Real Number System and Classifications

Real Number System and Classifications — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionNumber Line Representation of Real NumbersConcept Practice
2 marks~3 minCriterion A
A carpenter needs to mark a length of 5\sqrt{5} cm on a straight beam. She constructs a right triangle with legs of 2 cm and 1 cm, then uses a compass to transfer the hypotenuse length onto a number line scaled in centimetres.
a
Show that the hypotenuse of the triangle equals 5\sqrt{5} cm. [1]
b
Deduce between which two consecutive integers 5\sqrt{5} lies, and justify whether the carpenter's marked length is closer to the 2 cm or 3 cm graduation. [1]
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2QuestionUnderstanding Integers Fractions and DecimalsAssessment Practice
2 marks~3 minCriterion D
A nutritionist's energy bar recipe requires 34\dfrac{3}{4} cup of oats per batch of 6 bars. A gym supplier orders 10 bars.
a
Show that the adjusted quantity of oats needed for 10 bars is 1.25 cups. [1]
b
Standard measuring cups are marked in fractional increments: 14\dfrac{1}{4}, 13\dfrac{1}{3}, 12\dfrac{1}{2}, and 1 cup. Justify whether 1.25 cups can be measured exactly using these tools, and assess what this means for the reliability of the scaled recipe. [1]
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3QuestionNumber Line Representation of Real NumbersConcept Practice
2 marks~3 minCriterion D
A structural engineer checks whether three cable tension values — −2-\sqrt{2} kN (compression, modelled as negative), 1.51.5 kN, and π\pi kN — fall within a safe operating range of −1.5-1.5 kN to 3.53.5 kN.
a
Construct a number line and plot all three values, using decimal approximations accurate to two decimal places. [1]
b
Using your number line, justify whether all three tension values lie within the safe operating range, supporting your answer with a compound inequality. [1]
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4QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice
2 marks~3 minCriterion C
A school assigns jersey numbers from the set {2,3,4,5,6,7,8,9,10,11}\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}. A sports coordinator uses a Venn diagram with three overlapping circles to classify these numbers into natural numbers NN, prime numbers PP, and even numbers EE.
a
Explain one real-world situation, different from jersey numbers, in which a Venn diagram is used to classify items into overlapping categories. [1]
b
The number 2 belongs to both PP and EE. Justify whether placing 2 in the intersection of PP and EE accurately communicates its full membership in both sets to a student unfamiliar with set notation. [1]
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5QuestionNumber Line Representation of Real NumbersConcept Practice
2 marks~3 minCriterion A
A hiker starts at sea level (00 m), ascends to a peak at +450+450 m, then descends to a valley at −120-120 m.
a
On the number line below, plot the three elevations −120-120 m, 00 m, and +450+450 m. Calculate the total vertical distance traveled by the hiker. [1]
b
The hiker's support team dispatches a rescue vehicle only if the hiker's displacement from sea level exceeds 200200 m. The hiker's final position is −120-120 m. Advise the support team whether the rescue vehicle should be dispatched, justifying your answer using the hiker's displacement. [1]
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6QuestionClassifying Rational and Irrational NumbersAssessment Practice
2 marks~3 minCriterion A
A gardener calculates the exact diagonal of a square garden with side length 4 m as 424\sqrt{2} m. A measuring tape gives a reading of 5.66 m.
a
Explain why 424\sqrt{2} is irrational and 5.66 is rational, and why the measured value can only ever be a rational approximation of the exact diagonal. [1]
b
Justify whether replacing the exact diagonal with the measured approximation would affect the length of fencing needed for the garden's perimeter. [1]
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7QuestionSet Notation and Venn Diagrams for Number SetsConcept Practice
2 marks~3 minCriterion A
The Venn diagram shows the four numbers 2, −5, 13, 22,\ -5,\ \dfrac{1}{3},\ \sqrt{2} placed within the nested sets N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}.

Identify the elements that belong to Q∩Z\mathbb{Q} \cap \mathbb{Z} but not to N\mathbb{N}. Write your answer in set notation. [2]
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8QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice
4 marks~6 minCriterion A
A hospital database stores patient measurements. Five values are recorded:

−3,0.5,2,4,−23-3, \quad 0.5, \quad \sqrt{2}, \quad 4, \quad -\tfrac{2}{3}

The Venn diagram shows the universal set R\mathbb{R}, with nested subsets N⊂Z⊂Q\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}, and the disjoint region Q′\mathbb{Q}'. Each value is placed in its correct region.
a
State the meaning of the set {x∈Z∣x<0}\{x \in \mathbb{Z} \mid x < 0\}. [1]
b
List all elements from the diagram that belong to {x∈Z∣x<0}\{x \in \mathbb{Z} \mid x < 0\}, writing your answer in correct set notation. [1]
c
A database filter accepts only values satisfying {x∈Z∣x<0}\{x \in \mathbb{Z} \mid x < 0\}. Justify whether each recorded value passes or fails the filter, referring to the number classifications shown in the Venn diagram. [2]
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9QuestionUnderstanding Integers Fractions and DecimalsConcept Practice
3 marks~5 minCriterion A
A pharmacist measures three liquid volumes as fractions of a 1-litre bottle. The number line below shows the fill levels A, B, and C.

Point A is at 0.25 litres, point B at 0.5 litres, and point C at 0.75 litres.
a
Deduce the fraction of the bottle represented by point A, expressing your answer in simplest form. [1]
b
Justify why points A and C together fill exactly one complete bottle. [1]
c
The pharmacist states: "Any two of these three volumes can be combined to fill at least three-quarters of the bottle." Assess this claim by calculating all possible two-point combinations. [1]
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10QuestionComparing and Ordering Real NumbersAssessment Practice
4 marks~6 minCriterion D
An engineer is ordering six steel rods for a bridge. Their lengths, in metres, are:

l1=2,l2=π,l3=227,l4=1.414,l5=3.14,l6=3.14159l_1 = \sqrt{2}, \quad l_2 = \pi, \quad l_3 = \tfrac{22}{7}, \quad l_4 = 1.414, \quad l_5 = 3.14, \quad l_6 = 3.14159

The engineer approximates l1≈1.414l_1 \approx 1.414 and l2≈3.14l_2 \approx 3.14 when placing the order.
a
Arrange l1l_1 through l6l_6 in ascending order, showing the decimal values used to justify your ordering. [2]
b
Advise the engineer whether these approximations are sufficiently precise for safe use in bridge construction, referring to specific pairs of values whose ordering is affected and to the structural consequence of any error. [2]
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11QuestionComparing and Ordering Real NumbersConcept Practice
4 marks~6 minCriterion A
A sound engineer checks whether five audio gain ratios all fall within the "safe zone" of a mixing desk, defined as values strictly between 0.50.5 and 1.51.5.

The five ratios are: 35\dfrac{3}{5},  0.72\ 0.72,  2\ \sqrt{2},  π−2\ \pi - 2,  58\ \dfrac{5}{8}
a
Convert each ratio to decimal form, rounding to three decimal places where necessary. [1]
b
List the five ratios in order from least to greatest. [1]
c
The safe zone is the open interval (0.5, 1.5)(0.5,\ 1.5). Advise the engineer whether any gain settings require adjustment, justifying your answer using the decimal values from parts (a) and (b). [2]
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12QuestionUnderstanding Integers Fractions and DecimalsAssessment Practice
4 marks~6 minCriterion D
A civil engineer designs a circular fountain with a radius of 3.53.5 m. To estimate the area of the base, she uses the rational approximation π≈227\pi \approx \dfrac{22}{7}.
a
Calculate the area of the fountain's base using π=227\pi = \dfrac{22}{7}. Give your answer in m2^2. [2]
b
The engineer's supplier sells concrete in whole m2^2 units only. Using π≈3.14159\pi \approx 3.14159, calculate the more precise area, then advise the engineer whether the approximation 227\dfrac{22}{7} is sufficient for placing her concrete order. [2]
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13QuestionClassifying Rational and Irrational NumbersConcept Practice
4 marks~6 minCriterion A
A structural engineer models the natural frequency of a suspension bridge cable using the function f(x)=xf(x) = \sqrt{x}, where xx represents cable tension measured in kilonewtons (kN).

The engineer notes that only rational frequency values allow synchronisation with the bridge's monitoring system.
a
State the definition of a rational number. [1]
b
Explain why f(x)=xf(x) = \sqrt{x} produces an irrational value for most positive values of xx. [1]
c
Identify two values of xx for which f(x)f(x) is rational, and justify whether the engineer can always find a suitable tension value. [2]
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14QuestionNumber Line Representation of Real NumbersAssessment Practice
5 marks~8 minCriterion D
A surveyor models distances between landmarks along a straight road using a number line. Three landmarks are positioned at 00 m, 150150 m, and 320320 m. Each field measurement has a possible error of ±0.5\pm 0.5 m.
a
Show that the distance between the first and second landmarks is 150150 m, and calculate the distance between the first and third landmarks. [1]
b
Determine the range of possible true values for each distance calculated in part (a), given the measurement error. Show your working. [2]
c
Advise the project engineer whether this number line model is suitable for a construction project requiring precision within 0.40.4 m. [2]
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15QuestionUnderstanding Integers Fractions and DecimalsConcept Practice
2 marks~3 minCriterion B
A music streaming platform displays song lengths as fractions of a minute. A sound engineer notices that durations such as 19\dfrac{1}{9} min, 29\dfrac{2}{9} min, 39\dfrac{3}{9} min, and 49\dfrac{4}{9} min convert to 0.1‾0.\overline{1} min, 0.2‾0.\overline{2} min, 0.3‾0.\overline{3} min, and 0.4‾0.\overline{4} min respectively.
a
Deduce a general rule for the decimal expansion of n9\dfrac{n}{9}, where nn is a positive integer with 1≤n≤81 \leq n \leq 8. [1]
b
The platform flags any song duration that cannot be expressed as a terminating decimal. Using your rule, justify whether 79\dfrac{7}{9} min would be flagged, and state what this reveals about fractions with denominator 9. [1]

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16QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice
6 marks~9 minCriterion D
A local library holds 200 books classified into three sets: FF (fiction), NN (non-fiction), and AA (award-winning). The librarian records:

∣F∣=90,∣N∣=70,∣A∣=50|F| = 90, \quad |N| = 70, \quad |A| = 50
∣F∩N∣=30,∣F∩A∣=20,∣N∩A∣=15,∣F∩N∩A∣=10|F \cap N| = 30, \quad |F \cap A| = 20, \quad |N \cap A| = 15, \quad |F \cap N \cap A| = 10
a
Construct a Venn diagram showing all eight regions, including the region outside all three sets. [2]
b
Deduce the number of books belonging to exactly one category, and interpret what this result suggests about the library's classification system. [2]
c
Discuss two limitations of using this set-based model to represent a real library collection, explaining how each limitation reduces the model's usefulness. [2]
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17QuestionComparing and Ordering Real NumbersConcept Practice
2 marks~3 minCriterion B
A landscape architect is comparing the widths of two square garden plots. Plot A has area 5 m25 \text{ m}^2 and Plot B has area 6 m26 \text{ m}^2, so their widths are 5\sqrt{5} m and 6\sqrt{6} m respectively.

The decimal approximations of selected square roots are given below.

xx: 2, 3, 5, 62, \ 3, \ 5, \ 6
x\sqrt{x}: 1.414, 1.732, 2.236, 2.4491.414, \ 1.732, \ 2.236, \ 2.449
a
Analyse the relationship between the order of the values of xx and the order of the corresponding values of x\sqrt{x}. [1]
b
Justify which garden plot has the greater width, without evaluating either square root. [1]

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18QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice
6 marks~9 minCriterion C
A city planner categorises the integers from 1 to 20 as potential plot numbers. The universal set is U={x∈Z+∣1≤x≤20}U = \{x \in \mathbb{Z}^+ \mid 1 \leq x \leq 20\}. Plot numbers divisible by 3 are zoned for residential use (set AA), those divisible by 4 for commercial use (set BB), and those divisible by 5 for green space (set CC).
a
Construct a fully labelled three-set Venn diagram showing all elements of UU placed in their correct regions. [2]
b
Deduce n(A∩B)n(A \cap B) and n(A∪C)n(A \cup C), showing your method for each. [2]
c
Determine n((A∪B)∩C′)n((A \cup B) \cap C') and advise the planner whether the statement "Fewer than half of all residential or commercial plots are free from green-space zoning" is correct, justifying your advice with numerical evidence. [2]
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19QuestionUnderstanding Integers Fractions and DecimalsConcept Practice
2 marks~3 minCriterion A

Consider the numbers 0.75, 34\frac{3}{4}, 0.6, and 25\frac{2}{5}. Describe, using proper mathematical notation and clear language, the relationship between each decimal and its equivalent fraction. Explain how you would convert between the two forms.

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20QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice
6 marks~9 minCriterion C
A data analyst classifies numbers used in a financial model. The universal set ξ\xi contains 12 values:

ξ={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}\xi = \{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12\}

The analyst groups these into three nested sets — natural numbers N\mathbb{N}, integers Z\mathbb{Z}, and rational numbers Q\mathbb{Q} — where N⊂Z⊂Q\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}.
a
Construct a Venn diagram with three nested regions labelled N\mathbb{N}, Z\mathbb{Z}, and Q\mathbb{Q}. Place all 12 elements in the correct region. [2]
b
List the elements of each region:

\quad (i) N∩Z∩Q\mathbb{N} \cap \mathbb{Z} \cap \mathbb{Q}

\quad (ii) (Z∩Q)∖N(\mathbb{Z} \cap \mathbb{Q}) \setminus \mathbb{N}

\quad (iii) Q∖Z\mathbb{Q} \setminus \mathbb{Z} [2]
c
The analyst claims: "Because all values in the model are natural numbers, the regions outside N\mathbb{N} are always empty — so the classification adds no useful information."

Assess this claim, identifying at least one specific type of number that would arise in a realistic financial model and explaining which region it would populate. [2]
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