// Numbas version: finer_feedback_settings
{"name": "2. Calculation with numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2. Calculation with numbers", "tags": [], "metadata": {"description": "<p>Find reciprocal. Find a modulus. Find a factorial. Part of HELM Book 1.1</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h3>2 Calculation with numbers</h3>\n<!--l. 161\n-->\n<p class=\"noindent\">To perform calculations with numbers we use the <strong> operations </strong> , <!--l. 162\n--> \\(+,\\; -,\\;\\times\\;\\) and <!--l. 163\n--> \\(\\div\\) <!--l. 167\n--></p>\n<h4>2.1 Addition ( <!--l. 167\n--> \\(+\\) )</h4>\n<p class=\"noindent\"><!--l. 169\n--></p>\n<p class=\"noindent\">We say that <!--l. 169\n--> \\(4+5\\) is the <strong> sum </strong> of \\(4\\) and \\(5\\). Note that <!--l. 171\n--> \\(4+5\\) is equal to <!--l. 171\n--> \\(5+4\\) so that the order in which we write down the numbers does not matter when we are adding them. Because the order does not matter, addition is said to be <strong> commutative </strong> . This first property is called <strong> commutativity </strong> .</p>\n<p class=\"noindent\"><!--l. 176\n--></p>\n<p class=\"noindent\">When more than two numbers are to be added, as in <!--l. 177\n--> \\(4+8+9\\) , it makes no difference whether we add the \\(4\\) and \\(8\\) first to get <!--l. 178\n--> \\(12+9\\) , or whether we add the \\(8\\) and \\(9\\) first to get <!--l. 179\n--> \\(4+17\\) . Whichever way we work we will obtain the same result, \\(21\\). Addition is said to be <strong> associative </strong> . This second property is called <strong> associativity </strong> .</p>\n<p class=\"noindent\"><!--l. 182\n--></p>\n<h4>2.2 Subtraction ( <!--l. 182\n--> \\(-\\) )</h4>\n<p class=\"noindent\"><!--l. 184\n--></p>\n<p class=\"noindent\">We say that <!--l. 184\n--> \\(8-3\\) is the <strong> difference </strong> of <!--l. 185\n--> \\(8\\) and <!--l. 185\n--> \\(3\\) . Note that <!--l. 186\n--> \\(8-3\\) is <strong> not </strong> the same as <!--l. 187\n--> \\(3-8\\) and so the order in which we write down the numbers is important when we are subtracting them i.e. subtraction is not commutative. Subtracting a negative number is equivalent to adding a positive number, thus <!--l. 190\n--> \\(7-(-3)=7+3=10\\) .</p>\n<p class=\"noindent\"><!--l. 193\n--></p>\n<h4>2.3 The plus or minus sign ( <!--l. 193\n--> \\(\\pm\\) )</h4>\n<p class=\"noindent\"><!--l. 195\n--></p>\n<p class=\"noindent\">In engineering calculations we often use the notation <strong> plus or minus </strong> , <!--l. 197\n--> \\(\\pm\\) . For example, we write <!--l. 197\n--> \\(12\\pm 8\\) as shorthand for the two numbers <!--l. 198\n--> \\(12+8\\) and <!--l. 198\n--> \\(12-8\\) , that is \\(20\\) and \\(4\\). If we say a number lies in the range <!--l. 199\n--> \\(12\\pm 8\\) we mean that the number can lie between \\(4\\) and \\(20\\) inclusive.</p>\n<p class=\"noindent\"><!--l. 202\n--></p>\n<h4>2.4 Multiplication ( <!--l. 202\n--> \\(\\times\\) )</h4>\n<p class=\"noindent\"><!--l. 204\n--></p>\n<p class=\"noindent\">The instruction to multiply, or obtain the product of, the numbers \\(6\\) and \\(7\\) is written <!--l. 205\n--> \\(6\\times 7\\) . Sometimes the multiplication sign is missed out altogether and we write <!--l. 206\n--> \\((6)(7)\\) .</p>\n<p class=\"noindent\"><!--l. 208\n--></p>\n<p class=\"noindent\">Note that <!--l. 208\n--> \\((6)(7)\\) is the same as <!--l. 208\n--> \\((7)(6)\\) so multiplication of numbers is commutative. If we are multiplying three numbers, as in <!--l. 209\n--> \\(2\\times 3\\times 4\\) , we obtain the same result whether we multiply the \\(2\\) and \\(3\\) first to obtain <!--l. 211\n--> \\(6\\times 4\\) , or whether we multiply the \\(3\\) and \\(4\\) first to obtain <!--l. 211\n--> \\(2\\times 12\\) . Either way the result is \\(24\\). Multiplication of numbers is associative.</p>\n<p class=\"noindent\"><!--l. 214\n--></p>\n<p class=\"noindent\">Recall that when multiplying positive and negative numbers the sign of the result is given by the rules given in Key Point 1.</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h5>Key Point 1</h5>\n<h3>Multiplication</h3>\n<p class=\"noindent\">When multiplying numbers:</p>\n<div style=\"padding-left: 30px;\">\n<ul>\n<li>positive \\(\\times\\) positive \\(=\\) positive</li>\n<li>positive \\(\\times\\) negative \\(=\\) negative</li>\n<li>negative \\(\\times\\) positive \\(=\\) negative</li>\n<li>negative \\(\\times\\) negative \\(=\\) positive</li>\n</ul>\n</div>\n</div>\n<p class=\"noindent\"><!--l. 233\n--></p>\n<p class=\"noindent\">For example, <!--l. 233\n--> \\((-4)\\times 5=-20\\) , and <!--l. 233\n--> \\((-3)\\times (-6)=18\\) .</p>\n<p class=\"noindent\"><!--l. 236\n--></p>\n<p class=\"noindent\">When dealing with fractions we sometimes use the word &lsquo;of&rsquo; as in &lsquo;find <!--l. 237\n--> \\(\\frac12\\) of \\(36\\)&rsquo;. In this context &lsquo;of&rsquo; is equivalent to multiply, that is</p>\n<p class=\"noindent\"><!--l. 238\n--></p>\n<p class=\"noindent\"><!--l. 238\n--> \\(\\frac12\\enspace \\text{of}\\enspace 36\\quad\\text{is equivalent to}\\quad\\frac12\\times 36=18\\)</p>\n<p class=\"noindent\"><!--l. 242\n--></p>\n<h4>2.5 Division ( <!--l. 242\n--> \\(\\div\\) ) or ( <!--l. 242\n--> \\(/\\) )</h4>\n<p class=\"noindent\"><!--l. 244\n--></p>\n<p class=\"noindent\">The quantity <!--l. 244\n--> \\(8\\div 4\\) means <!--l. 245\n--> \\(8\\) divided by <!--l. 245\n--> \\(4\\) . This is also written as <!--l. 245\n--> \\(8/4\\) or <!--l. 245\n--> \\(\\frac84\\) and is known as the <strong> quotient </strong> of <!--l. 246\n--> \\(8\\) and <!--l. 246\n--> \\(4\\) . In the fraction <!--l. 247\n--> \\(\\frac84\\) the top line is called the <strong> numerator </strong> and the bottom line is called the <strong> denominator </strong> . Note that <!--l. 249\n--> \\(8/4\\) is not the same as <!--l. 249\n--> \\(4/8\\) and so the order in which we write down the numbers is important. Division is not commutative.</p>\n<p class=\"noindent\"><!--l. 253\n--></p>\n<p class=\"noindent\">When dividing positive and negative numbers, recall the following rules in Key Point 2 for determining the sign of the result:</p>\n<div class=\"framed\"><!--l. 263\n-->\n<p class=\"noindent\"></p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h5>Key Point 2</h5>\n<h3>Division</h3>\n<p class=\"noindent\">When dividing numbers:</p>\n<div style=\"padding-left: 30px;\">\n<ul>\n<li>\\(\\displaystyle{\\frac{\\text{positive}}{\\text{positive}} = \\text{ positive}}\\)</li>\n<li>\\(\\displaystyle{\\frac{\\text{positive}}{\\text{negative}} = \\text{ negative}}\\)</li>\n<li>\\(\\displaystyle{\\frac{\\text{negative}}{\\text{positive}} = \\text{ negative}}\\)</li>\n<li>\\(\\displaystyle{\\frac{\\text{negative}}{\\text{negative} }= \\text{ positive}}\\)</li>\n</ul>\n</div>\n</div>\n<!--l. 265\n-->\n<h4>2.6 The reciprocal of a number</h4>\n<!--l. 267\n-->\n<p class=\"noindent\">The <strong> reciprocal </strong> of a number is found by inverting it. If the number <!--l. 267\n--> \\(\\frac23\\) is <strong> inverted </strong> we get <!--l. 268\n--> \\(\\frac32\\) . So the reciprocal of <!--l. 269\n--> \\(\\frac23\\) is <!--l. 269\n--> \\(\\frac32\\) . Because we can write \\(4\\) as <!--l. 270\n--> \\(\\frac41\\) , the reciprocal of \\(4\\) is <!--l. 270\n--> \\(\\frac14\\) .</p>\n</div>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "2.6 reciprocal", "definition": "random(2..19)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "2.6 reciprocal", "definition": "random(2..19 except a)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "2.6 reciprocal", "definition": "random(-19..19 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "2.7 modulus", "definition": "random(-20..20)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "2.7 modulus", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "2.7 modulus", "definition": "random(1..20 except f)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "2.7 factorial", "definition": "random(11..13)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "2.7 modulus", "variables": ["d", "f", "g"]}, {"name": "2.6 reciprocal", "variables": ["a", "b", "c"]}, {"name": "2.7 factorial", "variables": ["h"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "2.6 Reciprocal", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the reciprocal of \\(\\frac{\\var{a}}{\\var{b}}\\)?</p>\n<p>Answer: [[0]]</p>\n<p>What is the reciprocal of \\(\\var{c}\\)?</p>\n<p>Answer: [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Fraction reciprocal", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{b}/{a}", "maxValue": "{b}/{a}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Integer reciprocal", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1/{c}", "maxValue": "1/{c}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "2.7 Modulus", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>2.7 The modulus notation ( <!--l. 289\n--> \\(|\\quad |\\) )</h4>\n<!--l. 292\n-->\n<p class=\"noindent\">We shall make frequent use of the modulus notation <!--l. 292\n--> \\(|\\quad |\\) . The <span class=\"ec-lmssbx-10x-x-120\"> modulus </span> of a number is the size of that number regardless of its sign. For example <!--l. 296\n--> \\(|4|\\) is equal to \\(4\\), and <!--l. 296\n--> \\(|-3|\\) is equal to \\(3\\). The modulus of a number is thus never negative.</p>\n<p class=\"noindent\">Give the modulus of the following numbers:</p>\n<p class=\"noindent\">\\(\\left|\\var{d}\\right| =\\)[[0]]</p>\n<p class=\"noindent\">\\(\\left|\\frac{\\var{f}}{\\var{g}}\\right| =\\)[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Integer modulus", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "abs({d})", "maxValue": "abs({d})", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Fraction modulus", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "abs({f}/{g})", "maxValue": "abs({f}/{g})", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "2.8 Factorial", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>2.8 The factorial symbol (!)</h4>\n<!--l. 321\n-->\n<p class=\"noindent\">Another commonly used notation is the <strong> factorial </strong> , denoted by the exclamation mark &lsquo;!&rsquo;. The number <!--l. 323\n--> \\(5!\\) , read &lsquo;five factorial&rsquo;, or &lsquo;factorial five&rsquo;, is a shorthand notation for the expression <!--l. 325\n--> \\(5\\times 4\\times 3\\times 2\\times 1\\) , and the number <!--l. 326\n--> \\(7!\\) is shorthand for <!--l. 326\n--> \\(7\\times 6\\times 5\\times 4\\times 3\\times 2\\times 1\\) . Note that <!--l. 327\n--> \\(1!\\) equals \\(1\\), and by convention <!--l. 328\n--> \\(0!\\) is defined as \\(1\\) also. Your scientific calculator is probably able to evaluate factorials of small integers. It is important to note that factorials only apply to positive integers.</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h5>Key Point 3</h5>\n<h3>Factorial notation</h3>\n<p class=\"noindent\">If \\(n\\) is a positive integer then \\(n!=n\\times (n-1)\\times (n-2)\\times\\dots 5\\times 4\\times 3\\times 2\\times 1\\)</p>\n</div>\n<!--l. 339\n-->\n<h5>Example 1</h5>\n<!--l. 340\n-->\n<ol class=\"enumerate1\" start=\"1\">\n<li class=\"enumerate\" id=\"x5-23002x1\">Evaluate \\(4!\\) and \\(5!\\) without using a calculator.</li>\n<li class=\"enumerate\" id=\"x5-23004x2\">Use your calculator to find <!--l. 342\n--> \\(10!\\).</li>\n</ol>\n<!--l. 353\n-->\n<h5>Solution</h5>\n<!--l. 353\n-->\n<ol class=\"enumerate1\" start=\"1\">\n<li class=\"enumerate\" id=\"x5-24002x1\">\\(4!=4\\times 3\\times 2\\times 1=24\\). Similarly \\(5!=5\\times 4\\times 3\\times 2\\times 1=120\\). Note that \\(5!=5\\times 4!=5\\times 24=120\\).</li>\n<li class=\"enumerate\" id=\"x5-24004x2\"><!--l. 353\n--> \\(10!=3,628,800\\).</li>\n</ol>\n<p>Use your calculator to find \\(\\var{h}!\\)</p>\n<p>\\(\\var{h}!=\\)[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Factorial", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{h}!", "maxValue": "{h}!", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "resources": []}]}], "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}