. A number like ; 5 ; 10 is also called a SURD. 4 or 100 is not a SURD but it is a rational number. By the Theorem of Pythagoras : A AC = + AC = 1 AC = 1 B C Page 4 M911 GRADE 10.indb 4 010/1/1 01:49:05 PM

5 This is the exact value for the length of AC and we should use 1 on our calculator for any further calculations. If we write 1 as,6 (correct to 1 dec. digit) or as,61 (correct to dec. digits) or as,606 (correct to dec. digits) then we are using a rational approximation for a number which is actually irrational and our final answer will not be exact. Did you follow how 1 was rounded off to a specified number of decimal digits? This is a skill you should have from junior school. You should also be able to give answers correct to the nearest unit, ten or hundred. Activity 1. Given, Write this number 1.1 correct to the nearest unit 1. correct to the nearest ten 1. correct to the nearest hundred 1.4 correct to 1 dec. digit 1.5 correct to dec. digits 1.6 correct to 5 dec. digits. Round 0 to.1 the nearest integer. one decimal digit Page 5 M911 GRADE 10.indb 5 010/1/1 01:49:06 PM

6 . three decimal digits We have now discussed all of the numbers in the real number system. You should now be able to see that this system can be represented in this picture form: REAL NUMBER SYSTEM 7 6 Rationals Integers ,1 4 Whole Natural 1,,, 4, , ,5 Irrationals ,17614 p You may now be asking if any other kinds of numbers exist. The answer is yes. Numbers which are not real are called non-real. Since every point on a line represents a real number, and every real number is a point of the line this means that non-real numbers cannot be located on a number line. Any number of the form a where a < 0 (a is a negative number) is called nonreal. So 10; 16; are all called non-real numbers, and we cannot place them on the real number line. Note: 1. a where a < 0 (a is a negative number) 8 = is real and rational 9 is real but irrational 7 = is real and rational 4. a where a < 0 (a is negative number) 4 16; 4 8; 4 81 are all non-real In general n a is non-real only if a < 0 and n is an even number. You are now ready for the assessment of this section. Page 6 M911 GRADE 10.indb 6 010/1/1 01:49:07 PM

7 Activity 4 1. Classify (this means put into a category) the following real numbers. The first two are done for you is a natural number, a whole number, an integer and a rational number is an integer and a rational number , p 1.7. Show all working:.1 Express 1, 4 as a common fraction. Express 0, 1 + 0,0 as a common fraction. Given p = b 4ac.1 Evaluate p if a = 4, b = 1 and c = 8. State whether p is rational or irrational Page 7 M911 GRADE 10.indb 7 010/1/1 01:49:09 PM

8 . Round p off to decimal digits..4 Show all working to establish between which two integers p lies. 4. Given E = x Give any value for x that would make E a rational number. 4. Give any value for x that would make E an irrational number. 4. Give any value for x that would make E a non-real number. 5.1 Choose and write down any rational number and any irrational number between 6 and 7. Use arrows to show where they would be represented on this number line Now write down a number between 6 and 7 which has a rational square root. 6. If x = ; y = and z = 16 decide whether each of the following algebraic expressions is real and rational, real and irrational or non-real x 6. x 6. x + y z 6.5 z 6.6 z + p Page 8 M911 GRADE 10.indb 8 010/1/1 01:49:10 PM

9 7. Give two different irrational numbers whose product is:. 7.1 rational 7. irrational 7. bigger than both numbers 7.4 bigger than the smaller and smaller than the bigger 7.5 smaller than both numbers Solutions to Activities Activity 1 1. irrational. rational. irrational 4. rational 5. irrational 6. rational 7. irrational 8. rational 9. irrational Activity. Now 4 < 5 < 9 (choose the two numbers either side of 4 < 5 < 9 5 which can easily be square rooted) < 5 <. Now 64 < 69 < < 69 < 81 8 < 69 < 9 so 69 lies between 8 and 9 4. Be careful! 5 < 4 < 16 5 < < 4 < 4 so 4 lies between 5 and Activity 1.1 Look at 10,7 this is 10 7 which is closer to Place your pen on the digit 1 decimal place after the decimal comma. Look to the right. This number is a 0 so we drop it and all the numbers to the right. So,107456,1 1.5 Place your pen on the digit decimals after the decimal comma. Look to the right. This number is a 7. Increase the digit in the second decimal place by 1. Now drop the 7 and all the numbers to the right. So,107456,11 Page 9 M911 GRADE 10.indb 9 010/1/1 01:49:11 PM

10 1. Same story So,107456,1075 The rule is : if the digit to the right is a 5 or larger than 5 (6, 7, 8, 9) then it changes the digit to the left up by = 5, to nearest integer ,5. 0 5,477 Activity 4 1. is rational is natural, whole, integer, rational 1.5 4,7 is rational 1.6 p is irrational 1.7 is irrational.1 Let x = 1, x = 14,444 99x = 1 x = 1 99 = 41 *. 0, 1 + 0, = 0, ,0 = 0,1 Let x = 0,1 10x = 1, 100x = 1, 90x = 1 * Remember the answer only will not x = 1 90 = 15 * get you full marks/credit..1 p = ( 1) 4(4)( 8) = 19. p is irrational. 11, , < 19 < < 19 < < 19 < 1 19 lies between 11 and 1* 4. E = x Try x = 0 E = 4 = this is rational or try x = 4 E = 16 = 4 this is rational or try x = 7 E = 5 = 5 this is rational and many more Page 10 M911 GRADE 10.indb /1/1 01:49:1 PM

11 4. Could give E = 1,,, 5, 6, 7 etc 4. Could give E =,, 4 etc , Rational choose 6 or any other (like 6 ; ; 6,15 ) Irrational choose 40 or any other (like 41 ; 7 ; 8 ) 5. 6 < 5 4 < 7 and 5 4 = 5 which is rational or 6 < < 7 and = 1 which is rational 5 The only way to get these answers is by trial and error. 6.1 irrational 6. rational 6. irrational 6.4 rational 6.5 non-real 6.6 irrational 7.1 Any like 18 = 6 4 = = 7. Any like = 6 5 = Any like = 6 10 = 5 = = 1 now < 1 < etc = now < 5 < 4 or = 1 now 1 < < 4 Page 11 M911 GRADE 10.indb /1/1 01:49:1 PM