Numbers are symbols that are used to count physical quantities and measure length, weight, temperature, and more. Numbers can be divided into two main types:
Imaginary Numbers
Imaginary numbers are special because they involve the square roots of negative numbers. For example, the square root of -1 is an imaginary number.
Why are they called imaginary numbers?
There is not a single real number whose square is negative. In other words, there is no real number that, when multiplied by itself, gives a negative result. For instance, there is no number that satisfies the equation x^2=−4.
Example includes √-4. In the simple form, we can write it as 2i because √-1 can be denoted by i. Therefore, √-4 = √-1 . √4 = i . 2 = 2i.
Real Numbers
Real numbers include all numbers that are not imaginary. They can be further divided into two categories: Rational Numbers and Irrational Numbers.
Rational Numbers
Rational numbers can be expressed as fractions (like 1/2). When you convert these fractions into decimal form, they can either terminate(end) or repeat.
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Terminating: A decimal that ends after a certain point (e.g., 0.5).
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Repeating: A decimal that goes on forever with a repeating pattern (e.g., 1/3=0.333…).
Types of Rational Numbers
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Fractions: These are numbers that fall between integer numbers. For example, 1/2 = 0.5(0.5 falls between 0 and 1 integers), -3/4 = -0.75(-0.75 falls between -1 and -2 integers)). A fraction consists of two parts: the numerator(top part) and the denominator(bottom part).
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Integers: These include numbers that are positive, negative, or zero (e.g., -3, -2, -1, 0, 1, 2, 3).
Integers can be further divided into:
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Whole Numbers: These are all non-negative integers starting from 0 (e.g., 0, 1, 2, 3…).
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Natural Numbers: These are positive integers starting from 1 (e.g., 1, 2, 3…).
Irrational Numbers
These are numbers that cannot be expressed as a ratio of two integers (i.e., in the form a/b). These numbers cannot be expressed in the form of a/b because when we have a ratio of two numbers, the decimal representation will always have a repeating pattern or a terminating value. However, the decimal representations of irrational numbers are always non-terminating or non-repetitive. An example of an irrational number is √2. The value of √2 is approximately 1.41421356230950… Here, you can see that the numbers after the decimal do not have a repetitive pattern, and they are also not terminating.
So, if √2 cannot be expressed as a complete number, does that make it an imaginary number? No, irrational numbers exist in nature. One example of an irrational number is pi (π). Pi is the ratio of the circumference of a circle to its diameter. It produces a non-repeating and non-terminating decimal value. However, circles exist and are real in nature, so these are not considered imaginary numbers.