which of these numbers can be classified as both rational and real?
a. 1/2
b. -1.016879413894
c. square root of 5
d. 0.89089908999
which is both a real number and an integer?
a. square root of 7
b. 0.15
c. -15
d. 1/3
what is an example of an irrational number
a. negative square root of 16
b. square root of 25
c. negative square root of 64
d. negative square root of 101
which statement is false?
a. every integer is a real number
b. every real number is a rational number
c. every positive rational number has a square root
d. every integer is a rational number
To determine which numbers can be classified as both rational and real, we need to understand the definitions of rational and real numbers.
A rational number is any number that can be expressed as a ratio of two integers (where the denominator is not zero). In other words, it can be written as a fraction.
A real number is a number that can be found on the number line. This includes rational numbers, as well as irrational numbers, which cannot be expressed as a fraction.
Let's analyze each question:
1) Which of these numbers can be classified as both rational and real?
a. 1/2: This is a rational number since it can be expressed as a fraction. It is also a real number since it can be found on the number line. Therefore, the answer is yes.
b. -1.016879413894: This is a decimal number and can be expressed as a fraction by putting it over a power of 10. For example, -1.016879413894 = -1016879413894/1000000000000. Hence, it is a rational number. It is also a real number. Therefore, the answer is yes.
c. Square root of 5: The square root of 5 is an irrational number since it cannot be expressed as a fraction. Therefore, it is not a rational number.
d. 0.89089908999: Similar to b, this decimal can be expressed as a fraction. It is a rational number and a real number. Therefore, the answer is yes.
Answer: The numbers a, b, and d can be classified as both rational and real.
2) Which is both a real number and an integer?
a. Square root of 7: The square root of 7 is an irrational number and is not an integer.
b. 0.15: This is a decimal number and can be expressed as a fraction. It is a rational number but not an integer.
c. -15: -15 is an integer since it is a whole number. It is also a real number since it can be found on the number line. Therefore, the answer is yes.
d. 1/3: This is a fraction, so it is a rational number. However, it is not an integer.
Answer: The number c, -15, is both a real number and an integer.
3) What is an example of an irrational number?
a. Negative square root of 16: The square root of 16 is 4, but since it is negative, it becomes -4. This is a rational number, not an irrational number.
b. Square root of 25: The square root of 25 is 5. This is a rational number since it can be expressed as 5/1. Therefore, it is not an irrational number.
c. Negative square root of 64: The square root of 64 is 8, but since it is negative, it becomes -8. This is a rational number, not an irrational number.
d. Negative square root of 101: The square root of 101 is an irrational number. By multiplying it by -1, it remains irrational.
Answer: The number d, negative square root of 101, is an example of an irrational number.
4) Which statement is false?
a. Every integer is a real number: This statement is true. Integers are whole numbers, including negative numbers, which can be found on the number line.
b. Every real number is a rational number: This statement is false. While every rational number is a real number, not every real number is rational. There are irrational numbers that cannot be expressed as fractions.
c. Every positive rational number has a square root: This statement is true. Every positive rational number has a positive square root.
d. Every integer is a rational number: This statement is true. Every integer can be expressed as a fraction with a denominator of 1.
Answer: The statement b, "Every real number is a rational number," is false.