ARITHMETIC

SEMEPXDOST 2020 - UPLB DOST Scholars' Society Study Notes

Table of Contents

  • Arithmetic
  • Sets
    • Definition of Sets
    • Notation
    • Examples
    • Subsets
    • Subset Notation
    • Numerical Sets
    • Set of All Real Numbers
    • Set of All Natural Numbers
    • Set of All Whole Numbers
    • Set of All Integers
    • Set of All Rational Numbers
    • Set of All Irrational Numbers
    • Set Builder Notation
    • Interval Notation
    • Exercises
  • Basic Applications of Arithmetic
    • Percentages
    • Averages
  • Ratio and Proportion
    • Definitions
    • Ratio Formula
    • Proportion Formula
    • Examples

Arithmetic

Sets
Definition of Sets
  • A set is defined as a collection of things where the members of this collection have a common property.
Notation
  • To represent a set, we list each element (or "member") separated by commas and enclose the entire listing in curly brackets:
    • Example: {x,y,z} or {x,y,z, …}
Examples
  • Sets can also consist of various objects. For example:
    • S = {socks, shoes, watches, shirts, …}
    • T = {index, middle, ring, pinky}
    • U = {d, o, s, t, s, e, m, e, p}
Subsets
Definition
  • A set "X" is said to be a subset of another set "Y" if all elements of set "X" are elements of set "Y".
Subset Notation
  • The subset relationship is represented mathematically as:
    • If A is a subset of B, we write ( A
      ightarrow B )
Numerical Sets
  • The following types of numerical sets are covered:
    • Set of all Integers
    • Set of all Real Numbers (denoted by ( ext{IR} ))
    • Set of all Complex Numbers
Set of All Real Numbers
Definition
  • The set of all real numbers is denoted by ( ext{IR} ). It includes five subsets:
Subsets of Real Numbers
  • Rational Numbers: Any number that can be expressed as ( \frac{x}{y} ) where ( x ) and ( y ) are integers and ( y \neq 0 ).
    • Examples include ( 1.375, -3, 0, \frac{1}{2} ).
  • Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers, such as ( \pi, \sqrt{2}, ) and ( e ).
Set of All Natural Numbers
Definition
  • The set of natural numbers (also known as counting numbers) is given by:
    • ( {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} )
Set of All Whole Numbers
Definition
  • The set of whole numbers includes all the elements of the natural numbers plus zero:
    • ( {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} )
Set of All Integers
Definition
  • The set of integers consists of all whole numbers and their negative counterparts:
    • ( {… -3, -2, -1, 0, 1, 2, 3, …} )
Set of All Rational Numbers
Definition
  • The set of rational numbers comprises numbers that can be expressed as the ratio of two integers:
    • ( { \frac{x}{y} | x, y \in \mathbb{Z}, y \neq 0 } )
Set of All Irrational Numbers
Definition
  • Irrational numbers are defined as numbers that cannot be expressed as a ratio of two integers.
Set Builder Notation
Definition
  • Set Builder Notation is a mathematical notation used to describe a set by enumerating its elements or stating the properties that its members must satisfy.
Examples of Set Builder Notation
  • Sets can be expressed concisely, for instance:
    • ( A = {1, 2, 3, 4, 5} )
    • ( A = {x | x \in \mathbb{N}, \ 1 \leq x \leq 5} )
Interval Notation
Definition
  • Interval notation is used to express the domain of a function. The symbols used include:
    • ( \cup ) - Union of two sets
    • ( ( ) ) - Open interval
    • ( [ ] ) - Closed interval
Examples of Interval Notation
  • To represent intervals, examples include:
    • ( {x | x \neq 0} )
    • ( (-2, 3] )
    • ( (-\infty, 0) \cup (0, \infty) )
Exercises
True or False Questions
  1. All integers are whole numbers. (True)
  2. Some rational numbers are integers. (True)
  3. Some whole numbers are irrational numbers. (False)
  4. All integers are rational. (True)
  5. If a number is rational, then it must be a whole number. (False)
  6. No whole numbers are integers. (False)
Translate to Builder Notation
  1. ( {2, 4, 6, 8, 10} \rightarrow {x : x \text{ is an even natural number less than 12})
  2. ( {2, 3, 5, 7, 11} \rightarrow {x : x \text{ is a prime number less than 12})
  3. ( {January, June, July} \rightarrow {x : x \text{ is a month whose name starts with letter J})
Translate to Interval Notation
  1. ( {x | 2 < x < 8} \rightarrow (2, 8))
  2. ( {x | -5 \leq x < 0} \rightarrow [-5, 0))
  3. ( {x | x > 3} \rightarrow (3, \infty))
  4. ( {x | x \leq -4} \rightarrow (-\infty, -4] )

Basic Applications of Arithmetic

Percent
Definition
  • Percentages express how large or small one quantity is relative to another quantity.
Explanation
  • A percentage is defined as a number or ratio as a fraction of 100:
    • ( 1 ext{%} = \frac{1}{100} )
Calculating Percentages
  • To calculate a percentage, turn the numbers into a ratio as a fraction of 100. For example, if 8 out of 15 students are boys, the fraction that represents this can be calculated as follows:
Example 1
  • For the calculation of 25% of 80:
    • The equation is:
    • ( 25 ext{%} \rightarrow \frac{25}{100} )
    • Then, multiply:
    • ( \frac{25}{100} \cdot 80 = 20 )
Averages
Definition
  • The arithmetic mean, or "average", is a measure of the middle or typical value of a dataset.
  • It is calculated by dividing the sum of a collection of numbers by the number of elements in that collection.
Example 1
  • To find the average of the numbers in the set {12, 25, 34, 17, 8, 42}:
    • Total = 12 + 25 + 34 + 17 + 8 + 42 = 138
    • Count = 6
    • Average = ( \frac{138}{6} = 23 )
Example 2
  • A small company with 8 employees has the following hourly wages:
    • Two employees are paid \$35 per hour
    • Two employees are paid \$27 per hour
    • Four employees are paid \$25 per hour
    • To find the average hourly wage:
    • Total Hourly Wages = 235 + 227 + 4*25 = 70 + 54 + 100 = 224
    • Average Hourly Wage = ( \frac{224}{8} = 28 )

Ratio and Proportion

Definitions
Ratio
  • Ratio refers to the quantitative relationship between two amounts, represented typically in the form of a fraction ( a:b ), meaning ( \frac{a}{b} ) where a and b are any two integers.
Proportion
  • Proportion asserts that two ratios are equal.
Ratio Formula
  • If we have two quantities or numbers, the formula for calculating the ratio is defined as:
    • ( a:b \rightarrow \frac{a}{b} )
Proportion Formula
  • For two ratios, if ( a:b ) and ( c:d ), the formula states:
    • ( \frac{a}{b} = \frac{c}{d} )
    • Here, ( b ) and ( c ) are termed as means, while ( a ) and ( d ) are referred to as extremes.
Example 1
  • Are ( 4:5 ) and ( 8:10 ) in proportion?
    • Calculation:
    • ( \frac{4}{5} = 0.8 )
    • ( \frac{8}{10} = 0.8 )
    • Since both ratios are equal, they are in proportion.
Example 2
  • Given a ratio of ( 2:3 ) and the sum of the two numbers equals 60:
    • Let the two numbers be represented by ( 2x ) and ( 3x ).
    • The total is: ( 2x + 3x = 60 )
    • Simplifying gives ( 5x = 60 ) leading to ( x = 12 ).
  • Hence, the two numbers are:
    • First Number: ( 2x = 2 \times 12 = 24 )
    • Second Number: ( 3x = 3 \times 12 = 36 )
    • Required numbers are 24 and 36.