Question

State which of the following statements are true or false. Give reasons for your answer. All fractions are rational numbers All rational numbers are fractions. (in) All rational numbers other than integers are fractions. (iv) Every integer is a rational number. Every rational number is an integer. (vi) Every integer is a whole number. (vii) Every rational number is a whole number.

Answer 1

1. All fractions are rational numbers. True. A rational number is of the form p/q where p & q are integers and q is nonzero.
2. All rational numbers are fractions. False. A value like 7 is rational but not a fraction.
3. All rational numbers other than integers are fractions. True. Refer to question 1 and question 2.
4. Every integer is a rational number. True. Something like 7 is the same as 7/1, showing it is rational. More generally, the integer p is the same as p/1 which shows it is rational.
5. Every rational number is an integer. False. The rational number 2/3 isn't an integer.
6. Every integer is a whole number. False. The set of integers is {...,-3,-2,-1,0,1,2,3,...} while the set of whole numbers is {0,1,2,3,...}; so for instance, -12 is an integer but not a whole number.
7. Every rational number is a whole number. False. 2/3 is rational but not a whole number.

In summary the answers are:

1. True
2. False
3. True
4. True
5. False
6. False
7. False

Answer 2

Fractions are a type of rational number, but not all rational numbers are fractions. Every integer is a rational number and a whole number, but not every rational number is an integer or a whole number.

Step-by-step explanation:

Statement (i) All fractions are rational numbers: True.

A fraction is a number which represents a part of a whole and can be written as a/b, where a is the numerator and b is the denominator. Rational numbers, on the other hand, are numbers that can be expressed as a fraction where the denominator is not zero. Therefore, all fractions are rational numbers because they can be written in the form of a ratio of two integers.

Statement (ii) All rational numbers are fractions: False.

While all fractions are rational numbers, not all rational numbers are fractions. A rational number can be written as a fraction or as a terminating or repeating decimal. For example, 0.5 is a rational number but not a fraction.

Statement (iii) All rational numbers other than integers are fractions: False.

Not all rational numbers that are not integers are fractions. Some rational numbers can be expressed as decimals which are neither terminating nor repeating, such as π (pi) or √2 (the square root of 2).

Statement (iv) Every integer is a rational number: True.

Every integer can be expressed as a fraction where the denominator is 1. For example, 5 can be written as 5/1, which is a fraction.

Statement (v) Every rational number is an integer: False.

Rational numbers can include fractions and decimals that are not integers. For example, 0.5 is a rational number but not an integer.

Statement (vi) Every integer is a whole number: True.

Integers are whole numbers that can be positive, negative, or zero. Therefore, every integer is a whole number.

Statement (vii) Every rational number is a whole number: False.

Rational numbers can include fractions and decimals that are not whole numbers. For example, 0.5 is a rational number but not a whole number.