From www.pearsonschool.com/IBprograms Ϫ of the rational numbers include __ 1 , 0.23, ___5 and 16. The number 0.23 is 2 2 3 If you know that the German Number and Algebra included in the list because it is equal to ___ 23 , and 16 is included because word for numbers is Zahlen, it Ϫ 100 . is clear why we use the symbol ___16 ___5 Ϫ it is equal to 1 . Note that 3 is equal to 1.66. All integers are therefore ޚ for integers. Assessment statements also included in the rational numbers, as are all decimal numbers where the decimal either terminates or repeats. 2.1 The sets of natural numbers, ގ; integers, ޚ; rational numbers, ޑ; and real numbers, ޒ. For more about integers, visit 2.2.1 Approximation: decimal places; signifi cant fi gures. Percentage errors. www.heinemann.co.uk/ Real numbers hotlinks, enter the express code 2.2.2 Estimation. ϫ k р Ͻ ʦ ޚ The real numbers include all numbers that can be represented by a 4310P and click on weblink 2.2. 2.3.1 Expressing numbers in the form a 10 where 1 a 10 and k . __ 2.3.2 Operations with numbers expressed in the form a ϫ 10k where decimal, including ␲, √5 , e and many more. If you look at the decimal 1 р a Ͻ 10 and k ʦ ޚ. expansions of these numbers, they never repeat, and they never terminate. 2.4 SI (Système International) and other basic units of measurement: for When you write down 3.14 to approximate ␲, you are rounding it off to example, gram (g), metre (m), second (s), litre (l), metre per second Ϫ1 2 decimal places. In general, any square root of a number that is a prime (ms ), Celsius and Fahrenheit scales. ޒ 2.7.5 Solutions of pairs of linear equations in two variables by use of a GDC. number will be irrational. Numbers that belong in but do not belong in 2.7.6 Solutions of quadratic equations: by factorizing; by use of a GDC. ޑ are called irrational numbers (the symbol for which is ޑЈ). For the fi rst 100 digits of ␲, visit www.heinemann.co.uk/ hotlinks, enter the express code Example 2.1 4310P and click on weblink 2.3. Which number groups (ގ, ޚ, ޑ or ޒ) include the following numbers? Artefacts, dating from 30 000 BC, Overview ␲ ___ 0.454 454 445…, __ , √16 , 0.000 000 002 can be found which show 2 notches cut into them similar By the end of this chapter you will be able to: to today’s tally marks. Obviously Solution • explain number concepts, including different number systems these notches were used for For 0.454 454 445…, although it is clear we have a pattern in the digits, no • show how the differences between these number systems affect counting, and are an example group of the digits repeats itself exactly, so this number is not a rational of an early form of written approximations and calculations ␲ number. It belongs in ޒ but not in ޑ. For __ , although this is a ratio of two language. How has the • express numbers in scientifi c notation or standard form 2 numbers, one of the two numbers is not an integer and there is no way development of number • understand and know how to use SI (Système International) units systems aff ected civilization? to simplify the fraction so that it will be an integer. This number would • solve systems of linear and quadratic equations. ___ also be grouped in ޒ. √ 16 is equal to 4, so this number would be most appropriately grouped in ގ and hence ޚ, ޑ and ޒ as well. For a complete defi nition 0.000 000 002 is a rational number as it can be written as ___________ 2 ; of real numbers, visit www. 2.1 Organization of numbers 1 000 000 000 heinemann.co.uk/hotlinks, therefore it should be grouped in ޑ and hence it belongs to ޒ as well. enter the express code 4310P and click on weblink 2.1. 2.1 The sets of natural numbers, ގ; integers, ޚ; rational numbers, ޑ; and real numbers, ޒ. Exercise 2.1 ގ Figure 2.1 The relationship The four main groups of numbers are the natural numbers ( ), the ޚ ޑ ޒ 1. List four examples of a number from each of: between the diff erent types of integers ( ), the rational numbers ( ) and the real numbers ( ). The a) natural numbers b) rational numbers numbers can be represented natural numbers are those numbers you use for counting, such as 0, 1, 2, c) integers d) real numbers visually using a Venn diagram. 3, etc. The integers include the natural numbers, as well as the negative a 2. Find two numbers a and b such that __ ϭ 1.9 which shows that 1.9 is rational. counting numbers, such as Ϫ2, Ϫ1, 0, 1, 2, etc. The rational numbers b include the integers (we will see why later) and all numbers that can be 3. Explain why the number 5.17 is rational. written as a fraction or, more properly, as a ratio of two integers. The real 4. List all of the natural numbers less than 10. numbers include all numbers that can be given a decimal representation. ޒޑޚ ގ 5. Give an example of a rational number between 3.1 and 3.2. Express your a solution in the form __ where a and b are integers. Rational numbers b The rational numbers include any number that can be written in the 6. Using Figure 2.1 as an example, draw a Venn diagram to____ represent. the categories of each of the following numbers: Ϫ4, 4.5, √ 2.25 , 0, 0.1, 1 form __ a where a and b are integers and b is not equal to 0. Some examples b 40 41 2 Number and Algebra From www.pearsonschool.com/IBprograms limited to three digits, or signifi cant fi gures, of accuracy. If we had 7. Look at the following list of numbers: ␲ ___ instead measured the table to the nearest metre and found it to be 2 metres Ϫ2.7, 35, 300 000, __ , √17 , 0 3 long, we could have only one signifi cant fi gure in any answers to further a) Which numbers are rational? calculations involving this length. b) Which numbers are integers? Table 2.1 This table shows c) Which numbers are irrational? Example measurement Number of signifi cant fi gures the relationship between 8. To which sets (ގ, ޚ, ޑ, ޑЈ, ޒ) does 210 belong? measurements and signifi cant 230.4 to nearest 0.1 cm 4 fi gures. 9. Indicate which of the following numbers are irrational: __ . ___ ___ ␲ √3 230 to nearest 1 cm 3 0.1, √19 , √ 64 , __ , 0.121 121 112 111 12…, ___ 3 2 . 3 __ 230 to nearest 10 cm 2 10. A is defi ned to be the numbers {4, Ϫ5, 6.3, 5.5, __ , √3 , 2␲, 0}. 2 a) List the elements in A but not in ޑ. 200 to nearest 100 cm 1 b) List the elements in A but not in ޚ. ގ c) List the elements in A but not in . If we want to fi nd the error in a measurement or calculation, we use the d) List the elements in A and in ޑЈ. Ϫ v______E vA formula percent error ϭ ϫ 100, where v is the exact value of e) Give three examples of numbers in ގ, and ޑ. | vE | E the measurement and vA is the approximate value of the measurement. 11. A is defi ned to be all of the prime numbers less than 20. B is defi ned to be all integers less than 20. Note that the vertical lines in this formula indicate that we must use the C is defi ned to be all numbers less than 10. absolute value of the result of the calculation, which means our answer a) Give an example of a number which is in A, B and C. will be positive. b) List all of the elements of A. At the time of writing, the c) Copy Figure 2.1 and then draw and label circles onto it to represent the sets Example 2.2 A, B and C. world record for the number If the height of a door is measured to be 220 cm, to the nearest centimetre, ␲ of digits of was 1 241 100 fi nd: 000 000 digits, set in 2002 by Yasumasa Kanada of Tokyo a) the minimum possible height of the door University. See b) the maximum possible height of the door www.heinemann.co.uk/ 2.2 Numbers in calculations hotlinks, enter the express code c) the maximum error in this measurement. 4310P and click on weblink 2.4. 2.2.1 Approximation: decimal places; signifi cant fi gures. Percentage Solution errors. a) Since we know the height of the door to the nearest centimetre, the most we could be wrong is 0.5 cm. Therefore, the minimum height is Estimation 219.5 cm. Advances in computer technology have allowed us a high degree of b) Using a similar argument as part a), the maximum height is 220.5 cm. Ϫ Ϫ accuracy in carrying out calculations, but this would not have been _______vE vA _______vE vA c) Percent error ϭ ϫ 100 or percent error ϭ ϫ 100 | vE | | vE | hundreds tens units tenths hundredths thousandths possible without some important mathematical developments earlier in Ϫ Ϫ 203.667 our history.