Which number can be added to a rational number to explain that the sum of two rational numbers is rational?
a: pi
b: 3 7/8
c: square root of 8
d: square root of 47
well, only b is rational ...
To determine the number that can be added to a rational number to ensure that the sum of two rational numbers is still rational, we need to understand the properties of rational numbers.
Rational numbers are numbers that can be expressed as a quotient or fraction of two integers, where the denominator is not equal to zero. In other words, rational numbers can be written in the form a/b, where a and b are integers and b is not equal to zero.
To add two rational numbers, we need to find a common denominator and then simply add the numerators. Since both rational numbers have a denominator of b, the sum of two rational numbers will be (a + c)/b, where a and c are the numerators.
Let's consider the options:
a: π (pi) is an irrational number. Irrational numbers cannot be expressed as a quotient or fraction of two integers, so adding π to a rational number will not guarantee a rational sum.
b: 3 7/8 can be expressed as (31/8) + (7/8) = (31 + 7)/8 = 38/8, which simplifies to 19/4. Since both numbers are rational, the sum is rational. Therefore, adding 3 7/8 to a rational number will indeed result in a rational sum.
c: √8 can be simplified to √(4 * 2), which further simplifies to 2√2. Adding 2√2 to a rational number will result in an irrational sum, as √2 is irrational.
d: √47 is an irrational number. Adding √47 to a rational number will result in an irrational sum.
Therefore, the number that can be added to a rational number to ensure that the sum of two rational numbers is rational is option b: 3 7/8.