System: You are ChatGPT, a large language model trained by OpenAI.
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User: 182 23 Number systemsINTRODUCTIONA number is a unit of mathematical data used to count, quantify, label and measure. We are used to using standard number systems such as the decimal system and over the next few chapters you will learn about binary and hexadecimal too. In this chapter we will look at how different types or sets of numbers can be used in different ways in computing.SPECIFICATION COVERAGE 3.5.1.1 Natural numbers3.5.1.2 Integer numbers3.5.1.3 Rational numbers3.5.1.4 Irrational numbers3.5.1.5 Real numbers3.5.1.6 Ordinal numbers3.5.1.7 Counting and measuring LEARNING OBJECTIVES In this chapter you will learn: •there are different types of numbers such as natural numbers,inte gers, rational and irrational numbers, real numbers and ordinals•that numbers are used in different ways to produce different outcomes •how an array is used to store numbers•the importance of numbers for counting and measuring. ●Natural numbersThese are the most recognisable type of number as they are the numbers thatwe use every day for counting and ordering. Typically these are numbersmade up of the decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is known asdecimal or base 10 as there are ten different digits that we use. Using asingle-digit number we can represent a maximum value of 9. We simplyadd a new digit, to create the numbers 10–99. We then continue addingdigits to create hundreds, thousands and so on.KEYWORD Natural number: a positive whole number including zero.183RATIONAL NUMBERSEach extra digit we add is worth ten times as much as the previous  digit, as it is base 10. This is an important concept and will help you to understand the binary system in the next chapter. We can use this system to represent an infinite range of numbers, the basic principle being that there are only actually ten different digits on which all the numbers are based. It is a common mistake to view natural numbers as 1 to 10, whereas in fact they are 0 to infinity. The inclusion of 0 as a natural number has implications in computing. For example, if you set a counter when programming, the first instance of the counter will be 0 rather than 1. The mathematical symbol for natural numbers is N or b so you might represent them as follows: b = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}●Integer numbersAn integer  is a whole number whose value can be positive or negative. Zero is a lso classed as an integer. A whole number is one that does notcontain a fractional part, which means there can be no fractions after the number and no decimal place values.
 Negative values are indicated by the minus sign (–). The relationshipbetween the negative and positive integers is that if you add the two you getback to zero. For example –3 + 3 = 0 or 3 + (–3) = 0. Integers are one of the standard data types that can be used when definingvariables and constants in programming languages. In theory, there are aninfinite number of integers that can be represented. In practice, differentlanguages have variations of integer types depending on the size of thenumber that needs to be stored. In Visual Basic for example, the Integer type uses four bytes allowing integers in the range –2 147 483 648 through2 147 483 647 to be created. It has another data type called Long to handle integers that will be outside this range.
 The mathematical symbol for integer numbers is Z or _ so you mightrepresent them as follows: _ = {..., –3, –2, –1, 0, 1, 2, 3, ...} ●Rational numbersA rational number  is one that can be expressed as a fraction, also knownas a quoti ent. Fractions are made up of two integers with the value being the ratio between the two. The ratio can be expressed as a fraction or asits decimal equivalent. For example we know ½ as half, and it can also berepresented as a decimal as 0.5. The top integer is called the numerator and the bottom integer is called thedenominator. The numerator can be any integer, which means that it can beany positive or negative whole number. The value of the fraction thereforecan be negative or positive and it can be greater or less than one, or zero.Fractions can also result in the value of one. For example 55 = 1 or 88 = 1. This means that by definition all integers are rational numbers. The only rule is that the denominator can be any integer apart from zero, as to divide by zero creates an undefined result. For example if you divided KEYWORD Rational number: any number that can be expressed as a fraction or ratio of integers.KEYWORD Integer: any whole positive or negative number including zero.23 Number systems 1843 by 0, the answer is not 0 as it is not possible to define how many times 0 goes into 3. This creates specific problems in computer programming as a program may not be able to continue when it comes across an undefined result. This is called a ‘divide by zero’ error and programmers need to be aware of it and trap it by writing appropriate code.
 The mathematical symbol for rational numbers is Q or a .
 ●Irrational numbersAn irrational number is any number that cannot be represented as a ratiof inte gers because the decimal equivalent would go on forever without repeating. A classic example of an irrational number is π. As a fraction, awidely used approximation is 227. In decimal form, the number is infinite so it has to be truncated (cut off) to a set number of decimal places, forexample to 3.14 or if more accuracy is required, to 3.1415926535.
 Some square roots and cube roots are irrational numbers. A square root can be represented as x = n2. If you start with a simple example, the square root of 16 is 4. If you try to calculate the square root of 2, 3 or 99, you willfind that it is impossible as there is no number that you can square that willexactly make 2, 3 or 99.A feature of an irrational number is that the values after the decimal placedo not repeat in a pattern. For example, there is no pattern to the decimalplaces in π. Recurring decimals do have a pattern and therefore are classedas rational numbers. For example, one-third = 0.333... recurring. The implication in computing is that where irrational numbers need to be handled, the programmer needs to decide on the level of precision that isrequired and therefore how much memory to allocate to storing the value.In simple terms, this might mean defining the number of places to use afterthe decimal point. Computers handle this in a specific way using fixed andfloating point numbers and there is more on this in Chapter 25.
 ●Real numbersA real number  is any positive or negative value and can include a fractional part. Integers, rational numbers and irrational numbers are all real numbers. The defining feature of a real number is that the fractionalpart can be any length, allowing the number to represent a measurement toany level of precision and accuracy required.
 For example on a scale between 1 and 10, real numbers can be used tobreak that down into units of 0.1. On a scale between 0 and 0.1, realnumbers can be used to break that down into units of 0.01 and so on.There is an infinite range of numbers that can be generated. Real numbers are used to measure continuous or infinitely changing values.
 For programmers, the issue is in defining how accurate and precise anumbers needs to be for the application. Consider the following examples: 1 A surveyor working on a road may need to take measurements ranging fromseveral kilometres to several metres, e.g. 10.15  km.2 An Olympic race timer for the 100  m may need to deal with a range from 8 to 15 se conds accurate to a thousandths of a second, e.g. 9.934 seconds.KEYWORDIrrational number: a number that cannot be represented as a fraction or ratio as the decimal form will contain infinite repeating values. KEYWORDReal number: any positive or negativenumber with or without a fractional part.185COUNTING AND MEASUREMENT3 A sensor taking readings of the core temperature in a nuclear power station may need to cope with numbers from 0 to 600 degrees accurate to ﬁve decimal places, e.g. 325.65744.
 As with irrational numbers, computers deal with real numbers using fixed or floating point techniques that are described in detail in Chapter 25. ●Ordinal numbers Ordinal numbers  are those that identify the position of something withina list. For example: first, second and third. They are often used with cardinal numbers , which identify the size of the list. For example, you might say you were third out of 20.
 In computing, ordinal numbers are used to identify the position (location)of data within an ordered set. Consider the following set: S = {’Anne’, ‘Asif’, ‘John’, ‘Mary’, ‘Wanda’, ...}This set is said to be well-ordered  as it has an internal structure that defines the relationship between the data items. In this case, the data is made up of names in ascending alphabetical order. The ordinal numberis one that shows the order of the data so in this case the first item S(1) =Anne, the second item S(2) = Asif, the third item S(3) is John and so on.
 This is a useful technique to locate and manipulate data within certain data structures such as lists, queues and arrays. Some programming languagessupport an ordinal data type, which will contain a value that can becounted and ordered.A one-dimensional array  called Register shows a list of names in a register: Allen Brown Christie Davali Ennis Ordinal numbers are used to assign the data as follows: Register(1) = "Allen" Register(2) = "Brown"A table called Exam  shows the results for each student: Allen Brown Christie Davali Ennis50 75 82 90 45 Ordinal numbers are used to extract the data as follows: Exam(1,2) = 50Exam(2,2) = 75●Counting and measurementAs we have seen, the basic use of numbers is to count and measure. We use different types of numbers in different ways depending on the task. Forexample, we: ●count using natural numbers as we only need to use positive whole numbers●measure using real numbers as the range of numbers may be positive or negative and may require a fractional part.KEYWORDS Ordinal number: a number used to id entify position relative to other numbers. Cardinal number: a number that identi fies the size of something.
 Well-ordered set: a group of rela ted numbers with a defined order.Array: a data structure where data it ems are grouped together under a single identifier and are then accessed based on their position.23 Number systems 186The use of natural numbers to count is common in programming. For example: ●a counter may be used to keep track of how many times a loop statementis rep eated ●the program counter in the processor keeps track of which instructioneeds to be processed next ●a natural number is used to identify the location of data within a data structure●a variable may be set up to keep count, for example, of the number of items in a stock control system, or the score in a computer game. The use of real numbers to measure is common in programming. For example: ●CNC machines handle measurements that vary from millimetres tometres and must work to a high degree of accuracy ●microwave cookers control and measure both time and temperature ●power stations use data control systems to optimise the production ofelectri city●robotics engineers use real-time measurements of the environment in which the robot is working.
 Practice questions can be found at the end of the section on page 228. TASKS1 What are the defining features of: a)natural numbers d)irra tional numbers b)integers e)real  numbers?c)rational numbers2 Why is it important for programmers to distinguish between the different types of numbers?3 Use an example to explain the difference between an ordinal number and a cardinal number.4 Explain why all integers are rational numbers. 5 Why does dividing a number by zero not result in a value of zero?6 What type of numbers would be needed to record the following data? a)the number of runners in a race b)the position of runners as they ﬁnish a racec)the temperature on race day d)the time it took to run the race7 Identify three computer applications that make use of: a)real numbers b)natur al numbers. STUDY / RESEARCH TASKS1 Find out about the irrational numbers called ‘Euler’s number’ and ‘the  Golden Ratio’.2 Apart from 2, 3 and 99, identify some other square roots that are  irrational numbers.3 Find out which number types are supported as data types in a  programming language of your choice.4 Why do programming languages often have several different data  types for an integer?5 The Ancient Greeks did not recognise the number zero. How did their  number systems work without it?KEY POINTS•It is important to understand that there are different typesof numbers. •We use the word decimal torefer to numbers that are base 10, that is, made up ofthe numbers 0 to 9. •Numbers are used to countand tomeasure (or quantify). •Numbers used to countare called ordinals and arean important concept incomputing, for example, whenidentifying the location ofitems within a list. •As a programmer, you need tochoose the right number type when working with data, e.g.whether to work with a numberas an integer or a real number.
User: Make the following text more clear.
Format it properly.
Do NOT change the meaning of the words or methods used to obtain the answer or anything like that.
This is for AQA A(S)-Level Computer Science. I repeat, do not alter the words as it could have an affect on the marks acquired from the marking scheme.
However, remove any redundant information or USELESS examples that are not needed for a student to know, or simplify unnecessary information (if it has a definition or keywords, it is not unnecessary).
If you can, use different values for the examples.
For things that will not score a mark or are generally useless, for example, "we will cover" or a fun fact, change the wording.
Link the 'sections' of your answer to the following spec-points (by title):
e.g. "**Topic** (Spec Points: a.b.c. Whatever)
3.5.1.1 Natural numbers
Content
Additional information
Be familiar with the concept of a natural number and the set ℕ of natural numbers (including zero).
ℕ = {0, 1, 2, 3, … }

3.5.1.2 Integer numbers
Content
Additional information
Be familiar with the concept of an integer and the set ℤ of integers.
ℤ = { …, -3, -2, -1, 0, 1, 2, 3, … }

3.5.1.3 Rational numbers
Content
Additional information
Be familiar with the concept of a rational number and the set ℚ of rational numbers, and that this set includes the integers.
ℚ is the set of numbers that can be written as fractions (ratios of integers). Since a number such as 7 can be written as 7/1, all integers are rational numbers.

3.5.1.4 Irrational numbers
Content
Additional information
Be familiar with the concept of an irrational number.
An irrational number is one that cannot be written as a fraction, for example √2.

3.5.1.5 Real numbers
Content
Additional information
Be familiar with the concept of a real number and the set ℝ of real numbers, which includes the natural numbers, the rational numbers, and the irrational numbers.
ℝ is the set of all 'possible real world quantities'.

3.5.1.6 Ordinal numbers
Content
Additional information
Be familiar with the concept of ordinal numbers and their use to describe the numerical positions of objects.
When objects are placed in order, ordinal numbers are used to tell their position. For example, if we have a well-ordered set S = {‘a’, ‘b’, ‘c’, ‘d’}, then ‘a’ is the 1st object, ‘b’ the 2nd, and so on.

3.5.1.7 Counting and measuring
Content
Additional information
Be familiar with the use of:
natural numbers for counting
real numbers for measurement.


You MUST make sure they are ALL met. Everything it says on there you should meet.

These are the available exam questions you MAY choose to add on to the notes, with 'des' explaining what it's about, and 'fil' meaning the Question ID.
{'S5$.1.a': [{'index1': '|B.B$|', 'index2': '|2$|', 'index3': '2$', 'index4': 'S5$.1.a', 'index5': '|', 'index6': 'Q23S201p2', 'qid': 'Q23S201p2', 'des': 'Natural number; number systems', 'index7': '', 'marks': '1 marks', 'time': '1', 'ttext': 'Number representation', 'fld': 'AA_COM', 'fil': 'Q23S201P2', 'year': 2023, 'res': '', 'flag1': 0}, {'index1': '|B.B$|', 'index2': '|2$|', 'index3': '2$', 'index4': 'S5$.1.a', 'index5': '|', 'index6': 'Q23S201p4', 'qid': 'Q23S201p4', 'des': 'Natural number; number systems', 'index7': '', 'marks': '1 marks', 'time': '1', 'ttext': 'Number representation', 'fld': 'AA_COM', 'fil': 'Q23S201P4', 'year': 2023, 'res': '', 'flag1': 0}], 'S5$.1.b': [], 'S5$.1.c': [], 'S5$.1.d': [{'index1': '|B.B$|', 'index2': '|2$|', 'index3': '2$', 'index4': 'S5$.1.d', 'index5': '|', 'index6': 'Q23S201p1', 'qid': 'Q23S201p1', 'des': 'Irrational numbers; number systems', 'index7': '', 'marks': '1 marks', 'time': '1', 'ttext': 'Number representation', 'fld': 'AA_COM', 'fil': 'Q23S201P1', 'year': 2023, 'res': '', 'flag1': 0}], 'S5$.1.e': [{'index1': '|B.B$|', 'index2': '|2$|', 'index3': '2$', 'index4': 'S5$.1.e', 'index5': '|', 'index6': 'Q23S201p3', 'qid': 'Q23S201p3', 'des': 'Real number; number system', 'index7': '', 'marks': '1 marks', 'time': '1', 'ttext': 'Number representation', 'fld': 'AA_COM', 'fil': 'Q23S201P3', 'year': 2023, 'res': '', 'flag1': 0}], 'S5$.1.f': [{'index1': '|B.B$|', 'index2': '|2$|', 'index3': '2$', 'index4': 'S5$.1.f', 'index5': '|', 'index6': 'Q23S201p5', 'qid': 'Q23S201p5', 'des': 'Ordinal number', 'index7': '', 'marks': '1 marks', 'time': '1', 'ttext': 'Number representation', 'fld': 'AA_COM', 'fil': 'Q23S201P5', 'year': 2023, 'res': '', 'flag1': 0}], 'S5$.1.g': []}

These are the content of said exam questions, delivered in form 'fil', JSON:

{'Q23S201P2': {'question': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Shade in <b>one</b> lozenge to indicate which of the following values is a natural number.</p>\r\n<p class="bottom">&nbsp;</p>\r\n<table class="data-table left1">\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>A</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded-left text-left"><img src="AA_COM/Q23S201P2_files_Q/img01.jpg"/>&nbsp;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P2_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>B</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded-left text-left"><img class="vert-middle vert-up-1" src="AA_COM/Q23S201P2_files_Q/img02.jpg"/>&nbsp;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P2_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>C</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">73</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P2_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>D</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#x2013;19</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P2_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n</table>\r\n<p class="mark">(Total 1 mark)</p></h5>', 'markscheme': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question"><b>Mark is for AO1 (understanding)</b></p>\r\n<p class="indent1new"><b>C</b> (73);</p>\r\n<p class="indent1new"><b>R</b>. More than one lozenge shaded.</p>\r\n<p class="mark">[1]</p></h5>', 'examinerreport': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Most students correctly identified the values from the multiple-choice answers.</p></h5>', 'notes': '', 'resources': ''}, 'Q23S201P4': {'question': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Shade in <b>one</b> lozenge to indicate which of the following symbols represents the set of numbers most suitable for counting the number of people in a room.</p>\r\n<p class="bottom">&nbsp;</p>\r\n<table class="data-table left1">\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>A</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#8469;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P4_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>B</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#8474;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P4_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>C</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#8477;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P4_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>D</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#8484;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P4_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n</table>\r\n<p class="mark">(Total 1 mark)</p></h5>', 'markscheme': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question"><b>Mark is for AO1 (knowledge)</b></p>\r\n<p class="indent1new"><b>A</b> (ℕ)<b>;</b></p>\r\n<p class="indent1new"><b>R</b>. More than one lozenge shaded.</p>\r\n<p class="mark">[1]</p></h5>', 'examinerreport': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Over three-quarters of students correctly identified ℕ (the symbol for the natural numbers) as the most suitable set. The most common incorrect answer was ℤ (the symbol for integers): the set of integers includes negative numbers so is not suitable for counting.</p></h5>', 'notes': '', 'resources': ''}, 'Q23S201P1': {'question': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Shade in <b>one</b> lozenge to indicate which of the following values is an irrational number.</p>\r\n<p class="bottom">&nbsp;</p>\r\n<table class="data-table left1">\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>A</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded-left text-left"><img src="AA_COM/Q23S201P1_files_Q/img01.jpg"/>&nbsp;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P1_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>B</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left"><img class="vert-middle vert-up-1" src="AA_COM/Q23S201P1_files_Q/img02.jpg"/>&nbsp;</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P1_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>C</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">73</p></td>\r\n\t\t<td class="col-3 no-border"><p class="padded"><img src="AA_COM/Q23S201P1_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n\t<tr>\r\n\t\t<td class="col-label no-border"><p class="padded"><b>D</b></p></td>\r\n\t\t<td class="col-6 no-border"><p class="padded text-left">&#x2013;19</p></td>\r\n\t\t<td class="col-3 no-border"><p class="paddede"><img src="AA_COM/Q23S201P1_files_Q/loz.jpg"/>&nbsp;</p></td>\r\n\t</tr>\r\n</table>\r\n<p class="mark">(Total 1 mark)</p></h5>', 'markscheme': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question"><b>Mark is for AO1 (understanding)</b></p>\r\n<p class="indent1new"><b>B</b> (&#x221A;2);</p>\r\n<p class="indent1new"><b>R</b>. More than one lozenge shaded.</p>\r\n<p class="mark">[1]</p></h5>', 'examinerreport': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Most students correctly identified the values from the multiple-choice answers.</p></h5>', 'notes': '', 'resources': ''}, 'Q23S201P3': {'question': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Define the set of real numbers.</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="mark">(Total 1 mark)</p></h5>', 'markscheme': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question"><b>Mark is for AO1 (knowledge)</b></p>\r\n<p class="indent1new">The set of all possible real-world quantities;</p>\r\n<p class="indent1new m-t-0">Includes all rational and irrational numbers;</p>\r\n<p class="indent1new m-t-0">A value that represents any quantity along an infinite number line;</p>\r\n<p class="indent1new m-t-0"><b>A.</b> All numbers excluding imaginary/complex numbers.</p>\r\n<p class="indent1new"><b>MAX 1</b></p>\r\n<p class="mark">[1]</p></h5>', 'examinerreport': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">Only two-thirds of students got the mark for this question. The most common answer was that real numbers included all rational and irrational numbers.</p></h5>', 'notes': '', 'resources': ''}, 'Q23S201P5': {'question': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">What is meant by the term <b>ordinal number</b>?</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="indent1new answer-toggle">_______________________________________________________________________</p>\r\n<p class="mark">(Total 1 mark)</p></h5>', 'markscheme': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question"><b>Mark is for AO1 (knowledge)</b></p>\r\n<p class="indent1new">Ordinal numbers are used to represent/describe the position/index of an object/entity placed in order/sequence;</p>\r\n<p class="indent1new"><b>A.</b> By example (1<sup>st</sup>, 2<sup>nd</sup>, 3<sup>rd</sup>, etc) as long as at least three given.</p>\r\n<p class="mark">[1]</p></h5>', 'examinerreport': '<h5 class=\'contentSeparator\' style=\'page-break-inside: avoid;\'><div class=\'qnumber-hidden\'></div><p class="question">This question was badly answered with only half the students gaining the mark for describing ordinal numbers and almost 8% of students not attempting this question. Incorrect answers gave vague descriptions relating to position, but failed to state that this was where an object/entity placed in the order/sequence.</p></h5>', 'notes': '', 'resources': ''}}

In the markscheme, a ';' typically means that a mark was gained.

If necessary for scoring marks, try to embed the exam-questions into your notes, sometimes without even referencing the fact that it is an exam question. For example, if an exam question is a definition or benefits, use the answer in your notes.
Remember. DO NOT ADD USELESS STUFF. Your task is to create the MOST EFFICIENT (SHORTEST [for an explanation] BUT STILL SCORES ALL MARKS) possible notes resource, which will score all marks easily, so its wording has to be mostly like that of the exam questions. By any means, if necessary, please include examples and exam questions! At the end you can have a 'try these exam questions' thing.
At the end of your answer, you will give a 'breakdown'/'summary' of each thing you covered which will tell the most consice FULLY MARK-SCORING explanation possible.

In summary, your answer must:

* Be efficient, useful
* For topics in an exam that may require some form of mathematical working, always show an example of a question.
* Be clear – provide clear, step-by-step examples/instructions when needed.
* Follow the entire specification and everything it says.
* Link your section to a spec point. E.g. **Topic** (Spec Points: a.b.c Whatever)
* Score ALL marks.
* Include exam questions at the end.
* Remove redundant, unnecessary and useless information that would not be brought up in an exam answer.
* Be helpful to the reader
* Include a summary at the end that goes through everything you need to score all marks.

At the end of the formatted text, after everything is done, fill out the JSON:
Remove the type clarification.
{
"chapter": (int),
"page": (int),
"chapter_name": (str),
"topics": (list)
}

Assistant: