Page 69 - ArithBook5thEd ~ BCC

√ √
It is important to note that − N is not the same as −N. In fact, the latter expression presents
√
a problem: do negative numbers have square roots? For example, what is −4? This number, if it
exists, must have absolute value 2, so it would have to be either 2 or −2. But

2
2
2 =(−2) =4 (not −4).
So neither 2 nor −2 is a square root of −4. A similar argument applies to any negative number.

Square roots of negative numbers are unde-
fined within the system of signed numbers.

It is possible to expand the set of signed numbers so as to remedy this defect, but we leave that for a
more advanced course.
√ √
Example 73. −5is undefined. But − 5is a negative number between −3and −2:
√
−3 < − 5 < −2.

2.8.1 Exercises

Find the square roots, or state that they are undefined.
√
1. 9
√
2. − 25
√
3. −25
√
4. − 49
√
5. 100
√
6. 0
Between what two consecutive integers do the following square roots lie?
√
11. 7
√
12. − 7
√
13. − 30

Insert the appropriate inequality (< or >) between the following pairs of numbers.
√ √
14. − 10 − 8
√ √
15. 12 15

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