Number System Chapter 1
Number Systems
Chapter 1
Following
are the key points that should be kept in mind while preparing this chapter in
perspective of Entrance test of Universities.
üLearn to write numbers using the symbols that were introduced in Egyptian
Civilization very long time ago.
üLearn How to Differentiate between a rational and an irrational number in both ways:
i) by their definitions and ii) in the form of decimals.
üMaster all the properties of Real Numbers related to equality and inequality both.
üCheck out the shortcuts related to powers of iota in Math_tip_01.Powers of Iota.
üDifferentiate between Complex numbers and Imaginary numbers. Numbers with non-zero Real part and zero or non-zero imaginary part are called complex Numbers and each Real number is a complex number with its imaginary part zero. Means that for a number to be complex, its real part should be non-zero while imaginary part may or may not be zero. On the other hand , Numbers whose imaginary parts are always non-zero and Real parts are always zero are called Imaginary Numbers. For example:" 2+3i " is a complex number but not an Imaginary number but " 3i " is an Imaginary number and also a complex number. So, Each Imaginary number is a complex number but each complex number is not an imaginary number. Similarly, Each Real number is a complex number but no Real number can be Imaginary number. Sometimes Imaginary numbers are also named as pure complex numbers.
ü Each real number is Self-conjugate. Means Conjugates of 3 and -3 are respectively 3 and -3.
üFormula for the Multiplicative Inverse of a complex number.
üIota is called Imaginary Unit.
üLearn to separate out Real (Re (z)) and Imaginary parts (Im (z)) of a complex number.
üTheorems on Complex numbers , Modulus , Argument(angle) and Polar Form of a complex number.
üDe'Moiveries Theorem to solve powers of Complex numbers. In this perspective following is a very important Example. Suppose 'z' is a complex number with 'r' modulus and 'x' argument (angle).Now if *z^3(means z-cubed)* equals iota and *r* is equal to one find 'x'. Now , By De'Moivries theorem : z^3 = r(cos3x + i sin3x) , put z^3=i and r=1 , we get : i = cos3x + isin3x . Now obviously we have to find such 3x at which cos3x is zero and sin3x is 1. So, finally we get that 3x = pi/2 which implies that *x = pi/6*.
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