two new questions about my algebra problem i'm all confused?

Question

true or false questions....
1. the rational numbers are a subset of irrational numbers.

2. if a number is expressed as a decimal, then it is rational.

3.every rational number is a real number.

4. if a number is rational, it cannot be irrational.

Please help me with this and i have more to come i'm just doing a lil at a time

Answer 1

1. False. Example. Rational numbers are numbers that can be expressed as ratios of integers. Suppose SQRT(2) is a rational number then it can be expressed in the form a/b = SQRT(2). This means that 2b^2 = a^2. But it is a fact that an integer value can only be factored into prime numbers in one way meaning that the previous statement can not be possible. The right hand side will have an even number of prime factors, since it is a number squared and that number can have either even or odd numbers of prime factors. The left side though will have an odd number. So this relationship can not be true.

2. False. Some are and some aren't. Pi can be expressed as a decimal but it is not rational. The defintion is that it can be expressed as a ratio of integers. So 5/7 = .717171... Pi can only be approximated by a ratio of integers.

3. Yes this is true. Real numbers are the union of the sets of both rational and irrational numbers.

4. True.

Answer 2

true or false questions....
1. the rational numbers are a subset of irrational numbers.
false


2. if a number is expressed as a decimal, then it is rational.
false

3.every rational number is a real number.
true

4. if a number is rational, it cannot be irrational.
true

Answer 3

1.)t
2.)f
3.)t
4.)t