You are given a list of numbers {-6, 6, - 7/6, 2.73773777…, 0, square root of 43}. Which numbers are rational? Irrational?
Which numbers are rational? Select all that apply.
A. 2.73773777…
B. 6
C. Square root of 43
D. - 7/6
E. -6
F. 0
I’m not trying to cheat I just need help explaining how to do this.
Correct Answers:
B: 6
D: - 7/6
E: -6
F: 0
Unless x is a perfect square, √x is irrational
is 43 a perfect square?
well, 6^2 = 36
7^2 = 49
so 43 cannot be a perfect square
any repeating decimal is rational
.737 = 737/999
rational means result of dividing two integers
A. is NOT - it looks like a repeating decimal at first but is not because number of sevens changes
B. 6/1 = 6, Yes
C. No. By the way the square root of any PRIME number fails
D yes, negative is ok
E yes
F yes, but remember you are not allowed to DIVIDE by 0
Any number which can be expressed as a fraction is rational
(notice the "ratio" part in "rational")
This includes all natural and whole numbers, all integers, and all fractions
all terminating decimals, and all decimals have a repeat in the sequence of decimals
e.g. 4.56 , 72.484848...
Irrationals, (not rational) are all those real numbers that cannot be expressed as an exact fraction. This would include all square roots of positive numbers except those square roots of perfect squares
e.g. √5 is irrational, but √36 is rational
so clearly C), √43 is irrational, as is A) 2.73773777... . In the decimal number there will never be an actual repeat of the decimals since we seem to be adding a 7 in each round, so it never repeats.
To determine which numbers in the given list are rational and which ones are irrational, we need to understand the definitions of these terms.
Rational numbers are numbers that can be expressed as a fraction or ratio of two integers. They can be positive, negative, or zero.
Irrational numbers, on the other hand, cannot be expressed as a fraction or ratio of two integers. They are non-repeating and non-terminating decimals. Common examples of irrational numbers include the square root of non-perfect square numbers (such as the square root of 2 or the square root of 7) and some decimal numbers like π (pi) and e (Euler's number).
Now, let's analyze each number in the given list:
A. 2.73773777…
This number appears to be a non-repeating decimal. Unless there is a pattern that repeats after a certain number of decimal places, it is most likely an irrational number. To be certain, we would need to check all decimal places for any repeating pattern.
B. 6
This is an integer, which can also be written as 6/1. Therefore, it is a rational number.
C. Square root of 43
To determine if the square root of 43 is rational or irrational, we need to assess if 43 is a perfect square (a number whose square root is an integer). Since 43 is not a perfect square, we can conclude that its square root is irrational.
D. -7/6
This is a fraction where both the numerator and denominator are integers. Therefore, it is a rational number.
E. -6
Similar to option B, -6 is an integer and can be expressed as -6/1, making it a rational number.
F. 0
Zero can be expressed as the fraction 0/1. Hence, it is a rational number.
To summarize, the rational numbers in the given list are: B (6), D (-7/6), E (-6), and F (0). The irrational numbers are: A (2.73773777…) and C (square root of 43).