Imaginary Numbers: Geometrical Interpretation, Complex Number

Numbers that are not real are known as imaginary numbers. Hero of Alexandria, a Greek mathematician, was the first to discover imaginary numbers. Gerolamo Cardano, an Italian mathematician, devised the rules for multiplying imaginary numbers. These figures can be used to calculate the square roots of negative values. The quadratic equation is of the form ax² + bx + c = 0, with b² – 4ac as the discriminant. Whenever the discriminant is less than zero, finding the square root becomes necessary for us.

Read Also: Multiplication and Division of Integers

Key Terms: Real Numbers, Rational Numbers, Composite number, Fractions, Prime Numbers, Irrational Numbers, Prime Numbers

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Imaginary Numbers Definition

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Imaginary numbers are those that, when squared, result in a negative number. In other terms, we can define an imaginary number as the square root of a negative number that has no tangible value. Unlike real numbers, imaginary numbers cannot be represented on a number line, but they are real in the sense that they are used in maths. 

Complex numbers are another name for imaginary numbers. In equations of quadratic planes, imaginary numbers appear when the imaginary numbers do not touch the x-axis. In advanced calculus, imaginary numbers are also very useful. The letter 'i' is used to represent imaginary numbers.

Let's try squaring some real numbers:

(−5)² = −5×−5 = 25

8² = 8×8 = 64

(1.3)² = 1.3×1.3 = 1.69

Here are some imaginary numbers examples:

√(-4) = √(-1) · √4 = i (2) = 2i

√(-3) = √(-1) · √3 = i √3

In the above examples, 2i and i √3 are imaginary numbers. Each of these numbers is a product of a non-zero real number and i as can be seen.

The video below explains this:

Quadratic Equations Detailed Video Explanation:

Read More: Difference Between Fraction and Rational Numbers

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Geometrical Interpretation of Imaginary Numbers

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A point (a, b) in the Argand plane is commonly used to represent a complex number a+bi. For example, a complex number 4-2i represents the point (4, -2) on the Argand plane. As a result, an imaginary number bi (which may be expressed as 0 + bi) indicates a point (0, b) on the plane, and hence a vertical axis point (imaginary axis). As a result, imaginary numbers always lie on the Argand plane's vertical axis. Here are a few examples.

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Complex Number

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Complex numbers in mathematics do not refer to complicated numbers; rather, they refer to the combination of two types of numbers to make a complex. Real and imaginary numbers are the two types of numbers that make up complex numbers. They are the building blocks of more obscure math, such as algebra. Complex numbers are used in a variety of real-world applications, such as electronics and electromagnetism.

a + bi represents a complex number, with the real number at the first and the imaginary number at the last.

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Calculating Imaginary Numbers

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We can do all of the same calculations with imaginary numbers as we can with real numbers.

Adding/Subtracting Imaginary Numbers

Adding and subtracting imaginary numbers works in the same way as combining like terms in algebra does.

• 2i + 3i = 5i
• 2i - 3i = -i

Multiplying Imaginary Numbers

We multiply imaginary numbers in the same way that we multiply algebraic terms. We may need to utilize the exponent's rule a^m × a^n = a^m+n in this case. However, we must remember that i² = -1 in this case. Here are a few examples.

1. (a+bi)(c+di) = (a+bi)c + (a+bi)di

= ac+bci+adi+bdi²

= (ac-bd)+i(bc+ad)

2. 2i × 3i = 6i² = 6(-1) = -6
3. 3i2 × -5i³ = -15i^5 = -15 (i²)² i = -15 (-1)² i = -15i

Dividing Imaginary Numbers

While dividing imaginary rules, we use the rule of exponents a^m / a^n = a^m-n. In the result after division, we usually do not keep "i" in the denominator. If we get so, then we use the rule 1/i = -i (this is because 1/i = 1/i · i/i = i/i² = i/(-1) = -i). Here are some examples:

• 4i/2i = 2
• 3i3 / 4i6 = 3/(4i3) = 3/(4(i2)(i)) = -3/(4i) = (-3/4)(-i) = 3i/4

Also Check:

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Things to Remember

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• When the minus sign comes out of the square root, it becomes i because i = √(-1).
• When "i" is the denominator (or in the case of negative powers of i), use the rule 1/i= -i.
• in = ir, where 'r' is the remainder obtained by dividing n by 4. Then we can apply one of the rules: i2 = -1, i3 = -i, i4 = 1.

Check Also: Study Notes for Complex Numbers

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