Mathematics for Computing by G. P. McKeown

By G. P. McKeown

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An octal number is a number written to base 8. Although binary numbers are ideally suited for handling by digital computers, they are rather cumbersome for handling by people. For example, 1658 10 =11001111010 2, that is, four decimal digits are replaced by eleven binary digits. However, since 23 = 8, any octal digit may be represented by three bits, for example, 18=0012 and 78=1112' etc. Thus, any binary number may be simply converted into its shorter octal equivalent, and any octal number into its binary equivalent.

Z+ denotes the set of non-negative integers and Z- the set of nonpositive integers. Thus Z+ =NU{O} Z-=Z\N Q = {p/qlpEZ, qEN} is the set of rational numbers. If XEZ, x/lEQ and since x/I = x, it follows that Z c Q. However, Q also contains elements not in Z such as 1/2, -1/3, 2/3. 1. Numbers which are not rational numbers form the set of irrational numbers, Q', and it is shown later that this set has infinite cardinality. 1 J2 is not a rational number. This may be expressed concisely as J2$Q Proof The result is shown by assuming J2EQ and then deducing a contradiction.

If OP represents O(=a+ib and -----'" --+ OQ represents P=c+id then QP represents O(-p=(a-c)+i(b-d). ~ More generally, it can be easily shown that if PoP 1 represents a ~ ) complex number 0(1' P 1 P z represents 0(2' P 2P 3 represents 0(3' ... - 1P. represents O(m then pop. represents 0( 1 + 0(2 + ... + 0(. 17). 18). An immediate result from this observation is as follows. 19. 20. The length of OR is equal to the length of OP since the length of OQ is 1. The inclination of OR is n/2 greater than that of 6P since the inclination of OQ is n/2.

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