By ed van der Geer at al Birkhaeuser

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**Additional resources for Arithmetic Algebraic Geometry**

**Example text**

2 5 2−5 (b) From the example above you may be tempted to think that a c a−c − = b d b−d provided none of the denominators equals 0. Give an example to show that this is not true. 64 Suppose b = 0 and d = 0. Explain why a c ad − bc − = . b d bd 20 chapter 0 The Real Numbers worked-out solutions to Odd-Numbered Exercises Do not read these worked-out solutions before attempting to do the exercises yourself. Otherwise you may mimic the techniques shown here without understanding the ideas. For Exercises 1–4, determine how many diﬀerent values can arise by inserting one pair of parentheses into the given expression.

This means that if the answer to part (a) is “true”, then you should explain why a < bc whenever 0 < a < b d and 0 < c < d; if the answer to part (a) is “false”, then you should give an example of numbers a, b, c, and d such that 0 < a < b and 0 < c < d but a b ≥ . d c |a + b| ≤ |a| + |b| for all real numbers a and b. 80 Show that if a and b are real numbers such that |a + b| < |a| + |b|, then ab < 0. 81 Show that |a| − |b| ≤ |a − b| for all real numbers a and b. 3 Inequalities, Intervals, and Absolute Value 35 worked-out solutions to Odd-Numbered Exercises 1 Evaluate |−4| + |4|.

Thus not every real number is a rational number. In other words, not every point on the real line corresponds to a rational number. Irrational number A real number that is not rational is called an irrational number. √ We have just shown that 2 is an irrational number. The real numbers π and e, which we will encounter in later chapters, are also irrational numbers. Once we have found one irrational number, ﬁnding others is much easier, as shown in the next two examples. ” —Sherlock Holmes 6 chapter 0 The Real Numbers example 1 The attitude of the ancient Greeks toward irrational numbers persists in our everyday use of “irrational” to mean “not based on reason”.

### Arithmetic Algebraic Geometry by ed van der Geer at al Birkhaeuser

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