The sum of two rational numbers is always a rational number. Rational numbers are defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero. When you add two rational numbers, the result can also be expressed as a quotient of two integers, thus making the sum rational.
the sum of two rational numbers is always the result of what type of number??
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the sum of a rational and irrational number will result in what type of number?
The sum of a rational number and an irrational number will always result in an irrational number.
To understand why, consider a rational number \( r \) and an irrational number \( x \). If we assume that their sum \( r + x \) is a rational number, we can rearrange this to find \( x \) as follows:
\[ x = (r + x) - r \]
Since \( r + x \) is assumed to be rational and \( r \) is also rational, the difference of two rational numbers \((r + x) - r\) would also be rational. This would imply that \( x \) is rational, which contradicts the assumption that \( x \) is irrational.
Therefore, the sum \( r + x \) must be irrational.
the product of a nonzero rational number and an irrational number will result in what type of number?
The product of a nonzero rational number and an irrational number will always result in an irrational number.
To understand this, let's denote the nonzero rational number as \( r \) (where \( r \neq 0 \)) and the irrational number as \( x \). If we assume that the product \( r \times x \) is rational, we can manipulate the expression as follows:
\[ r \times x = q \]
where \( q \) is a rational number. Then, if we solve for \( x \), we get:
\[ x = \frac{q}{r} \]
Since \( q \) is rational and \( r \) is a nonzero rational number, the division \( \frac{q}{r} \) is also rational. This would mean that \( x \) is rational, which contradicts the assumption that \( x \) is irrational.
Thus, the product \( r \times x \) must be irrational.
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