Tutorial: Real numbers
(This topic is also in Section 0.1 in Finite Mathematics and Applied Calculus) #[I don't like this new tutorial. Take me back to the older tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#
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,\ \ $ $-3,\ \ ...$
- $0.0,\ \ $ $1.0,\ \ $ $-1.0,\ \ $ $2.0,\ \ $ $-2.0,\ \ $ $3.0,\ \ $ $-3.0,\ \ ...$
- $\displaystyle \frac{5}{2} = 2.5$ or $\displaystyle 2.50000\cdots,\ \ $ $\displaystyle \frac{4}{3} = 1.3333\cdots,\ \ $ and $\displaystyle \ \frac{-21}{130} = -0.1\ 615384\ 615384 \cdots.$
- $\sqrt{2} = 1.4142135623730951\cdots,$ $\pi = 3.141592653589793\cdots,$ and $\ e = 2.718281828459045\cdots.$
- $0.1\ 01\ 001\ 0001\ 00001\cdots$
Examples
Operations on the real numbers
The five most common operations on the set of real numbers are:
$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†
2. Exponents
3. Multiplication and division
4. Addition and subtraction
†
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
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.
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)
Suggested video for this topic: Video by Math Meeting
%%Examples:
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):
Note
#[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 a*b. | -4*5*2+6=-34 x*y-6*x |
#[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 $\dfrac{a}{b}.$ (a*b)/c = a*b/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 |
- 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.]#
Now try the exercises in Section 0.1 in Finite Mathematics and Applied Calculus.
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Copyright © 2019 Stefan Waner and Steven R. Costenoble