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Properties of Rational Numbers Quiz - CBSE Class 8 Mathematics

Free practice quiz: Properties of Rational Numbers (CBSE Class 8 Mathematics). 18 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 18 multiple-choice questions on Properties of Rational Numbers, and 18 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Closure of Rational Numbers

  • A rational number has the form p/q, where p and q are integers and q is not zero.
  • Adding, subtracting or multiplying two rational numbers always gives a rational number, so these operations are closed.
  • Division is not closed, because dividing by zero is not defined.
Sum of two rationals
\[\frac{p}{q}+\frac{r}{s}=\frac{ps+qr}{qs},\; q,s\neq 0\]
Product of two rationals
\[\frac{p}{q}\times\frac{r}{s}=\frac{pr}{qs},\; q,s\neq 0\]
Division needs a non-zero divisor
\[\frac{p}{q}\div\frac{r}{s}=\frac{ps}{qr},\; q,r,s\neq 0\]

Common mistake: Saying rationals are closed under division. They are not, because a rational number divided by zero is undefined.

Commutative and Associative Properties

  • Commutative: changing the order of two numbers does not change the sum or product.
  • Associative: changing the grouping of three numbers does not change the sum or product.
  • Neither property holds for subtraction or division.
Commutative (addition)
\[a+b=b+a\]
Commutative (multiplication)
\[a\times b=b\times a\]
Associative (addition)
\[(p+q)+r=p+(q+r)\]
Associative (multiplication)
\[(p\times q)\times r=p\times(q\times r)\]

Common mistake: Thinking a − b = b − a. In fact a − b = −(b − a), so the two are equal only when a = b.

Identities and Inverses

  • 0 is the additive identity and 1 is the multiplicative identity, because each leaves a number unchanged.
  • The additive inverse of a is −a. The multiplicative inverse (reciprocal) of p/q is q/p.
  • Zero has no multiplicative inverse.
Additive identity
\[a+0=0+a=a\]
Multiplicative identity
\[a\times 1=1\times a=a\]
Additive inverse
\[a+(-a)=0\]
Multiplicative inverse
\[\frac{p}{q}\times\frac{q}{p}=1,\; p,q\neq 0\]

Common mistake: Changing the sign when finding a reciprocal. The reciprocal of a negative number stays negative; only the additive inverse changes the sign.

Distributive Property

  • Multiplication distributes over addition and subtraction.
  • Multiply the outside number by every term inside the bracket, then add or subtract the results.
Over addition
\[x\times(y+z)=x\times y+x\times z\]
Over subtraction
\[x\times(y-z)=x\times y-x\times z\]

Common mistake: Multiplying only the first term inside the bracket, for example writing x(y + z) = xy + z.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. Which property is illustrated by the equation (1/2) + (3/4) = (3/4) + (1/2)?

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    Answer: A. Commutative Property

    The commutative property states that the order of numbers in addition does not affect the sum.

  2. If x = 1/3, what is the value of x * 3?

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    Answer: C. 1

    Substitute x with 1/3. (1/3) * 3 = 3/3 = 1 This demonstrates the multiplicative identity property where any number multiplied by its multiplicative inverse equals 1.

  3. If a and b are rational numbers, which property states that a * (b * c) = (a * b) * c ?

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    Answer: B. Associative Property of Multiplication

    The associative property of multiplication states that the grouping of the numbers does not affect the product.

  4. Show that the distributive property holds for rational numbers a = 1/2, b = 1/3, c = 1/4 using an example

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    Answer: C. a(b+c) = ab + ac

    Substitute the values and perform the calculations to show that a(b+c) equals ab + ac.

  5. Provide a counterexample to show that subtraction is not commutative for rational numbers.

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    Answer: D. Counterexample is readily available

    Simply provide an example showing (a-b) ≠ (b-a) for rational numbers a and b (e.g., 1-2 ≠ 2-1).

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. What is the additive inverse of 2/3?
  2. Simplify: (2/5) * (5/2)
  3. Which property is shown in [(1/2) + (1/3)] + (1/4) = (1/2) + [(1/3) + (1/4)]?
  4. Find the multiplicative inverse of -4/7
  5. Explain why 0 is the additive identity for rational numbers.
  6. Prove that the set of rational numbers is closed under addition.
  7. Is the set of rational numbers closed under division? Explain.
  8. Prove that the multiplicative inverse of a non-zero rational number is also a rational number.
  9. Construct a detailed proof demonstrating the associative property of multiplication for rational numbers.
  10. Explain how the properties of rational numbers are crucial for solving equations and simplifying expressions involving fractions.
  11. Discuss the implications of the closure property of rational numbers under addition and multiplication in the context of number systems.
  12. Formulate a proof showing that if a and b are rational numbers, then (a+b)/2 is also a rational number.
  13. Prove or disprove: The set of rational numbers forms a field under the operations of addition and multiplication. Justify your answer.

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