Imaginary Numbers Definition With Easy Examples

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History of Imaginary Numbers

Imaginary numbers were first developed by a Greek mathematician named Hero of Alexandria. The formulas for multiplying fictitious numbers were created later in 1572 by the Italian mathematician Gerolamo Cardano. The square roots of negative numbers can be calculated using these numbers.

Let’s discover more about imaginary numbers, including their meaning, examples, and graphical explanations.

What are Imaginary Numbers?

Imaginary numbers are those that, when squared, produce a negative number. They can also be thought of as the square root of negative numbers. An imaginary number is created by multiplying a non-zero real integer by the imaginary unit “i” (sometimes referred to as “iota”), where i = √(-1) or i² = -1.

Try squaring some real numbers now:

  • (−3)² = −3×−3 = 9
  • 4²= 4×4 = 16
  • (1.3)² = 1.3×1.3 = 1.69

Do any of the outcomes have a negative value?
No. In other words, any real number’s square is always positive.

What integer, then, yields a square that is negative?
It is a fictitious quantity. When using the quadratic formula to solve quadratic equations, we frequently encounter the square root of negative numbers in math. Using imaginary numbers in these situations is required.

Examples of imaginary numbers are shown below:

  • √(-9) = √(-1) · √9 = i (3) = 3i
  • √(-5) = √(-1) · √5 = i √5

3i and i √5 are imaginary numbers in the examples above. Each of these numbers, as can be seen, is the result of multiplying a non-zero real number by i. We can therefore arrive at the following rule for imaginary numbers:
√(-x) = i √x

These are a subset of complex numbers, which are created by adding together a real number and an imaginary number. In other words, a complex number has the formula a + ib, where a and b are both real numbers and a bi is an imaginary number.


Graphical Representation of Imaginary Numbers


Q1. What is 2i equal to?

Ans. As we know that i²=-1 so

Q2. Is 0 an imaginary number?

Ans. Zero is considered to be both a real and imaginary number.

Q3. Is √2 an imaginary number?

Ans. No, √2 is an irrational number.
Assume that √2 is a rational number. Let’s write √2 = m/ n, where m and n are two integers. Let’s also suppose that this fraction is in the lowest terms, which means that m and n must share a component because the fraction cannot be further simplified.
√2 = m/ n
Let’s square all sides, then multiply both sides by n²:
2 = m²/ n²
2 × n² = m²
Given that m² is an even number (a multiple of 2) and that an odd number squared is always an odd number, m must likewise be an even number. Since m is an even number, we can write m = 2x a for any other integer a.
2 × n² = (2 × a)²
2 × n² = 4 × a²
n² = 2 × a²
And now that we’ve established that m is even, we can conclude that n is also even by applying the same logic.
Since m and n are both even, the fraction m/n can be simplified by dividing the numerator and denominator by 2. M/N is therefore not in the lowest terms. Remember that we stated at the beginning that the fraction m/n is in the lowest terms; hence, we have a contradiction.
This proves that our earlier assumption that √2 is rational was incorrect. √2 is an irrational number.

If you want to know about Integers, then please check this article – 3 Types of Integers, Their Easy Explanations and Definition.

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