No, 1/0 (one divided by zero) is not indeterminate. It is undefined.
Here's why:
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Undefined vs. Indeterminate: These terms are often confused, but they have distinct meanings in mathematics.
- Undefined: An operation is undefined if there is no meaningful value that can be assigned to it. Division by zero falls into this category. There is no number which, when multiplied by zero, equals one.
- Indeterminate: An expression is indeterminate if it can take on multiple values depending on the context or how the limit is approached. A common example is 0/0.
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Why 1/0 is Undefined:
- Division is the inverse operation of multiplication. So, 1/0 = x means 0 * x = 1.
- There is no real number x that satisfies the equation 0 * x = 1. Any number multiplied by zero equals zero, not one.
- Therefore, dividing 1 by 0 is not a valid mathematical operation, and the result is undefined.
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Why 0/0 is Indeterminate:
- Consider the expression 0/0 = y. This means 0 * y = 0.
- Any number multiplied by zero equals zero. Therefore, any value of y could satisfy this equation. Because there isn't a single, well-defined solution, 0/0 is considered indeterminate.
In summary, while both 1/0 and 0/0 involve zero, they have different classifications: 1/0 is undefined because it has no solution, while 0/0 is indeterminate because it has infinitely many possible solutions.
