2. Exponents (ab)"=a"b (a/b) 3. Fractional Exponents where ais the positive gth root of a if a>0 and the negative gth root of a if a is negative and q is odd. Accordingly, the five rules of exponents given above (for integers) are also valid if m and n are fractions, 4. Irrational Exponents If anexponent is irrational, e.g, V2, the quantity, such as a is the limit of the sequence a · Operations with zero 0"=0 5. Logarithms If x, y, and b are positive and b#1
2 2. Exponents For integers m and n, a a a a a a a a ab a b a b n m n m n m n m n m n m m m m = = = = + − / / ( ) ( ) ( )m m m = a b/ 3. Fractional Exponents a a p q q p / / = ( ) 1 where a1/q is the positive qth root of a if a > 0 and the negative qth root of a if a is negative and q is odd. Accordingly, the five rules of exponents given above (for integers) are also valid if m and n are fractions, provided a and b are positive. 4. Irrational Exponents If an exponent is irrational, e.g., 2, the quantity, such as a 2 , is the limit of the sequence a a a 1 4 1 41 1 414 . . . , , , . . . . Operations with Zero 0 0 1 m 0 = = ; a 5. Logarithms If x, y, and b are positive and b ≠ 1, •
Change of Base(a≠1) 6. Factorial The factorial of a positive integer n is the product of all the positive integers less than or equal to the inte- ger n and is denoted n! Thus, t!=12.3 Factorial 0 is defined 0= 1 Stirlings A (See also 9. 2.) 7. Binomial Theorem For positive integer
3 log log log log log log lo b b b b b b xy x y x y x y ( ) ( / ) = + = − g log log log log log b p b b b b b x p x x x b N = = − = = ( / )1 1 1 0 ote b x x : log . b = Change of Base (a ≠ 1) log log log b a b x x a = 6. Factorials The factorial of a positive integer n is the product of all the positive integers less than or equal to the integer n and is denoted n! Thus, n! = ⋅ ⋅ ⋅…⋅ 1 2 3 n. Factorial 0 is defined 0! = 1. Stirling’s Approximation lim ! n n n e n n →∞ ( / ) 2π = (See also 9.2.) 7. Binomial Theorem For positive integer n, • •
(x+y)"=x2+nx"y+ n(n-1) nn-1)( n-2)x2-3 + 8. Factors and Expansion (a+b)2=a2+2ab+b2 (a+b)3=a3+3a2b+3ab2+b3 (a-b3)=(a-b)( (a3+b3)=(a+b)(a2-ab+b2) 9. Progression in which the receding term a,a+d,a+2d,,a+(n-1)d. If the last term is denoted l=a+(n-1)d, then the sUIn Is s=-(a+D)
4 ( ) ( ) ! ( ( ) x y x nx y n n x y n n n n n n n + = + + − + − − −1 1 −2 2 2 1 2) 3 3 3 1 ! . x y nxy y n n n − − +… + + 8. Factors and Expansion ( ) ) ( ) a b a ab b a b a ab b a b a + = + + − = − + + = + 2 2 2 2 2 2 3 3 2 2 3 ( a b ab b a b a a b ab b a b 2 2 3 3 3 2 2 3 2 2 3 3 3 + + − = − + − − = ( ( ) ) (a b a b a b a b a ab b a b a − + − = − + + + = ) ) ( ) ) ( ) ( ( ( ) ( 3 3 2 2 3 3 + − + b a ab b ) ( ) 2 2 9. Progression An arithmetic progression is a sequence in which the difference between any term and the preceding term is a constant (d): a a d a d a n d , , ,..., . + + + − 2 1( ) If the last term is denoted l a n d [ ( ) = + −1 ], then the sum is s n = +a l 2 ( )
a geometric progression is a sequence in which the ratio of any term to the preceding terms is a constant r. Thus for n terms a ar. aIr. The sum is 10. Complex Numbers A complex number is an ordered pair of real num- bers(a, b) Equality:(a, b)=(c, d) if and only if a=c and b=d Addition: (a, b)+(c, d)=(a+c, b+d) Multiplication: (a, b)(c, d)=(ac-bd, ad +bc) The first element of(a, b)is called the real part; the second, the imaginary part. An alternate notation for (a, b)is a+ bi, where i=(1, 0), and i(0, 1)or0+1 is written for this complex number as a convenience With this understanding, i behaves as a number. (2-3)(4+i)=8-12i+2i-32=11-10i.The onjugate of a bi is a-bi, and the product of omplex number and its conjugate is a+b. Thus, quotients are computed by multiplying numerator nd denominator by the conjugate of t 2+3i(4-2i)(2+3i)14+8i7+4i 4+2i(4-2i)(4+2i)20
5 A geometric progression is a sequence in which the ratio of any term to the preceding terms is a constant r. Thus, for n terms, a ar ar ar n , , ,..., 2 −1 The sum is S a ar r n = − 1− 10. Complex Numbers A complex number is an ordered pair of real numbers (a, b). Equality: ( , ) ( , ) a b c d = if and only if a = c and b = d Addition: ( ) ( , ) , a,b c d a c b d + = + + ( ) Multiplication: ( )( , ) ( a,b c d ac bd ad bc = − + , ) The first element of (a, b) is called the real part; the second, the imaginary part. An alternate notation for (a, b) is a + bi, where i 2 = −( ), , 1 0 and i (0, 1) or 0 + 1i is written for this complex number as a convenience. With this understanding, i behaves as a number, i.e., ( )( )2 3 4 8 12 2 3 11 10 2 − + = − + − = − i i i i i i. The conjugate of a + bi is a bi − , and the product of a complex number and its conjugate is a b 2 2 + . Thus, quotients are computed by multiplying numerator and denominator by the conjugate of the denominator, as illustrated below: 2 3 4 2 4 2 2 3 4 2 4 2 14 8 20 + 7 + = − + − + = + = i i i i i i ( ) ( ) i ( ) ( ) + 4 10 i
11. Polar Form The complex number x+ iy may be represented by a plane vector with components x and y: (see Figure 1. 1 ) Then, given two complex numbers 1)and z,=5 (cose, +i sine,), the product and quotient are Product: z,z,=rIcos(0,+0,)+i sin(e, +02) isin(1-02) Powers:="=((cos0+sinoR FIGURE 1.1 Polar form of complex number
6 11. Polar Form The complex number x + iy may be represented by a plane vector with components x and y: x iy r i + = + (cos sin θ θ) (see Figure 1.1). Then, given two complex numbers z r i 1 1 1 1 = + (cos ) θ θ sin and z r i 2 2 = + (cos sin ), 2 2 θ θ the product and quotient are: Product: z z r r i 1 2 1 2 = + + + [cos( ) sin( )] 1 2 1 2 θ θ θ θ Quotient: z z r r i 1 2 1 2 / / [ ] = − + − ( ) cos( ) sin( ) 1 2 1 2 θ θ θ θ Powers: z r i r n i n n n n = + = + [ ( )] [ ] cos sin cos sin θ θ θ θ 0 x y r P (x,y) θ Figure 1.1 Polar form of complex number