Because a is rational, there exist integers m and n such that a=m/n.
Because b is rational, there exist integers p and q such that b=p/q.
Consider a+b. a+b=(m/n)+(p/q)=(mq/nq)+(pn/mq)=(mq+pn)/(nq).
(mq+pn) is an integer because the product of two integers is an integer, and the sum of two integers is an integer. nq is an integer since the product of two integers is an integer. Because a+b equals the quotient of two integers, a+b is rational.
If both numbers are rational then x plus y is a rational number
1) Adding an irrational number and a rational number will always give you an irrational number. 2) Multiplying an irrational number by a non-zero rational number will always give you an irrational number.
Yes, it is.
That simply isn't true. The sum of two irrational numbers CAN BE rational, but it can also be irrational. As an example, the square root of 2 plus the square root of 2 is irrational.
Integers are whole numbers. Rational numbers can be fractions / decimals. But it is NEVER a whole number E.G. of rational numbers : 3/4 or 1.5
Yes. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
If both numbers are rational then x plus y is a rational number
Since the sum of two rational numbers is rational, the answer will be the same as for the sum of an irrational and a single rational number. It is always irrational.
The way in which the binary functions, addition and multiplication, are defined on the set of rational numbers ensures that the set is closed under these two operations.
Yes.A mixed number is a whole number plus a fraction.Fractions are rational, that is of the form c/dWhole numbers are rational - put them over 1 to form a[n improper] fraction a/b, eg 5 = 5/1The sum of two rational numbers is rational.
The rational numbers form a field. In particular, the sum or difference of two rational numbers is rational. (This is easy to check directly). Suppose now that a + b = c, with a rational and c rational. Since b = c - a, it would have to be rational too. Thus you can't ever have a rational plus an irrational equalling a rational.
1) Adding an irrational number and a rational number will always give you an irrational number. 2) Multiplying an irrational number by a non-zero rational number will always give you an irrational number.
Yes, it is.
Yes, the sum is always rational.
That simply isn't true. The sum of two irrational numbers CAN BE rational, but it can also be irrational. As an example, the square root of 2 plus the square root of 2 is irrational.
Integers are whole numbers. Rational numbers can be fractions / decimals. But it is NEVER a whole number E.G. of rational numbers : 3/4 or 1.5
No, never.