which of these numbers can be classified as both real and irrational.
A) - 1.52
B) 8.3232…
C) √49
D) 4.215864…
In the given options, the number that can be classified as both real and irrational is:
C) √49
The square root of 49 is a real number because it is a positive number (assuming the principal square root is taken) and real numbers include all positive and negative numbers.
Additionally, √49 is irrational because it cannot be expressed as a fraction or a terminating or repeating decimal. It is an irrational number because the square root of a non-perfect square is always an irrational number.
To determine which of these numbers can be classified as both real and irrational, let's first understand the definitions of real numbers and irrational numbers:
1. Real numbers: Real numbers are all the numbers on the number line. They include rational numbers (numbers that can be written as fractions) and irrational numbers (numbers that cannot be written as fractions).
2. Irrational numbers: Irrational numbers are numbers that cannot be expressed as a simple fraction or ratio of two integers. They are non-repeating and non-terminating decimals.
Now, let's analyze each option:
A) -1.52: This number is a simple decimal and can be expressed as a fraction (-152/100), so it is a rational number. Therefore, it is not classified as irrational.
B) 8.3232…: This number is a repeating decimal, indicated by the ellipsis (…). Repeating decimals can be written as fractions, so it is rational. Therefore, it is not classified as irrational.
C) √49: The square root of 49 is 7. Since 7 is a whole number and can be expressed as a fraction (7/1), it is rational. Therefore, it is not classified as irrational.
D) 4.215864…: This number is a non-repeating decimal with no discernible pattern. It cannot be expressed as a fraction, so it is irrational. Therefore, it can be classified as both real and irrational.
In conclusion, the only number from the given options that can be classified as both real and irrational is D) 4.215864….