Real Numbers
Real numbers, symbolised as R, form a set that includes both rational and irrational numbers. They can be visualised on the number line and are crucial in representing quantities and facilitating various mathematical operations.
Rational Numbers
Rational numbers are those that can be expressed as a fraction of two integers, where the numerator and the denominator are integers, and the denominator is not zero.
- Examples: 3/4, -5/2, 7
- Properties:
- Closure: The sum, difference, and product of two rational numbers are rational.
- Inverse: The inverse of a rational number (except zero) is rational.
- Density: Between any two rational numbers, there exists another rational number.
Decimal Representation
Rational numbers can be expressed as either terminating or repeating decimals. For instance, 1/4 = 0.25 (terminating) and 1/3 = 0.333... (repeating).
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers and have non-terminating, non-repeating decimal expansions.
- Examples: sqrt(2), pi
- Properties:
- Non-Expressible: Cannot be expressed as exact decimals or fractions.
- Density: Between any two irrational numbers, there exists another irrational number.
Surds
Surds are a subset of irrational numbers, which are the roots of numbers that are not whole numbers. For example, sqrt(3) and sqrt(5) are surds.
Example Question 1: Identifying Real Numbers
Identify which of the following numbers are rational and which are irrational: 5/3, sqrt(3), 6.75, pi.
Solution:
- 5/3 is rational.
- sqrt(3) is irrational.
- 6.75 is rational.
- pi is irrational.
Complex Numbers
Complex numbers introduce an extension to the real number system by incorporating imaginary numbers, which are defined as multiples of the square root of -1, denoted by i.
Imaginary Unit
- Definition: i = sqrt(-1).
- Properties:
- i2 = -1
- i3 = -i
- i4 = 1
Form of Complex Numbers
A complex number is expressed as a + bi, where a and b are real numbers, and a is the real part, while bi is the imaginary part of the complex number.
Operations with Complex Numbers
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
- Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Example Question 2: Complex Number Operations
Calculate the sum and product of 3 + 4i and 1 - 2i.
Solution:
- Sum: (3 + 4i) + (1 - 2i) = 4 + 2i
- Product: (3 + 4i)(1 - 2i) = (3 + 8) + (4 - 6)i = 11 - 2i
Magnitude of Complex Numbers
The magnitude (or modulus) of a complex number z = a + bi is the distance from the origin to the point representing the complex number in the complex plane.
Formula
|z| = sqrt(a2 + b2)
Example Question 3: Finding Magnitude
Find the magnitude of the complex number 5 + 12i.
Solution:
|5 + 12i| = sqrt(52 + 122) = sqrt(25 + 144) = sqrt(169) = 13
FAQ
The magnitude of a complex number, often denoted as |z|, represents the distance of the point representing the complex number from the origin in the complex plane. The complex plane is a two-dimensional coordinate system where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. The magnitude is calculated using the Pythagorean theorem: |z| = sqrt(a2 + b2), where z = a + bi. In graphical or geometric interpretations, such as in vector analysis or phasor diagrams in electrical engineering, the magnitude gives the length of the vector or phasor, providing a scalar quantity that represents the size or quantity in the complex number.
Complex numbers find numerous applications in various fields, including engineering, physics, and computer science. In electrical engineering, for instance, complex numbers are used to analyse AC circuits. The voltage and current in an AC circuit are often represented as complex numbers, where the real part represents the amplitude, and the imaginary part represents the phase shift. In fluid dynamics, complex analysis is used to describe potential flow in two dimensions. Moreover, in computer graphics, complex numbers are used to perform transformations such as rotation, scaling, and translation on images. Thus, complex numbers provide a powerful mathematical tool in solving real-world problems.
Yes, a complex number can be a real number if the imaginary part of the complex number is zero. A complex number is expressed in the form a + bi, where a and b are real numbers. If b = 0, then the complex number is a + 0i, which is simply a, a real number. In this case, the complex number is said to be purely real. Similarly, if a = 0, the complex number is said to be purely imaginary. Thus, the set of real numbers is a subset of the set of complex numbers, and every real number is also a complex number with an imaginary part of zero.
The concept of magnitude, when applied to real numbers, is analogous to the absolute value. For a real number x, its magnitude (or absolute value) is denoted as |x| and represents the distance of x from zero on the real number line, disregarding direction. For positive x, |x| = x, and for negative x, |x| = -x. When we extend this concept to complex numbers, the magnitude represents the distance from the origin to the point representing the complex number in the complex plane, calculated as |z| = sqrt(a2 + b2) for z = a + bi. Thus, the magnitude provides a non-negative scalar value representing the "size" of a real or complex number.
The square roots of negative numbers are considered imaginary due to the definition of the imaginary unit, i, which is defined as the square root of -1 (i = sqrt(-1)). In the real number system, there is no number that, when squared, results in a negative number, since the square of any non-zero real number is positive. Therefore, to extend the number system and allow for the square roots of negative numbers, mathematicians introduced imaginary numbers. Imaginary numbers enable solutions to certain equations that have no solutions in the real numbers, such as x2 + 1 = 0, and are crucial in advanced mathematics and physics.
Practice Questions
The number sqrt(7) + sqrt(11) is irrational. This is because the square roots of prime numbers (like 7 and 11) are irrational. When you add or subtract an irrational number and a rational number, the result is irrational. Furthermore, the sum of two irrational numbers can be rational only if the two numbers are additive inverses of each other, which is not the case here. Therefore, sqrt(7) + sqrt(11) is irrational.
To find the product of two complex numbers, we use the distributive property (also known as the FOIL method for binomials). So, (3 + 4i)(1 - 2i) = 3 * 1 + 3 * (-2i) + 4i * 1 + 4i * (-2i) = 3 - 6i + 4i - 8i2 Since i^2 = -1, = 3 - 2i + 8 = 11 - 2i Thus, the product of z1 and z2 is 11 - 2i.
