1 divided by 0 is not equal to infinity in the context of standard arithmetic. It is undefined. However, it approaches infinity when dealing with limits.
The Undefined Nature of Division by Zero
In the realm of real numbers, division by zero is undefined. Here's why:
- Basic Division: Division is the inverse operation of multiplication. For example, 6 divided by 3 equals 2 because 2 multiplied by 3 equals 6.
- Zero as a Multiplier: Any number multiplied by zero equals zero. Therefore, there is no real number that, when multiplied by zero, equals any number other than zero itself.
- The Problem: If we attempt to divide 1 by 0, we are essentially looking for a number that, when multiplied by zero, would equal one. But, as stated earlier, no such number exists.
Therefore, in the set of real numbers, 1 divided by 0 is undefined, as the reference states: "the division by the number 0 is undefined among the set of real numbers."
Infinity in Limits
The concept of 1 divided by 0 equaling infinity is only applicable when we explore limits. Limits describe the behavior of a function as it approaches a particular value.
The Limit Approach
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Imagine a sequence where we divide 1 by successively smaller numbers:
- 1 / 1 = 1
- 1 / 0.1 = 10
- 1 / 0.01 = 100
- 1 / 0.001 = 1000
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As the denominator gets closer and closer to zero, the result gets increasingly large.
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In limit terms, we say that the limit of 1/x, as x approaches zero from the positive side, is positive infinity. We denote this as: lim (x->0+) 1/x = +∞.
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Similarly, as x approaches zero from the negative side, the limit is negative infinity: lim (x->0-) 1/x = -∞.
Table of Values
| Value of x | 1/x |
|---|---|
| 1 | 1 |
| 0.1 | 10 |
| 0.01 | 100 |
| 0.001 | 1000 |
| 0.0001 | 10000 |
| 0.00001 | 100000 |
| -0.00001 | -100000 |
| -0.0001 | -10000 |
| -0.001 | -1000 |
| -0.01 | -100 |
| -0.1 | -10 |
| -1 | -1 |
As you can see, when x approaches 0 from the positive side (right), 1/x increases and goes to positive infinity. When x approaches 0 from the negative side (left), 1/x decreases and goes to negative infinity.
Key Takeaways
- Undefined in Real Numbers: Division by zero is fundamentally undefined within the standard rules of real number arithmetic.
- Infinity in Limits: The concept of 1 divided by 0 equaling infinity arises only when dealing with limits. This means that the function 1/x approaches infinity as x approaches zero, but it never actually equals infinity.
