#set document(title: "10.13 The Real Number System", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 10.13#h(0.6em)The Real Number System === Subsets of the Real Numbers The numbers associated with points on a number line are called the #strong[real numbers]. The set of real numbers is denoted by #math.equation(block: false, alt: "R")[$R$]. You are already familiar with several types, or subsets, of real numbers: - The set #math.equation(block: false, alt: "N")[$N$] of #strong[natural], or #strong[counting numbers], as its name suggests, consists of the numbers #math.equation(block: false, alt: "1 , 2 , 3 , 4 , and so on ,")[$1 , 2 , 3 , 4 , … ,$] where "#math.equation(block: false, alt: "and so on")[$…$]" indicates that the list continues without end. - The set #math.equation(block: false, alt: "W")[$W$] of #strong[whole numbers] consists of the natural numbers and zero: #math.equation(block: false, alt: "0 , 1 , 2 , 3 and so on")[$0 , 1 , 2 , 3 …$]. - The set #math.equation(block: false, alt: "Z")[$Z$] of #strong[integers] consists of the natural numbers, their negatives, and zero: #math.equation(block: false, alt: "and so on , minus 3 , minus 2 , minus 1 , 0 , 1 , 2 , 3 , and so on")[$… , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , …$]. All of these numbers are subsets of the rational numbers. === Rational Numbers A number that can be expressed as the quotient of two integers #math.equation(block: false, alt: "the fraction a over b")[$display(frac(a, b))$] where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], is called a #strong[rational number]. The integers are rational numbers, and so are common fractions. Some examples of rational numbers are #math.equation(block: false, alt: "5 , minus 2 , 0 , the fraction 2 over 9 , the square root of 16 ,")[$5 , − 2 , 0 , display(frac(2, 9)) , sqrt(16) ,$] and #math.equation(block: false, alt: "the fraction minus 4 over 17")[$display(frac(− 4, 17))$]. The set of rational numbers is denoted by #math.equation(block: false, alt: "Q")[$Q$]. Every rational number has a decimal form that either terminates or repeats a pattern of digits. For example, #math.equation(block: true, alt: "the fraction 3 over 4 equals 3 divided by 4 equals 0.75 , a terminating decimal")[$frac(3, 4) = 3 ÷ 4 = 0.75 , " " "a " "terminating decimal"$] and #math.equation(block: true, alt: "the fraction 2 over 37 equals 9 divided by 37 equals 0.243243243 and so on")[$frac(2, 37) = 9 ÷ 37 = 0.243243243 …$] where the pattern of digits #math.equation(block: false, alt: "243")[$243$] is repeated endlessly. We use the #strong[repeater bar] notation to write a repeating decimal fraction: #math.equation(block: true, alt: "the fraction 9 over 37 equals 0.243 bar")[$frac(9, 37) = 0. limits(243)^(―)$] === Irrational Numbers Some real numbers #emph[cannot] be written in the form #math.equation(block: false, alt: "the fraction a over b")[$display(frac(a, b))$] , where #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are integers. For example, the number #math.equation(block: false, alt: "the square root of 2")[$sqrt(2)$] is not equal to any common fraction. Such numbers are called #strong[irrational numbers]. Examples of irrational numbers are #math.equation(block: false, alt: "the square root of 15 , π ,")[$sqrt(15) , π ,$] and #math.equation(block: false, alt: "minus the cube root of 7")[$− root(3, 7)$]. The decimal form of an irrational number never terminates, and its digits do not follow a repeating pattern, so it is impossible to write down an exact decimal equivalent for an irrational number. However, we can obtain decimal #emph[approximations] correct to any desired degree of accuracy by rounding off. A graphing calculator gives the decimal representation of #math.equation(block: false, alt: "π")[$π$] as #math.equation(block: false, alt: "3.141592654")[$3.141592654$]. This is not the #emph[exact] value of #math.equation(block: false, alt: "π")[$π$], but for most calculations it is quite adequate. Some #math.equation(block: false, alt: "n")[$n$]th roots are rational numbers and some are irrational numbers. For example, #math.equation(block: true, alt: "the square root of 49 , the cube root of the fraction 27 over 8 , and 81 to the power 1 / 4")[$sqrt(49) , " " " " " " root(3, frac(27, 8)) , " " " " " and " " " " " 81^(1 / 4)$] are rational numbers because they are equal to #math.equation(block: false, alt: "7 , the fraction 3 over 2 ,")[$7 , display(frac(3, 2)) ,$] and #math.equation(block: false, alt: "3")[$3$], respectively. On the other hand, #math.equation(block: true, alt: "the square root of 5 , the cube root of 54 , and 7 to the power 1 / 5")[$sqrt(5) , " " " " " " root(3, 54) , " " " " " and " " " " " 7^(1 / 5)$] are irrational numbers. We can use a calculator to obtain decimal approximations for each of these numbers: #math.equation(block: true, alt: "the square root of 5 approximately equals 2.236 , the cube root of 54 approximately equals 3.826 , and 7 to the power 1 / 5 approximately equals 1.476")[$sqrt(5) ≈ 2.236 , " " " " " " root(3, 54) ≈ 3.826 , " " " " " " " and " " " " " " " 7^(1 / 5) ≈ 1.476$] The subsets of the real numbers are related as shown in Figure. Every natural number is also a whole number, every whole number is an integer, every integer is a rational number, and every rational number is real. Also, every real number is either rational or irrational. #figure(figph[real numbers], alt: "real numbers", caption: none) #examplebox("Example 1")[][ + #math.equation(block: false, alt: "2")[$2$] is a natural number, a whole number, an integer, a rational number, and a real number. + #math.equation(block: false, alt: "the square root of 15")[$sqrt(15)$] is an irrational number and a real number. + The number #math.equation(block: false, alt: "π")[$π$], whose decimal representation begins #math.equation(block: false, alt: "3.14159 and so on")[$3.14159 …$] is irrational and real. + #math.equation(block: false, alt: "3.14159")[$3.14159$] is a rational and real number (which is close but not exactly equal to #math.equation(block: false, alt: "π")[$π$]). ] === Properties of the Real Numbers The real numbers have several useful properties governing the operations of addition and multiplication. If #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$] represent real numbers, then each of the following equations is true: - #math.equation(block: false, alt: "a plus b equals b plus a, Commutative properties; a b equals b a")[$& a + b = b + a & & #hide($b l a n k b l a n k$) "Commutative properties" \ & a b = b a$] - #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis, Associative properties; open parenthesis a b close parenthesis equals a open parenthesis b c close parenthesis")[$& ( a + b ) + c = a + ( b + c ) & & "Associative properties" \ & ( a b ) = a ( b c )$] - #math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c, Distributive property")[$& a ( b + c ) = a b + a c & & #hide($b l a n k 0$) "Distributive property"$] - #math.equation(block: false, alt: "a plus 0 equals a, Identity properties; a times 1 equals a")[$& a + 0 = a & & #hide($b l a n k b l a n k 0000$) "Identity properties" \ & a ⋅ 1 = a$] These properties do not mention subtraction or division. But we can define #emph[subtraction] and #emph[division] in terms of addition and multiplication. For example, we can define the difference #math.equation(block: false, alt: "a minus b")[$a − b$] as follows: #math.equation(block: true, alt: "a minus b equals a plus open parenthesis minus b close parenthesis")[$a − b = a + ( − b )$] where #math.equation(block: false, alt: "minus b")[$− b$], the #strong[additive inverse] (or #strong[opposite]) of #math.equation(block: false, alt: "b")[$b$], is the number that satisfies #math.equation(block: true, alt: "b plus open parenthesis minus b close parenthesis equals 0")[$b + ( − b ) = 0$] Similarly, we can define the quotient #math.equation(block: false, alt: "the fraction a over b")[$display(frac(a, b))$]: #math.equation(block: true, alt: "the fraction a over b equals a open parenthesis the fraction 1 over b close parenthesis open parenthesis b not equal to 0 close parenthesis")[$frac(a, b) = a ( frac(1, b) ) #hide($b l a n k$) ( b ≠ 0 )$] where #math.equation(block: false, alt: "the fraction 1 over b")[$display(frac(1, b))$], the #strong[multiplicative inverse] (or #strong[reciprocal]) of #math.equation(block: false, alt: "b")[$b$], is the number that satisfies #math.equation(block: true, alt: "b times the fraction 1 over b equals 1 open parenthesis b not equal to 0 close parenthesis")[$b ⋅ frac(1, b) = 1 #hide($b l a n k$) ( b ≠ 0 )$] Division by zero is not defined. #examplebox("Example 2")[][ Use the commutative and associative laws to simplify the computations. + #math.equation(block: false, alt: "24 plus 18 plus 6")[$24 + 18 + 6$] + #math.equation(block: false, alt: "4 times 27 times 25")[$4 ⋅ 27 ⋅ 25$] #solutionbox[ + Apply the commutative law of addition. #math.equation(block: true, alt: "24 plus 18 plus 6, equals open parenthesis 24 plus 6 close parenthesis plus 18; equals 30 plus 18 equals 48")[$24 + 18 + 6 & = ( 24 + 6 ) + 18 \ & = 30 + 18 = 48$] + Apply the commutative law of multiplication. #math.equation(block: true, alt: "4 times 27 times 25, equals open parenthesis 4 times 25 close parenthesis times 27; equals 100 times 27 equals 2700")[$4 ⋅ 27 ⋅ 25 & = ( 4 ⋅ 25 ) ⋅ 27 \ & = 100 ⋅ 27 = 2700$] ] ] === Order Properties of the Real Numbers Real numbers obey properties about order, that is, properties about inequalities. The familiar inequality symbols, #math.equation(block: false, alt: "less than")[$<$] and #math.equation(block: false, alt: "greater than")[$>$], have the following properties: - If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are any real numbers, then one of three things is true: #math.equation(block: true, alt: "a less than b , or a greater than b , or a equals b")[$a < b , " " " " " or " " " " " a > b , " " " " " or " " " " " a = b$] - (Transitive property) For real numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$], #math.equation(block: true, alt: "if a less than b and b less than c , then a less than c")[$"if " a < b " " " and " " " b < c , " " " then " " " a < c$] We also have three properties that are useful for solving inequalities: - If #math.equation(block: false, alt: "a less than b")[$a < b$], then #math.equation(block: false, alt: "a plus c less than b plus c")[$a + c < b + c$]. - If #math.equation(block: false, alt: "a less than b")[$a < b$] and #math.equation(block: false, alt: "c greater than 0")[$c > 0$], then #math.equation(block: false, alt: "a c less than b c")[$a c < b c$]. - If #math.equation(block: false, alt: "a less than b")[$a < b$] and #math.equation(block: false, alt: "c less than 0")[$c < 0$], then #math.equation(block: false, alt: "a c greater than b c")[$a c > b c$]. #examplebox("Example 3")[][ + If #math.equation(block: false, alt: "x less than y")[$x < y$] and #math.equation(block: false, alt: "y less than minus 2")[$y < − 2$], then #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$] + #math.equation(block: false, alt: "π less than 3.1416")[$π < 3.1416$], so #math.equation(block: false, alt: "10 π less than 31.416")[$10 π < 31.416$]. + #math.equation(block: false, alt: "the fraction 1 over 3 greater than 0.33")[$display(frac(1, 3)) > 0.33$], so #math.equation(block: false, alt: "minus the fraction 1 over 3 less than minus 0.33.")[$− display(frac(1, 3)) < − 0.33 .$] ] === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Real number - Multiplicative inverse - Additive inverse - Distributive property - Whole number - Natural number - Reciprocal - Opposite - Irrational number - Integers - Counting number - Transitive property - Commutative property - Terminating decimal - Rational number - Identity property - Associative property - Repeater bar ==== SKILLS Practice each skill in the exercises listed. + Identify types of numbers: \#1–12 + Write the decimal form of a fraction: \#13–20 + Use the properties governing arithmetic operations: \#21–40 + Use the properties of order: \#41–46 === Exercises A.13 For Problems 1-12, name the subsets of the real numbers to which the number belongs #math.equation(block: true, alt: "minus the fraction 5 over 8")[$− display(frac(5, 8))$] Rationals #math.equation(block: true, alt: "137")[$137$] #math.equation(block: true, alt: "the square root of 8")[$sqrt(8)$] Irrationals #math.equation(block: true, alt: "2.71828 and so on")[$2.71828 …$] #math.equation(block: true, alt: "minus 36")[$− 36$] Integers #math.equation(block: true, alt: "the square root of 49")[$sqrt(49)$] #math.equation(block: true, alt: "0")[$0$] Whole numbers #math.equation(block: true, alt: "0.0357 bar")[$0.0 limits(357)^(―)$] #math.equation(block: true, alt: "13289 bar")[$13 limits(289)^(―)$] Whole numbers #math.equation(block: true, alt: "the square root of the fraction 4 over 9")[$sqrt(display(frac(4, 9)))$] #math.equation(block: true, alt: "2 π")[$2 π$] Irrationals #math.equation(block: true, alt: "the fraction 13 over 7")[$display(frac(13, 7))$] For Problems 13-20, write the rational number in decimal form. Does the decimal terminate or does it repeat a pattern? #math.equation(block: true, alt: "the fraction 3 over 8")[$display(frac(3, 8))$] #math.equation(block: false, alt: "0.375")[$0.375$], terminates #math.equation(block: true, alt: "the fraction 5 over 6")[$display(frac(5, 6))$] #math.equation(block: true, alt: "the fraction 2 over 7")[$display(frac(2, 7))$] #math.equation(block: false, alt: "0.285714 bar")[$0. limits(285714)^(―)$], repeats a pattern #math.equation(block: true, alt: "the fraction 43 over 11")[$display(frac(43, 11))$] #math.equation(block: true, alt: "the fraction 7 over 16")[$display(frac(7, 16))$] #math.equation(block: false, alt: "0.4375")[$0.4375$], terminates #math.equation(block: true, alt: "the fraction 5 over 12")[$display(frac(5, 12))$] #math.equation(block: true, alt: "the fraction 11 over 13")[$display(frac(11, 13))$] #math.equation(block: false, alt: "0.846153 bar")[$0. limits(846153)^(―)$], repeats a pattern #math.equation(block: true, alt: "the fraction 25 over 6")[$display(frac(25, 6))$] For Problems 21-30, fill in the blank according to the indicated property. Commutative property #math.equation(block: true, alt: "7 plus 10 equals 10 plus bar")[$7 + 10 = 10 + underline(#h(3em))$] #math.equation(block: true, alt: "7")[$7$] Associative property #math.equation(block: true, alt: "open parenthesis 6 times 4 close parenthesis times 3 equals 6 times open parenthesis 4 times bar close parenthesis")[$( 6 ⋅ 4 ) ⋅ 3 = 6 ⋅ ( 4 ⋅ underline(#h(3em)) )$] Associative property #math.equation(block: true, alt: "open parenthesis 3 plus 6 close parenthesis plus 9 equals bar plus open parenthesis 6 plus 9 close parenthesis")[$( 3 + 6 ) + 9 = underline(#h(3em)) + ( 6 + 9 )$] #math.equation(block: true, alt: "3")[$3$] Commutative property #math.equation(block: true, alt: "open parenthesis 8 times 12 close parenthesis equals bar times 8")[$( 8 ⋅ 12 ) = underline(#h(3em)) ⋅ 8$] Commutative property #math.equation(block: true, alt: "36 times 147 equals bar times 36")[$36 ⋅ 147 = underline(#h(3em)) ⋅ 36$] #math.equation(block: true, alt: "147")[$147$] Commutative property #math.equation(block: true, alt: "13 plus 87 equals 87 plus bar")[$13 + 87 = 87 + underline(#h(3em))$] Associative property #math.equation(block: true, alt: "open parenthesis 17 times 2 close parenthesis times 5 equals 17 times open parenthesis bar times bar close parenthesis")[$( 17 ⋅ 2 ) ⋅ 5 = 17 ⋅ ( underline(#h(3em)) ⋅ underline(#h(3em)) )$] #math.equation(block: true, alt: "2 times 5")[$2 ⋅ 5$] Associative property #math.equation(block: true, alt: "open parenthesis 44 plus 12 close parenthesis plus 8 equals 44 plus open parenthesis bar plus bar close parenthesis")[$( 44 + 12 ) + 8 = 44 + ( underline(#h(2em)) + underline(#h(2em)) )$] Commutative property #math.equation(block: true, alt: "open parenthesis 5 plus 9 close parenthesis plus 4 equals open parenthesis 9 plus bar close parenthesis plus 4")[$( 5 + 9 ) + 4 = ( 9 + underline(#h(3em)) ) + 4$] #math.equation(block: true, alt: "5")[$5$] Commutative property #math.equation(block: true, alt: "open parenthesis 8 times 9 close parenthesis times 3 equals open parenthesis 9 times bar close parenthesis times 3")[$( 8 ⋅ 9 ) ⋅ 3 = ( 9 ⋅ underline(#h(3em)) ) ⋅ 3$] For Problems 31-40, use the commutative and associative properties to compute mentally. #math.equation(block: true, alt: "47 plus 28 plus 3")[$47 + 28 + 3$] #math.equation(block: true, alt: "78")[$78$] #math.equation(block: true, alt: "12 plus 147 plus 8")[$12 + 147 + 8$] #math.equation(block: true, alt: "26 plus 37 plus 3 plus 4")[$26 + 37 + 3 + 4$] #math.equation(block: true, alt: "70")[$70$] #math.equation(block: true, alt: "55 plus 32 plus 5 plus 8")[$55 + 32 + 5 + 8$] #math.equation(block: true, alt: "2 times 7 times 5")[$2 ⋅ 7 ⋅ 5$] #math.equation(block: true, alt: "70")[$70$] #math.equation(block: true, alt: "15 times 6 times 2")[$15 ⋅ 6 ⋅ 2$] #math.equation(block: true, alt: "50 times 13 times 2")[$50 ⋅ 13 ⋅ 2$] #math.equation(block: true, alt: "1300")[$1300$] #math.equation(block: true, alt: "4 times 26 times 25")[$4 ⋅ 26 ⋅ 25$] #math.equation(block: true, alt: "4 times 6 times 5 times 5")[$4 ⋅ 6 ⋅ 5 ⋅ 5$] #math.equation(block: true, alt: "600")[$600$] #math.equation(block: true, alt: "8 times 8 times 5 times 5")[$8 ⋅ 8 ⋅ 5 ⋅ 5$] For Problems 41-46, fill in the blank with the correct symbol: \<, \>, or #math.equation(block: false, alt: "equals")[$=$]. #math.equation(block: true, alt: "minus 0.667 bar minus the fraction 2 over 3")[$− 0.667 underline(#h(3em)) " " − display(frac(2, 3))$] #math.equation(block: true, alt: "less than")[$<$] #math.equation(block: true, alt: "the square root of 2 bar 1.4")[$sqrt(2) underline(#h(3em)) " " 1.4$] If #math.equation(block: false, alt: "x greater than 8")[$x > 8$], then #math.equation(block: false, alt: "x minus 7 bar 1.")[$x − 7 underline(#h(2em)) " " 1.$] #math.equation(block: true, alt: "greater than")[$>$] If #math.equation(block: false, alt: "x less than minus 6")[$x < − 6$], then #math.equation(block: false, alt: "x minus 6 bar minus 12.")[$x − 6 underline(#h(2em)) " " − 12 .$] If #math.equation(block: false, alt: "x greater than minus 2")[$x > − 2$], then #math.equation(block: false, alt: "minus 9 x bar 18.")[$− 9 x underline(#h(2em)) " " 18.$] #math.equation(block: true, alt: "less than")[$<$] If #math.equation(block: false, alt: "x less than minus 4")[$x < − 4$], then #math.equation(block: false, alt: "3 x bar minus 12.")[$3 x underline(#h(2em)) " " − 12 .$]