Inverse of 3 mod 11: (3\times 4 = 12\equiv 1), so inverse is 4. Multiply both sides by 4: (x \equiv 20 \equiv 9 \pmod11). Check: (3\times 9=27\equiv 5) ✓. Chapter 4 – Real Numbers Exercise 4.1 Prove: if (x) is real and (x^2 < 1), then (-1 < x < 1).
Find all cube roots of (-8).
Solve (3x \equiv 5 \pmod11).
(x^2 < 1 \Rightarrow x^2 -1 < 0 \Rightarrow (x-1)(x+1) < 0). Product negative iff one factor positive, the other negative. Case 1: (x-1<0) and (x+1>0) → (x<1) and (x>-1) → (-1<x<1). Case 2: (x-1>0) and (x+1<0) impossible (would require (x>1) and (x<-1)). Thus (-1<x<1). Concise Introduction To Pure Mathematics Solutions Manual
Multiply numerator and denominator by conjugate (1+i): [ \frac(2+3i)(1+i)(1-i)(1+i) = \frac2+2i+3i+3i^21+1 = \frac2+5i-32 = \frac-1+5i2 = -\frac12 + \frac52i ] Inverse of 3 mod 11: (3\times 4 =