Q1: Which of the following is the additive identity for rational numbers? (a) 1 (b) -1 (c) 0 (d) Any rational number
Solution:
Ans: (c) Explanation: Zero is the additive identity because adding zero to any rational number leaves it unchanged, that is, c + 0 = 0 + c = c.
Q2: What is the standard form of the rational number -16/(-24)? (a) -2/-3 (b) 2/3 (c) -2/3 (d) 16/24
Solution:
Ans: (b) Explanation: Dividing both numerator and denominator by their HCF 8 and making denominator positive gives 2/3, which is the standard form.
Q3: Which property states that for any rationals a and b, a + b = b + a? (a) Associativity (b) Closure (c) Commutativity (d) Distributivity
Solution:
Ans: (c) Explanation: The commutativity property states that changing the order of operands does not change the result in addition and multiplication of rational numbers.
Q4: What is the reciprocal of -3/5? (a) 3/5 (b) -3/5 (c) 5/3 (d) -5/3
Solution:
Ans: (d) Explanation: The reciprocal of a rational number p/q is q/p. For -3/5, the reciprocal is -5/3, maintaining the same sign.
Q5: Which of the following numbers is a rational number? (a) Only integers (b) Only fractions (c) Numbers in the form p/q where p and q are integers and q ≠ 0 (d) Only whole numbers
Solution:
Ans: (c) Explanation: A rational number is defined as any number that can be written in the form p/q, where p and q are integers and q is not zero.
Fill in the Blanks
Q1: A rational number is in standard form when the numerator and denominator are _____ and the denominator is positive.
Solution:
Ans: coprime
Q2: The multiplicative identity for rational numbers is _____.
Solution:
Ans: 1
Q3: Two rational numbers whose sum is zero are called _____ of each other.
Solution:
Ans: additive inverses
Q4: Rational numbers are closed under addition, subtraction, multiplication and division except division by _____.
Solution:
Ans: zero
Q5: Between any two distinct rational numbers there are _____ many rational numbers.
Solution:
Ans: infinitely
True or False
Q1: Zero has a reciprocal.
Solution:
Ans: False Explanation: Zero has no reciprocal because there is no rational number that when multiplied by zero results in one.
Q2: A positive rational number is always greater than a negative rational number.
Solution:
Ans: True Explanation: Any positive rational number lies to the right of zero and is always greater than any negative rational number.
Q3: Subtraction of rational numbers is commutative.
Solution:
Ans: False Explanation: Only addition and multiplication of rational numbers are commutative. Subtraction does not satisfy the commutative property for rational numbers.
Q4: The numbers 2/3 and 4/6 are equivalent rational numbers.
Solution:
Ans: True Explanation: Multiplying or dividing both numerator and denominator by the same non-zero integer gives equivalent rational numbers like 2/3 and 4/6.
Q5: A rational number and its reciprocal always have the same sign.
Solution:
Ans: True Explanation: When finding the reciprocal of a rational number, only the numerator and denominator swap positions, so the sign remains the same.
Match the Following
Column A
Column B
1. Numbers used for counting
A. Integers
2. Whole numbers together with their negatives
B. Multiplicative identity
3. Zero is the
C. Natural numbers
4. One is the
D. Reciprocal
5. Multiplicative inverse of a rational number
E. Additive identity
Solution:
Ans:
1 - C: Counting numbers are called natural numbers, represented as 1, 2, 3, 4 and so on in the number system.
2 - A: Integers are defined as whole numbers together with their negatives, such as -3, -2, -1, 0, 1, 2, 3.
3 - E: Zero is the additive identity because adding zero to any rational number leaves it unchanged in value completely.
4 - B: One is the multiplicative identity because multiplying any rational number by one leaves it unchanged in its value.
5 - D: The reciprocal or multiplicative inverse of a rational number p/q is q/p, which gives product 1 when multiplied.
Short Answer Questions
Q1: Define a rational number and explain when it is said to be in standard form.
Solution:
Ans: A rational number is any number that can be written in the form p/q, where p and q are integers and q is not equal to zero. A rational number is in standard form when the numerator and denominator are coprime, meaning their greatest common divisor is 1, and the denominator is positive. To convert to standard form, divide both by their HCF.
Q2: Explain the closure property of rational numbers with examples.
Solution:
Ans: The closure property states that a set is closed under an operation if applying that operation to members of the set always yields a member of the same set. Rational numbers are closed under addition, subtraction, multiplication and division except division by zero. For example, adding two rational numbers like 1/2 and 1/3 gives 5/6, which is also rational.
Q3: How do you add two rational numbers with different denominators? Explain with an example.
Solution:
Ans: To add rational numbers with different denominators, first convert them to equivalent fractions with the same denominator, usually the least common denominator. Then add the numerators and keep the common denominator. For example, to add 1/2 and 1/3, convert to 3/6 and 2/6 respectively, then add to get 5/6 as the final answer.
Q4: What is the difference between additive inverse and multiplicative inverse? Give examples.
Solution:
Ans: The additive inverse of a number x is -x such that x + (-x) equals zero. For example, the additive inverse of 2 is -2. The multiplicative inverse or reciprocal of a rational number p/q is q/p such that their product equals 1. For example, the multiplicative inverse of 2/3 is 3/2, and their product is 1.
Q5: How can you find rational numbers between two given rational numbers?
Solution:
Ans: To find rational numbers between two given rational numbers, make their denominators equal using LCM and list intermediate numerators as fractions. Alternatively, take the mean or average of two numbers repeatedly to obtain more numbers between them. For example, between 1/4 and 1/2, the mean is 3/8, which lies between them, showing infinitely many exist.
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