Is this a rational, irrational, or not a real number the square root of 49 and the square root of -81
sqrt49 is Rational.
sqrt(-81) = 9i, Imaginary.
To determine whether numbers are rational, irrational, or not real, we need to understand some concepts.
1. Rational numbers: These are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Rational numbers include integers, fractions, and terminating or repeating decimals.
2. Irrational numbers: These are numbers that cannot be expressed as a fraction of two integers. Irrational numbers have decimal representations that neither terminate nor repeat.
3. Not real numbers: These numbers do not exist on the real number line. Examples include complex numbers, which have both a real and imaginary part.
Now let's apply these concepts to the square roots you mentioned:
1. √49: The square root of 49 is 7 because 7 × 7 = 49. Since 7 is an integer and can be expressed as the fraction 7/1, √49 is a rational number.
2. √-81: The square root of -81 is not a real number. This is because no real number, when squared, can result in a negative value. However, we can define a new number system called complex numbers, where the square root of negative numbers is possible. In the case of √-81, we can write it as 9i, where i represents the imaginary unit (√-1). Thus, √-81 is a complex number, or more specifically, an imaginary number.
In summary:
- √49 is a rational number.
- √-81 is an imaginary number.