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This is called the commutative law of addition. For all real numbers a and b, and for all complex numbers a + jb and c + jd, the following two equations hold: a+b=b+a (a + jb) + (c + jd) = (c + jd) + (a + jb) COMMUTATIVE LAW OF MULTIPLICATION When two real or complex numbers are multiplied, it does not matter in which order the product is performed. This is called the commutative law of multiplication. For all real numbers a and b, and for all complex numbers a + jb and c + jd, the following two equations hold: ab = ba (a + jb)(c + jd) = (c + jd)(a + jb) ASSOCIATIVE LAW OF ADDITION When three real or complex numbers are added, it does not matter how the addends are grouped.

ROUNDING Rounding is the preferred method of rendering numbers in shortened form. In this process, when a given digit (call it r) is deleted at the right-hand extreme of an expression, the digit q to its left (which becomes the new r after the old r is deleted) is not changed if 0 ≤ r ≤ 4. If 5 ≤ r ≤ 9, then q increases by 1 (round it up). Most electronic calculators use rounding. 8 ≈4 PRECEDENCE Mathematicians agree on a certain order in which operations should be performed when they appear together in an expression.

This is derived from the Theorem of Pythagoras in basic plane geometry. The vector forms the hypo- 23 CHAPTER 1 Numbering Systems jy a + jb Direction d tu ni ag M e x Fig. 1-8. A number in the complex plane can be defined as a vector, having a specific magnitude (or length) and a specific direction. tenuse (longest side) of the right triangle. In the example of Fig. 1-9, this side is shown with length equal to c. The other two sides have lengths a and b. The formula for the absolute value of a + jb in this case is: | a + jb | = (a2 + b2)1/2 jy a + jb b c x a c2 = a2 + b2 c = a2 + b2 Fig.