Natural numbers are closed under ____ and under ____.
Addition and multiplication
Step 1: Recall the set
Natural numbers are: \(1, 2, 3, 4, \dots\)
Step 2: What does “closed under” mean?
If we do an operation on any two natural numbers and the result is always a natural number, we say the set is “closed under” that operation.
Step 3: Check Addition
Add two natural numbers:
\(2 + 3 = 5\)
\(7 + 8 = 15\)
Both answers (\(5\) and \(15\)) are natural numbers.
So, natural numbers are closed under addition.
Step 4: Check Multiplication
Multiply two natural numbers:
\(4 \times 6 = 24\)
\(1 \times 9 = 9\)
Both answers (\(24\) and \(9\)) are natural numbers.
So, natural numbers are closed under multiplication.
Step 5: (Helpful contrast)
Subtraction is not always safe:
\(5 - 7 = -2\) (not a natural number)
Division is not always safe:
\(7 \div 2 = 3.5\) (not a natural number)
Conclusion: The blanks are addition and multiplication.