Real Number System and Classifications

Real Number System and Classifications — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionSet 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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2QuestionNumber Line Representation of Real NumbersAssessment 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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3QuestionNumber 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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4QuestionUnderstanding 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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5QuestionNumber Line Representation of Real NumbersConcept Practice
2 marks~3 minCriterion A
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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6QuestionSet 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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7QuestionUnderstanding 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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8QuestionClassifying Rational and Irrational NumbersAssessment Practice
2 marks~3 minCriterion C
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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9QuestionComparing 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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10QuestionUnderstanding 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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11QuestionClassifying Rational and Irrational NumbersConcept Practice
2 marks~3 minCriterion A
A manufacturing process requires a cutting tool to move exactly π\pi metres along a curved track. The control software can only accept terminating or repeating decimal inputs, so an engineer enters 3.14 metres instead.
a
Justify whether 3.14 is a rational or irrational number. [1]
b
Advise the engineer whether the input of 3.14 metres is mathematically exact for the required distance, referring to the classification of π\pi in your response. [1]

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12QuestionSet 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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13QuestionClassifying Rational and Irrational NumbersConcept Practice
2 marks~3 minCriterion C
A square tile has an area of 144 cm2144 \text{ cm}^2. An architect states that the side length of this tile is a rational number.
a
State the definition of a rational number. [1]
b
Determine the side length of the tile and justify whether the architect's statement is correct. [1]

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14QuestionClassifying Rational and Irrational NumbersAssessment Practice
4 marks~6 minCriterion B
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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15QuestionSet Notation and Venn Diagrams for Number SetsConcept Practice
2 marks~3 minCriterion A
A temperature monitoring system flags integer temperatures (in °C) within the set

A={x∈Z∣−3≤x<2}.A = \{ x \in \mathbb{Z} \mid -3 \leq x < 2 \}.
a
List all elements of AA. [1]
b
A sensor records readings of −2 °C-2\text{ °C} and 2 °C2\text{ °C}. Deduce whether each reading would be flagged by the system, and justify what the boundary condition x<2x < 2 tells you about which temperatures are included in the alert range. [1]

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16QuestionComparing 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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17QuestionNumber Line Representation of Real NumbersConcept Practice
2 marks~3 minCriterion A
A quality-control engineer checks measurements on a production line. Two recorded values are 0.3750.375 mm and 2\sqrt{2} mm.

State which of these values can be expressed as a terminating or repeating decimal, and justify your answer by explaining the property of real numbers that determines this. [2]

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18QuestionNumber 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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19QuestionUnderstanding Integers Fractions and DecimalsConcept Practice
2 marks~3 minCriterion A
A recipe requires 0.3750.375 of a cup of olive oil per serving.
a
Show that 0.375=380.375 = \dfrac{3}{8}. [1]
b
A cook has exactly 13\dfrac{1}{3} of a cup of olive oil remaining. Justify whether this is sufficient for one serving, using fractions. [1]

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20QuestionClassifying Rational and Irrational NumbersAssessment Practice
6 marks~9 minCriterion C
A music producer stores a 3-minute song as a WAV (lossless) file of 30 MB and as an MP3 (lossy) file of 3 MB.
a
Calculate the compression ratio MP3 sizeWAV size\dfrac{\text{MP3 size}}{\text{WAV size}} and express your answer as a simplified fraction. [2]
b
A second algorithm targets a compression ratio of 110\dfrac{1}{\sqrt{10}}. Explain why applying this ratio to the 30 MB WAV file produces a compressed file size that is irrational. [2]
c
The producer must choose one format to archive all future recordings. Using the compression ratios from parts (a) and (b), advise the producer which format to select for long-term archiving, and explain what is permanently lost by choosing the smaller file size. [2]
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