Algebra/Closure

Closure

Closure is a property that is defined for a set of numbers and an operation. This Wikipedia article gives a description of the closure property with examples from various areas in math. As an Algebra student being aware of the closure property can help you solve a problem. For instance a problem might state "The sum of two whole numbers is 24." With practice you will be able to see that the possible set of numbers will be either all odd (e.g. (1,23),(3,21), ... etc.) or all even (e.g. (2,22), (4,20), ... etc.). The problem might not explicitly state the idea of whole numbers. It might state that two sides of a square sum to 24. If you remember working a problem like this before you know that the sides of a square need to be equal and you divide by two. The author of the problem might want to be trickier and say that two sides of an equilateral triangle sum to 24 and then ask for the perimeter of the triangle. In this case you might want to write the equation for the perimeter of an equilateral triangle. This might make it easier for you to see that again you just need to divide 24 by 2 to find the length of one side and plug it into the equation.

Exercises For Closure

Consider each statement and try to come up with an example that proves it is false. If you can do that write down your example and mark the answer false. Otherwise mark the answer true.

Natural Numbers

1. ________ Addition is closed under the Natural Numbers.
2. ________ Subtraction is closed under the Natural Numbers.
3. ________ Multiplication is closed under the Natural Numbers.
4. ________ Division is closed under the Natural Numbers.
5. ________ Exponentiation is closed under the Natural Numbers.
6. ________ Roots are closed under the Natural Numbers.

Whole Numbers

1. ________ Addition is closed under the Whole Numbers.
2. ________ Subtraction is closed under the Whole Numbers.
3. ________ Multiplication is closed under the Whole Numbers.
4. ________ Division is closed under the Whole Numbers.
5. ________ Exponentiation is closed under the Whole Numbers.
6. ________ Roots are closed under the Whole Numbers.

Integers

1. ________ Addition is closed under the Integers.
2. ________ Subtraction is closed under the Integers.
3. ________ Multiplication is closed under the Integers.
4. ________ Division is closed under the Integers.
5. ________ Exponentiation is closed under the Integers.
6. ________ Roots are closed under the Integers.

Rational Numbers

1. ________ Addition is closed under the Rational Numbers.
2. ________ Subtraction is closed under the Rational Numbers.
3. ________ Multiplication is closed under the Rational Numbers.
4. ________ Division is closed under the Rational Numbers.
5. ________ Exponentiation is closed under the Rational Numbers.
6. ________ Roots are closed under the Rational Numbers.

Irrational Numbers

1. ________ Addition is closed under the Irrational Numbers.
2. ________ Subtraction is closed under the Irrational Numbers.
3. ________ Multiplication is closed under the Irrational Numbers.
4. ________ Division is closed under the Irrational Numbers.
5. ________ Exponentiation is closed under the Irrational Numbers.
6. ________ Roots are closed under the Irrational Numbers.

Real Numbers

1. ________ Addition is closed under the Real Numbers.
2. ________ Subtraction is closed under the Real Numbers.
3. ________ Multiplication is closed under the Real Numbers.
4. ________ Division is closed under the Real Numbers.
5. ________ Exponentiation is closed under the Real Numbers.
6. ________ Roots are closed under the Real Numbers.

Answers