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Are irrational numbers a part of the real number system?
Yes.
Why is a real number is a component of the real number system?
It would be a bit stupid to call a system the real number system if real numbers were not a part of it!
Do integers numbers have decimals?
No, integers are whole numbers that do not have any decimal or fractional parts. Integers include positive numbers, negative numbers, and zero. Decimals are used to represent numbers that fall between integers on the number line.
What is real about real numbers?
These numbers are real in contrast to complex numbers which contain an element which is called imaginary. All this arises because the square of any "normal" number is non-negative. So it is not possible to take the square root of a negative number. In some cases, though, there are mathematical advantages in being able to do so. So mathematicians introduce i, the square root of -1. Numbers involving i are said to be imaginary and, by way of contrast, those that don't, are said to be real. A number comprising a real part and an imaginary part is called a complex number.
What is it called if something is positive or negative?
Numbers that can be positive or negative include the integers, the rational numbers, the real numbers, and the complex numbers. All integers are rational numbers (numbers that can be written as a fraction, like 2/1), but most rational numbers are not integers -- like -1/2. (2/1, a rational, can be written as 2, an integer). The real numbers include all the rationals, plus many, many more numbers that can't be written as ratios or fractions, such as the square root of 2, pi, and the euler constant, e. As with the rational numbers and integers, there are as many negative real numbers as there are positive ones. Finally, we have the complex numbers. These include all of the real numbers, plus the roots of negative real numbers. Complex numbers are written in two parts -- a real part, plus an "imaginary" part (which is just as "real" as the real part, but is called "imaginary" for historical reasons). For example, 1 + i is a complex number with positive real and imaginary parts, while -1 - i is a complex number with negative real and imaginary parts. Positive and negative number systems are clearly very important in mathematics and in everyday life. They are all distinguished by the fact that they include magnitudes less than zero, as well as greater than zero (magnitudes of complex numbers are more complicated because complex numbers can have both positive and negative parts in one complex number!) There is also the term "non-zero" which refers to values that are positive or negative but not a value that is neither. It is a very important mathematical term since many functions (reciprocals, for example) can only have non-zero domains.
