Tutorial: Real numbers

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(This topic is also in Section 0.1 in Finite Mathematics and Applied Calculus)

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Note It is assumed that you are familiar with integers, fractions, and decimals, and that you know how to add, subtract, multiply and divide them.

Real numbers

The real numbers are the numbers that can be written in decimal notation, including those that require an infinite decimal expansion. We consider three important types of real numbers: integers, rational numbers, and irrational numbers:

Integers, rational numbers, irrational numbers

Integers: These are the whole numbers; positive, negative, and zero:

$0,\ \ $ $1,\ \ $ $-1,\ \ $ $2,\ \ $ $-2,\ \ $ $3,\ \ $ $-3,\ \ ...$ As decimals, they can be written as

$0.0,\ \ $ $1.0,\ \ $ $-1.0,\ \ $ $2.0,\ \ $ $-2.0,\ \ $ $3.0,\ \ $ $-3.0,\ \ ...$ and so on.

Rational numbers: These are the numbers that can be represented as fractions of integers. Their decimal expansions either terminate (in zeros) or repeat indefinitely after some point; for instance,

$\displaystyle \frac{5}{2} = 2.5$ or $\displaystyle 2.50000\cdots,\ \ $ $\displaystyle \frac{4}{3} = 1.3333\cdots,\ \ $ and $\displaystyle \ \frac{-21}{130} = -0.1\ 615384\ 615384 \cdots.$ Note that integers are automatically rational numbers; for instance, 4 can be written as $\dfrac{4}{1}.$

Irrational numbers: These are all the real numbers that are not rational. Their decimal expansions never terminate nor repeat; for instance,

$\sqrt{2} = 1.4142135623730951\cdots,$ $\pi = 3.141592653589793\cdots,$ and $\ e = 2.718281828459045\cdots.$ The number

$0.1\ 01\ 001\ 0001\ 00001\cdots$is also irrational; even through its decimal expansion follows a pattern, it is non-repeating.

Note It is impossible to be certain whether or not a decimal expansion eventually repeats or not by just looking at part of it as shown here. For instance, who's to say that the decimal expansion of $\sqrt{2}$ above does not start to repeat after, say, the 1,000th decimal place? All we can say by examining part of the decimal expansion is that it appears to be non-repeating. Bear this in mind when you do the quiz questions below.

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Examples

Operations on the real numbers

The five most common operations on the set of real numbers are:•Addition • Subtraction • Multiplication • Division • Exponentiation."Exponentiation" means the raising of a real number to a power; for instance, $2^3 = 2 \cdot 2 \cdot 2 = 8$.

When we write an expression involving two or more of these operations, such as

$4 + 3 - 4 \div 3 \times 2^3 \quad $ #[or][o]# $\quad 6 + 12(7 - 3 \cdot 6)/4 \cdot 5 \quad$ #[or][o]# $\quad \dfrac{12 - (-2)}{2 \cdot 4^2 - 5^2}$

we use rules to decide on the order in which we do the operations:

Standard order of operations

1. Parentheses and fraction bars†

Use the standard order of operations shown here to calculate the values of all expressions inside parentheses or brackets first, working from the innermost parentheses out. When dealing with a fraction bar, think of the entire numerator and denominator as being enclosed in parentheses, so calculate the numerator and denominator separately.

2. Exponents

Raise all numbers to the indicated powers.

3. Multiplication and division

Do all the multiplications and divisions from left to right. Note on division: When division of integers leads to a fraction, it is often best to leave the fraction in reduced form rather than approximating by a decimal. (So, sometimes there is no calculation to do, as in $2/3,$ for instance.)

4. Addition and subtraction

Do the remaining additions and subtractions from left to right.

† Fraction bars are the horizontal lines separating the numerator and denominator in a fraction, as in $\dfrac{3-4}{6}$. The division signs $\div$ and $/$ do not count as fraction bars.

Remembering the order of operations: PEMDAS

P \gap[20] \t Parentheses and fraction bars \\ E \gap[20] \t Exponents \\ MD \gap[20] \t Multiplication and Division (from left to right) \\ AS \gap[20] \t Addition and Subtraction (from left to right)

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%%Examples:

Now some calculations for you to do:

Entering formulas with technology

Any good calculator or computer application will respect the standard order of operations. When entering formulas, we must always take particular care with division, exponentiation, and the use of parentheses.

Entering formulas

The following conventions apply to most forms of technology, such as spreadsheets, graphing calculators and, in large part, to computer programming langugages (although the method for entering exponents can vary quite a lot from prgramming langauge to programming language):

#[Operation][Operación]#	#[Symbol][Símbolo]#	#[Examples][Ejemplos]#
#[Addition, Subtraction, Negative][Suma, Resta, Negativo]#	The usual symbols: + and −	-3+5-8=-6 3-x+y
#[Multiplication][Multiplicación]#	The asterisk: . Enter $a \times b$ as ab.	-452+6=-34 xy-6x
#[Exponentiation][Exponenciación]#	The caret: ^. Enter $a^b$ as a^b. If the exponent includes sums, differences, and/or products, enclose it in parentheses. #[Enter][Ingresa]# $a^{b+c}$ #[as][como]# a^(b+c) #[Enter][Ingresa]# $a^{b}+c$ #[as][como]# a^b+c	2^3=8 2^x+y
Parentheses	() only; never square brackets [] or braces {}. Thus, for instance, enter $2[(4 + 3)/2]$ as 2*((4+3)/2).	(2(3+5)-2)/2=7 (2(x+y))^4
Redundant parentheses	Parentheses are only necessary to change the order of operations in a formula you enter; otherwise they do nothing. (a/b) = a/b and represents $\dfrac{a}{b}.$ (a)/(b) = a/b and represents $\dfrac{a}{b}.$ (ab)/c = ab/c and represents $\dfrac{ab}{c} = a\dfrac{b}{c}.$ (a^b)/c = a^b/c and represents $\dfrac{a^b}{c}.$	(1+3)/(2) = (1+3)/2 = $\dfrac{1+3}{2}$ #[but][pero]# 1+3/2 = $1 + \dfrac{3}{2}.$ (3^(4))/(2) = 3^4/2 = $\dfrac{3^4}{2}$ #[but][pero]# 3^(4/2) = $3^{4/2}.$ 1-(3^(4x)) = 1-3^(4x) = $1 - 3^{4x}$ #[but][pero]# 1-3^4x = $1 - 3^4x.$
Division	There are no fraction bars in technology formulas. For the division symbol use the slash /. If the numerator or denominator includes sums, differences, and/or products, enclose it in parentheses. Enter $\dfrac{a}{b}$ as a/b. Enter $\dfrac{a}{b+c}$ as a/(b+c) Enter $\dfrac{a+b}{c}$ as (a+b)/c Enter $\dfrac{a+b}{c+d}$ as (a+b)/(c+d)	4/(4+5)=4/9 4/4+5=6 (12+6)/3=6 12+6/3=14

Note

Popular graphing calculators use a shorter symbol for negative, but spreadsheets and programming languages always use the same symbol for negative and minus.
Popular graphing calculators allow you to omit the asterisks in products, but spreadsheets and programming languages do not.

#[Intervals][Intervalos]#

#[Subsets of the set of real numbers which happen to be unbroken segments are called intervals, and show up quite often and so we have a compact notation for them.][A subconjuntos del conjunto de los números reales que resultan ser segmentos continuos, nos llamamos intervalos, y se encunetra frecuentemente, por lo que tenemos una notación compacta para representarlos.]#

#[Interval Notation ][Notación de intervalo]#

#[Here is a list of types of intervals along with examples:][Lo siguiente es una lista de varios tipos de intervalos con ejemplos:]#

	#[Interval][Intervalo]#	#[Description][Descripción]#	#[Picture][Dibujo]#	%Example
#[Closed][Cerrado]#	$[a, b]$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a \leq x \leq b$		$[0, 10]$ #[Includes $0$ and $10$][Incluya $0$ y $10$]#
#[Open][Abierto]#	$(a, b)$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a < x < b$		$(-1, 5)$ #[Excludes $-1$ and $5$][Excluya $-1$ y $5$]#
#[Half-open][Semibierto]#	$(a, b]$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a < x \leq b$		$(-3, 1]$ #[Excludes $-3,$ includes $1$][Excluya $-3,$ incluya $1$]#
	$[a, b)$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a \leq x < b$		$[-4, 0)$ #[Includes $-4,$ excludes $0$][Incluya $-4,$ excluya $0$]#
#[Infinite][Infinito]#	$[a, +\infty)$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a \leq x < +\infty$		$[-3, +\infty)$ #[Includes $-3$][Incluya $-3$]#
	$(a, +\infty)$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $a < x < +\infty$		$(0, +\infty)$ #[Excludes $0$][xEcluya $0$]#
	$(-\infty, b]$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $-\infty < x \leq b$		$(-\infty, 2]$ #[Includes $2$][Incluya $2$]#
	$(-\infty, b)$	#[Set of real numbers $x$ with][Conjunto de números reales $x$ tal que]# $-\infty < x < b$		$(-\infty, -1)$ #[Excludes $-1$][Excluya $-1$]#
	$(-\infty, +\infty)$	#[Set of all real numbers][Conjunto de todos los números reales]# $-\infty < x < +\infty$		$(-\infty, +\infty)$

Now try the exercises in Section 0.1 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.