Real NumbersNumbers

Numbers are either Real or Imaginary.

Real numbers are either Rational or Irrational. All real numbers can be placed on a real number line.


Real numbers

Real Number Line

Rational numbers include any number that can be written as a fraction or a ratio of two integers, \LARGE \inline \fn_cs \frac{p}{q} where \LARGE \inline \fn_cs q \neq 0.

Irrational numbers cannot be written as a ratio or fraction. When you write an irrational number as a decimal, it does not end in 0 or it does not repeat. Example: \large \inline \fn_cs \sqrt{3} = 1.7320508075688772935274463415059…

The \large \inline \fn_cs \sqrt{3} can be placed on the real number line in the space between 1 and 2.

Rational numbers are subdivided into fractions or integers. On a number line the space in between two integers contains all the fractions or decimal numbers that are between the two integers.

Fractions are either repeating or terminating. When you write a fraction as a decimal and it ends in 0 it is called a terminating decimal. If it repeats it is called a repeating decimal.


\large \inline \fn_cs \frac{1}{2} = 0.50 and \large \inline \fn_cs \frac{7}{4} = 1.750 are terminating decimals.

\large \inline \fn_cs \frac{4}{3} = 1.333… or \large \inline \fn_cs 1.\overline{3} and \large \inline \fn_cs \frac{12}{99} = 0.1212121212 or \large \inline \fn_cs 0.\overline{12} are repeating decimals.

Integers are rational numbers that make a complete unit. They include all the negative units, 0, and all the positive units.


-4, \large \inline \fn_cs \sqrt{9}= \pm 3, 12/4 = 3, and 0/10 = 0 are all integers.

Whole numbers are zero and all the positive integers. {0,1,2,3,…}

Natural numbers are all the positive integers or the numbers you first used to learn to count. {1,2,3,..

The number 5 is a natural number, a whole number, an integer, a rational number and a real number.

Classify each of the following numbers: \large \inline \fn_cs \pi,-7, and 5/3

R – real

I – imaginary

Ra – rational

Ir – irrational

F – fraction

Term D – terminating decimal

Repeat D – repeating decimal

In – integer

W – whole number

N – natural number

\large \inline \fn_cs \pi is irrational and real [Ir and R]

7 is a natural number, a whole number, an integer, a rational number and a real number [N,W,In, Ra, R]

5/3 is a fraction, 1.6666… is a repeating decimal, a rational number and a real number [F, Repeat D, Ra, R]

Filed under: Pre-requisites for Pre-CalculusUncategorized

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