Let x and y be non-zero, rational numbers and let z be an irrational number. In the first column of the table below is an expression. In the second column, determine whether the value of each expression is rational or irrational
Expression
x + y
y + z
xy
xz
Value of expression
Rational or irrational
Expression Value of expression Rational or irrational
x + y depends on the values of x and y
y + z irrational
xy depends on the values of x and y
xz irrational
To determine whether the value of each expression is rational or irrational, we need to consider the properties of rational and irrational numbers.
1. x + y:
Since x and y are non-zero rational numbers, their sum (x + y) will also be a rational number. Therefore, the value of this expression is rational.
2. y + z:
Here, y is a non-zero rational number and z is an irrational number. When we add a rational number to an irrational number, the result will always be irrational. Therefore, the value of this expression is irrational.
3. xy:
When we multiply two rational numbers, the result is always a rational number. Therefore, the value of this expression is rational.
4. xz:
In this expression, x is a non-zero rational number and z is an irrational number. When we multiply a rational number by an irrational number, the result will always be irrational. Therefore, the value of this expression is irrational.
In summary:
- x + y: Rational
- y + z: Irrational
- xy: Rational
- xz: Irrational
To determine whether the value of each expression is rational or irrational, we need to understand the properties of rational and irrational numbers:
- Rational numbers can be expressed as a fraction (a/b), where a and b are integers and b is not equal to zero.
- Irrational numbers cannot be expressed as a fraction and have non-terminating, non-repeating decimal representations.
Let's go through each expression in the table:
Expression: x + y
To determine whether the sum of two rational numbers is rational or irrational, we look at the sum of the fractions. Since x and y are both rational numbers, their sum x + y can also be expressed as a fraction. Therefore, the value of this expression is rational.
Value of expression: Rational
Expression: y + z
Since y is rational and z is irrational, the sum y + z will be a combination of a rational number and an irrational number. When adding a rational and an irrational number, the result is always an irrational number.
Value of expression: Irrational
Expression: xy
When multiplying two rational numbers, the result is always a rational number. Therefore, the value of this expression is rational.
Value of expression: Rational
Expression: xz
When multiplying a rational number and an irrational number, the result is always an irrational number. Therefore, the value of this expression is irrational.
Value of expression: Irrational
To summarize:
Expression | Value of expression
----------------------------------------------
x + y | Rational
y + z | Irrational
xy | Rational
xz | Irrational