This document describes a conversation between Ni Suiti and her granddaughter Si Nessa about numbers in the land of Numberland. Si Nessa tells Ni Suiti that in the Bichromic province, numbers can be either black or red. Addition and multiplication of numbers follows rules determined by their colors. Later, Ni Suiti discusses the properties of 2-color numbers with her husband Ki Algo, who shows that they form a mathematical structure called a ring. However, 2-color numbers differ from real numbers in that they contain idempotent elements and zero divisors.
The document discusses complex numbers. It explains that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined where i^2 = -1. A complex number is defined as a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or multiplied by treating i as a variable and using rules like i^2 = -1. Examples show how to solve equations and perform operations with complex numbers.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It provides details on:
1) Converting between number systems using methods like the place value method or remainder method. For example, converting between binary, octal, and hexadecimal systems involves grouping bits or replacing digits with their base-n equivalents.
2) Representing negative numbers in binary, including through sign-magnitude and two's complement representations. The two's complement of a binary number is calculated by complementing each bit and adding 1.
3) Hexadecimal arithmetic which works similarly to decimal arithmetic but with 16 symbols (0-9 and A-F) instead of 10 symbols.
Números Complejos:
1. Representación gráfica de los números complejos.
2. Conjugado y opuesto de un número complejo.
3. Operaciones con números complejos.
The document discusses different number systems used in mathematics. It begins by explaining that a number system is defined by its base, or the number of unique symbols used to represent numbers. The most common system is decimal, which uses base-10. Other discussed systems include binary, octal, hexadecimal, and those used historically by cultures like the Babylonians. Rational and irrational numbers are also defined. Rational numbers can be written as fractions of integers, while irrational numbers cannot.
The document contains questions related to CAT, MAT, GMAT entrance exams. It discusses various topics like probability, permutations and combinations, averages, ratios etc. and provides solutions to sample questions in 3-4 sentences each. The overall document aims to help exam preparation by providing practice questions on common quantitative topics.
The document discusses various types of numbers including natural numbers, whole numbers, and integers. It provides examples and explanations related to properties of these numbers. Some key points include:
- Natural numbers start from 1 and do not include 0, negative numbers, or decimals.
- Whole numbers include all natural numbers and 0.
- Integers include whole numbers and their negatives.
- Examples are provided to illustrate properties like divisibility, perfect squares, and solving word problems involving sums and products of numbers.
- The last part discusses Donkey's stable number based on his true and false answers to questions about divisibility, being a square, and the first digit. It is determined his number must
The document discusses different numeric representation systems used in computing, including decimal, binary, octal, and hexadecimal. It explains key concepts such as positional notation, where the position of a digit determines its value; radix/base, which is the set of digits used; and weighting factors, where each digit position is multiplied by a power of the base to determine its value. Integer, floating point, and fixed point numbers are also defined in terms of how many bits are used for integer and fractional portions.
The document discusses complex numbers. It explains that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined where i^2 = -1. A complex number is defined as a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or multiplied by treating i as a variable and using rules like i^2 = -1. Examples show how to solve equations and perform operations with complex numbers.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It provides details on:
1) Converting between number systems using methods like the place value method or remainder method. For example, converting between binary, octal, and hexadecimal systems involves grouping bits or replacing digits with their base-n equivalents.
2) Representing negative numbers in binary, including through sign-magnitude and two's complement representations. The two's complement of a binary number is calculated by complementing each bit and adding 1.
3) Hexadecimal arithmetic which works similarly to decimal arithmetic but with 16 symbols (0-9 and A-F) instead of 10 symbols.
Números Complejos:
1. Representación gráfica de los números complejos.
2. Conjugado y opuesto de un número complejo.
3. Operaciones con números complejos.
The document discusses different number systems used in mathematics. It begins by explaining that a number system is defined by its base, or the number of unique symbols used to represent numbers. The most common system is decimal, which uses base-10. Other discussed systems include binary, octal, hexadecimal, and those used historically by cultures like the Babylonians. Rational and irrational numbers are also defined. Rational numbers can be written as fractions of integers, while irrational numbers cannot.
The document contains questions related to CAT, MAT, GMAT entrance exams. It discusses various topics like probability, permutations and combinations, averages, ratios etc. and provides solutions to sample questions in 3-4 sentences each. The overall document aims to help exam preparation by providing practice questions on common quantitative topics.
The document discusses various types of numbers including natural numbers, whole numbers, and integers. It provides examples and explanations related to properties of these numbers. Some key points include:
- Natural numbers start from 1 and do not include 0, negative numbers, or decimals.
- Whole numbers include all natural numbers and 0.
- Integers include whole numbers and their negatives.
- Examples are provided to illustrate properties like divisibility, perfect squares, and solving word problems involving sums and products of numbers.
- The last part discusses Donkey's stable number based on his true and false answers to questions about divisibility, being a square, and the first digit. It is determined his number must
The document discusses different numeric representation systems used in computing, including decimal, binary, octal, and hexadecimal. It explains key concepts such as positional notation, where the position of a digit determines its value; radix/base, which is the set of digits used; and weighting factors, where each digit position is multiplied by a power of the base to determine its value. Integer, floating point, and fixed point numbers are also defined in terms of how many bits are used for integer and fractional portions.
1. The document defines different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It discusses their properties and how to represent them on a number line.
2. Rational numbers can be terminating or non-terminating recurring decimals. Irrational numbers have non-terminating, non-recurring decimals. Surds are irrational roots of rational numbers.
3. The document covers operations and rationalization of like surds, binomial quadratic surds, and the definition and properties of absolute value.
The document defines cardinality as the number of elements in a set. It provides examples of finite sets and their cardinalities. It discusses bijective functions, equinumerous sets, cardinal and ordinal numbers. It defines countable, uncountable, denumerable, and finite sets. It proves that the set of real numbers is uncountable, and discusses the continuum hypothesis.
A number system is a method for writing and representing numbers using digits or symbols in a consistent way. It allows unique representation of numbers and performing arithmetic operations. The main types of number systems are decimal, binary, octal, and hexadecimal, which use bases of 10, 2, 8, and 16 respectively. Number systems are used daily for tasks like making phone calls, budgeting, cooking, using elevators, shopping, and more. [/SUMMARY]
quantitative aptitude, maths
applicable to
Common Aptitude Test (CAT)
Bank Competitive Exam
UPSC Competitive Exams
SSC Competitive Exams
Defence Competitive Exams
L.I.C/ G. I.C Competitive Exams
Railway Competitive Exam
University Grants Commission (UGC)
Career Aptitude Test (IT Companies) and etc.
The document discusses operations on real numbers, including:
1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted.
2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different.
3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.
This document defines key concepts related to real numbers and sets. It discusses the properties of real numbers, including their characteristics as being ordered, integral, and infinite. It also defines natural numbers, integers, rational numbers, and irrational numbers. The document then covers basic set operations like union, intersection, difference, symmetric difference, and complement. It concludes by defining absolute value and describing inequalities and properties of real numbers like closure of addition/multiplication.
The document discusses different types of infinite sets and their cardinal numbers. It provides examples of sets that are countable, like the natural numbers and integers, by showing a one-to-one correspondence between each set and a subset. However, the set of real numbers is uncountable because there is an infinite number of decimals between any two real numbers, making it impossible to list them in a one-to-one fashion. Infinite sets can be defined as those that can be placed in a one-to-one correspondence with a proper subset of themselves.
The document discusses various math problems and their solutions using algebraic methods. It introduces concepts like writing word problems algebraically by denoting the unknown as a variable like x. It explains how to solve problems by doing the inverse operations in reverse order, and discusses how this approach can be written algebraically. The document also discusses the history of algebra and its origins from ancient Egyptian and Arab mathematicians.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
This document introduces complex numbers by representing real numbers as vectors on a number line. It defines the imaginary unit i as the operator that rotates a vector 90 degrees, making it possible to take the square root of negative numbers. Complex numbers are then represented as points in a plane, allowing them to be written as z = x + iy, where x is the real component and y is the imaginary component. Key properties of complex numbers like addition and multiplication are explained geometrically and through polar coordinates, with the unit circle playing an important role.
500 most asked apti ques in tcs, wipro, infos(105pgs)PRIYANKKATIYAR2
This document provides 100 numerical aptitude questions and solutions that are commonly asked in campus recruitment drives by companies like Infosys, TCS, CTS, Wipro and Accenture. The questions cover topics such as number systems, permutations, combinations, time and work problems, percentages, profit and loss, and geometry. Shortcuts and tips are provided to solve problems more quickly. The questions are divided into parts for each company and an index provides the topic distribution of questions for each company.
Convert Numbers to Words with Visual Basic.NETDaniel DotNet
The document discusses converting numbers to words in British English. It begins by covering single and two-digit numbers using lookup tables. For three-digit numbers, it distinguishes between cases where the last two digits are 00 or not. It then explains how to handle numbers with more digits by grouping them into threes and applying the rules recursively with powers of 1000 (thousand, million, billion, etc.). Visual Basic code is provided to implement the number-to-words conversion.
A number system represents numbers using digits or symbols in a consistent manner. It allows unique representation of numbers and performing arithmetic operations. Common number systems include decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Each position in a number represents a power of the base. For example, in the decimal number 1457, the 1 is in the thousands place with a value of 1000.
Conversion binary to decimal, Decimal to binary , octal to binary, hexadecimal to binary, binary to hex , decimal to hex ................ All conversion .
This document defines sets and set operations like union, intersection, difference, and symmetric difference. It discusses types of numbers like natural numbers, integers, rational numbers, irrational numbers, and real numbers. It also covers absolute value and absolute value inequalities. The key topics covered are the definition of a set, set operations and their symbols, classifications of different number types, and how to solve absolute value inequalities.
This document contains a presentation on how mathematics is used in everyday life. It provides examples of how math is applied in art, business, cooking, music, and other domains. It also summarizes the contributions of important mathematicians like Pythagoras, Ramanujan, and properties of rational numbers, squares, cubes, and their roots. The presentation aims to demonstrate the pervasive and practical role of mathematics in various activities.
This document is a summer math review packet for students entering 8th grade. It provides objectives and practice problems for key 8th grade math topics including order of operations, ratios and proportions, solving equations and inequalities, integers, fractions decimals and percents, geometry, statistics, mean median and mode, coordinate system and transformations, and GCF/LCM. The packet contains 50 total practice problems across these 10 math domains to help review and reinforce skills over the summer.
The document discusses the real number system. It defines rational and irrational numbers, and provides examples of each. Rational numbers can be written as fractions, while irrational numbers can only be written as non-terminating and non-repeating decimals. The document also covers operations like addition, subtraction, multiplication, and division on integers, using rules like keeping or changing signs depending on whether the signs are the same or different.
1. The document discusses real numbers and their properties, including subsets such as rational and irrational numbers.
2. Key topics covered include using a number line to graph and order real numbers, properties of number operations like closure and commutativity, and defining operations like addition, subtraction, multiplication and division.
3. Unit analysis is also introduced to check that units make sense when performing number operations for real-life applications.
The document discusses the history and development of the Arabic numeral system. It explains that the numerals originated from the Phoenicians but were popularized by Arabs. It then provides a theory about how the shapes of the numerals may have been derived from representing different numbers of angles, showing examples. The document also briefly outlines some key developments in the history of algebra.
The document provides notes on representing and operating on integers in math. It discusses using colored tiles to represent positive and negative integers, and how integers can be added or subtracted using tiles or a number line. Examples show modeling integer addition and subtraction using tiles and a number line. Students are provided practice problems for representing, adding, and subtracting integers.
Multiple Choice Type your answer choice in the blank next to each.docxadelaidefarmer322
Multiple Choice:
Type your answer choice in the blank next to each question number.
_____1.
Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
_____2.
Graph the ellipse and locate the foci.
A.
foci at (0,
6) and (0, -6)
C.
foci at (
, 0) and (-
, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0,
5) and (0, -5)
_____3.
Solve the system by the substitution method.
2y - x = 5
x2 + y2 - 25 = 0
A.
B.
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
_____4.
Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x - 3,
f(x)
A. 5
B. 25
C. 2
D. 22
_____5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C.
∅
D. {(2, -7, -1)}
_____6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z
= -5
x - y + 3z
= -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
_____7.
A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Graph an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the graph to quadrant I only.
A.
C.
B.
D.
Short Answer Questions:
Type your answer below each question. Show your work.
8
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
9
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month a.
1. The document defines different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It discusses their properties and how to represent them on a number line.
2. Rational numbers can be terminating or non-terminating recurring decimals. Irrational numbers have non-terminating, non-recurring decimals. Surds are irrational roots of rational numbers.
3. The document covers operations and rationalization of like surds, binomial quadratic surds, and the definition and properties of absolute value.
The document defines cardinality as the number of elements in a set. It provides examples of finite sets and their cardinalities. It discusses bijective functions, equinumerous sets, cardinal and ordinal numbers. It defines countable, uncountable, denumerable, and finite sets. It proves that the set of real numbers is uncountable, and discusses the continuum hypothesis.
A number system is a method for writing and representing numbers using digits or symbols in a consistent way. It allows unique representation of numbers and performing arithmetic operations. The main types of number systems are decimal, binary, octal, and hexadecimal, which use bases of 10, 2, 8, and 16 respectively. Number systems are used daily for tasks like making phone calls, budgeting, cooking, using elevators, shopping, and more. [/SUMMARY]
quantitative aptitude, maths
applicable to
Common Aptitude Test (CAT)
Bank Competitive Exam
UPSC Competitive Exams
SSC Competitive Exams
Defence Competitive Exams
L.I.C/ G. I.C Competitive Exams
Railway Competitive Exam
University Grants Commission (UGC)
Career Aptitude Test (IT Companies) and etc.
The document discusses operations on real numbers, including:
1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted.
2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different.
3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.
This document defines key concepts related to real numbers and sets. It discusses the properties of real numbers, including their characteristics as being ordered, integral, and infinite. It also defines natural numbers, integers, rational numbers, and irrational numbers. The document then covers basic set operations like union, intersection, difference, symmetric difference, and complement. It concludes by defining absolute value and describing inequalities and properties of real numbers like closure of addition/multiplication.
The document discusses different types of infinite sets and their cardinal numbers. It provides examples of sets that are countable, like the natural numbers and integers, by showing a one-to-one correspondence between each set and a subset. However, the set of real numbers is uncountable because there is an infinite number of decimals between any two real numbers, making it impossible to list them in a one-to-one fashion. Infinite sets can be defined as those that can be placed in a one-to-one correspondence with a proper subset of themselves.
The document discusses various math problems and their solutions using algebraic methods. It introduces concepts like writing word problems algebraically by denoting the unknown as a variable like x. It explains how to solve problems by doing the inverse operations in reverse order, and discusses how this approach can be written algebraically. The document also discusses the history of algebra and its origins from ancient Egyptian and Arab mathematicians.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
This document introduces complex numbers by representing real numbers as vectors on a number line. It defines the imaginary unit i as the operator that rotates a vector 90 degrees, making it possible to take the square root of negative numbers. Complex numbers are then represented as points in a plane, allowing them to be written as z = x + iy, where x is the real component and y is the imaginary component. Key properties of complex numbers like addition and multiplication are explained geometrically and through polar coordinates, with the unit circle playing an important role.
500 most asked apti ques in tcs, wipro, infos(105pgs)PRIYANKKATIYAR2
This document provides 100 numerical aptitude questions and solutions that are commonly asked in campus recruitment drives by companies like Infosys, TCS, CTS, Wipro and Accenture. The questions cover topics such as number systems, permutations, combinations, time and work problems, percentages, profit and loss, and geometry. Shortcuts and tips are provided to solve problems more quickly. The questions are divided into parts for each company and an index provides the topic distribution of questions for each company.
Convert Numbers to Words with Visual Basic.NETDaniel DotNet
The document discusses converting numbers to words in British English. It begins by covering single and two-digit numbers using lookup tables. For three-digit numbers, it distinguishes between cases where the last two digits are 00 or not. It then explains how to handle numbers with more digits by grouping them into threes and applying the rules recursively with powers of 1000 (thousand, million, billion, etc.). Visual Basic code is provided to implement the number-to-words conversion.
A number system represents numbers using digits or symbols in a consistent manner. It allows unique representation of numbers and performing arithmetic operations. Common number systems include decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Each position in a number represents a power of the base. For example, in the decimal number 1457, the 1 is in the thousands place with a value of 1000.
Conversion binary to decimal, Decimal to binary , octal to binary, hexadecimal to binary, binary to hex , decimal to hex ................ All conversion .
This document defines sets and set operations like union, intersection, difference, and symmetric difference. It discusses types of numbers like natural numbers, integers, rational numbers, irrational numbers, and real numbers. It also covers absolute value and absolute value inequalities. The key topics covered are the definition of a set, set operations and their symbols, classifications of different number types, and how to solve absolute value inequalities.
This document contains a presentation on how mathematics is used in everyday life. It provides examples of how math is applied in art, business, cooking, music, and other domains. It also summarizes the contributions of important mathematicians like Pythagoras, Ramanujan, and properties of rational numbers, squares, cubes, and their roots. The presentation aims to demonstrate the pervasive and practical role of mathematics in various activities.
This document is a summer math review packet for students entering 8th grade. It provides objectives and practice problems for key 8th grade math topics including order of operations, ratios and proportions, solving equations and inequalities, integers, fractions decimals and percents, geometry, statistics, mean median and mode, coordinate system and transformations, and GCF/LCM. The packet contains 50 total practice problems across these 10 math domains to help review and reinforce skills over the summer.
The document discusses the real number system. It defines rational and irrational numbers, and provides examples of each. Rational numbers can be written as fractions, while irrational numbers can only be written as non-terminating and non-repeating decimals. The document also covers operations like addition, subtraction, multiplication, and division on integers, using rules like keeping or changing signs depending on whether the signs are the same or different.
1. The document discusses real numbers and their properties, including subsets such as rational and irrational numbers.
2. Key topics covered include using a number line to graph and order real numbers, properties of number operations like closure and commutativity, and defining operations like addition, subtraction, multiplication and division.
3. Unit analysis is also introduced to check that units make sense when performing number operations for real-life applications.
The document discusses the history and development of the Arabic numeral system. It explains that the numerals originated from the Phoenicians but were popularized by Arabs. It then provides a theory about how the shapes of the numerals may have been derived from representing different numbers of angles, showing examples. The document also briefly outlines some key developments in the history of algebra.
The document provides notes on representing and operating on integers in math. It discusses using colored tiles to represent positive and negative integers, and how integers can be added or subtracted using tiles or a number line. Examples show modeling integer addition and subtraction using tiles and a number line. Students are provided practice problems for representing, adding, and subtracting integers.
Multiple Choice Type your answer choice in the blank next to each.docxadelaidefarmer322
Multiple Choice:
Type your answer choice in the blank next to each question number.
_____1.
Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
_____2.
Graph the ellipse and locate the foci.
A.
foci at (0,
6) and (0, -6)
C.
foci at (
, 0) and (-
, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0,
5) and (0, -5)
_____3.
Solve the system by the substitution method.
2y - x = 5
x2 + y2 - 25 = 0
A.
B.
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
_____4.
Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x - 3,
f(x)
A. 5
B. 25
C. 2
D. 22
_____5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C.
∅
D. {(2, -7, -1)}
_____6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z
= -5
x - y + 3z
= -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
_____7.
A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Graph an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the graph to quadrant I only.
A.
C.
B.
D.
Short Answer Questions:
Type your answer below each question. Show your work.
8
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
9
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month a.
This document outlines the mathematics curriculum for the first year of secondary school. It includes 6 units covering topics like integers, angles, rational numbers, polygons, functions, and statistics/probability. For each unit it lists the main concepts that will be covered. The document then provides examples of questions and activities related to integers, including operations, absolute value, opposites, factorization, and the use of integers to represent real-world situations.
The document contains a series of math word problems and questions. It tests ordering numbers from least to greatest, finding averages, converting between percentages and decimals, solving multi-step word problems, identifying geometric shapes and properties, and determining logical sequences and patterns. The questions cover a wide range of elementary and middle school math concepts.
The document discusses the development and properties of complex numbers. Integers were originally used to count whole objects, then fractions were developed to represent portions of wholes. Real numbers were created to represent all numbers that can be written as decimals. However, some equations like x^2=-1 do not have real solutions. To solve these, an imaginary number i=√-1 was defined, where i^2=-1. A complex number is defined as a number of the form a+bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or subtracted by treating i as a variable.
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1. DIALOGUE ON 2-COLOR
NUMBER
Armahedi Mahzar (c) 2008
Part 1:
2-Color Numbers
Ni Suiti was in the veranda when her granddaughter Si Nessa come to tell her
experiences when she went to Numberland brought by her aunt Mak Retitia. She told Ni
Suiti that the Numberland is a wonderful land. In one region of the island he found out
that in there there are colorful numbers.
Si Nessa: Grandma, I have never known that numbers in the amazing Numberland
where numbers have colors.
Ni Suiti: Do you mean that we can find numbers with the same value but has different
colors?
Si Nessa: Yes, but more interesting is their properties. When two numbers meld into
another number then they become another colored number.
Ni Suiti: What is the color of the new number?
Si Nessa: They get a new color or stay in their original color according to the way they
meld: addition or multiplication.
Ni Suiti: What is the rules for color change for the addition?
Si Nessa: If the colors are similar, the result is similar color
Ni Suiti: So, red one plus red three is red five.
2 + 3 = 5
Black two plus Black one is Black three.
2 + 1 = 3
2. But what if we add two numbers of
different colors?
Si Nessa: It yields a duet which is a pair of numbers of different colors
Ni Suiti: How many kinds of colored number are there in Numberland?
Si Nessa: I was in two colored number region of the land. They call it Bichromic
province which is populated by two kind of number: Black and Red. But there are many
other regions which is occupied by numbers with more than two colors. I have never been
there.
Ni Suiti: Now what is the multiplication rules of of two colored numbers?
Si Nessa: That's really simple. Black number does not change the color of the number it
multiply. Red number change it.
This rule can be simplified to:
Similar colors are multiplied to Black
Different colors are multiplied to Red.
Ni Suiti: That means
Black 2 times Red 3 is equal to Red 6
2 X 3 = 6
and Red 2 times Black 3 is equal to Red 6
2 X 3 = 6
and Red 2 times Red 3 is equal to Black 6
2 X 3 = 6
Si Nessa: That's correct. But Dichromatic province is more populated by pairs of
numbers of different colors called duets.
Ni Suiti: I wonder what are the rules of duet melding
Si Nessa: If it is addition, then any colored number of the first duet will add to the
member of the other duet of the same color. In short: Equal colored number add up the
same color
Ni Suiti:
You mean
2 + 3 +
1 + 2 =
3. 3 + 5
Si Nessa: You are right grandma.
Ni Suiti: Now how can we multiply two duets?
Si Nessa: We multiply every colored number of the first duet to every colored number
of the second duet and add all the results up. In short: Add up all possible multiplications
Ni Suiti:
(1 + 2) X (3 + 2) =
=1 X 3 + 1 X 2 + 2 X 3 + 2 X 2 =
= 3 + 2 + 6 + 4 =
= 7 + 8
Si Nessa: Right again Grandma
Ni Suiti: What about negative numbers
Si Nessa: They're all has the same rules.
Ni Suiti: So we can have Red minus 2 = minus Red 2 = - 2
Si Nessa: Good Grandma!
Ni Suiti: So we will make a colored number dissapear if we add it to its negative.
Red 3 + Red minus 3 =
Si Nessa: Yes. That's the magic of colored numbers
.
Ni Suiti: In summary,
For single color number addition of similar color singlet yields similar colored singlet.
Different color numbers add to a duet.
Adding colorful number duets is adding their members colorwise.
Multiplying by black singlet does not change color.
Multiplying by red singlet changes the color.
Mutiplying duets is adding all the multiplication of of their colored members.
Si Nessa: Good Grandma. But, sorry I have to go home now. Because I have to prepare
my self for tomorrow Journey to the another province of the Numberland: the Bichromic
Two. Bye, now.
4. Part 2:
2-Color Arithmetic
Ni Suiti was in in the company of his husband, Ki Algo, the grandfather of Si Nessa.
She told him about the discoveries of their granddaughter in the Bichromic One. She like
to know his opinion to his granddaughter's discoveries. Then Ni Suiti told Ki Algo
about the rules of color transformation due to the arithmetic operations as it was told by
Si Nessa
Ki Algo: I am surprised, but I think the table for addition is the following
+ c d
a a + c a + d
b c + b
(b +
d)
Ni Suiti: That's cool.
Ki Algo: From the table it can be shown that the multiplication has the following
property:
If a, b and c are colored number then
(1) a + b = b + a
(2) a + (b + c) = (a + b) + c
Ni Suiti: So the ordering of the addition does not matter.
Ki Algo: The multiplication for colored number singlets is summarized in the following
table
X c d
a ac (ad)
b cb (bd)
Ni Suiti: That's also cool.
Ki Algo: From the table it can be shown that the multiplication has the following
property:
If a, b and c are colored number then
(1) ab = ba
5. (2) a(bc) = (ab)c
(3a) a(b+c) = ab + ac
(3b) (a+b)c = ac + bc
Ni Suiti: So the muliplication is indifferent of ordering of terms and it is both left and
right distributive to addition.
Ki Algo: It can easily proven that the 2-color number system has a multiplicative unit:
Black 1 or 1
1 (x + y) = (x + y) 1 for any duet x + y
Ni Suiti: So is both left and right unit.
Ki Algo: I can also prove that there also have an additive unit: Zero.
Zero = x - x which has the following property
Zero + a = a + Zero = a
Ni Suiti: I think zero is colorless
Ki Algo: From the table I can derive the formula for multiplying two colored number
duets
(Black a + Red b)(Black c + Red d) = Black (ac+bd) + Red (ad+bc)
If the duet x + y is abbreviated as (x, y) ,then the rule of multiplication is
(a,b)(c,d) = (ac+bd, ad+bc)
Ni Suiti: Simple formula to represent the long table. But the wonderful colors is lost.
What a pity.
Ki Algo: Conclusively, the 2-color numbers form what the mathematician called Ring.
Of course The mathematician Ring is not some thing you can wear in your finger, it is a
collection of numbers with two compositions (+ and .) which follow certain axioms.
Dichromic numbers form a Ring because for all 2-color numbers a, b and c follow the
following eight Axioms
Four Axioms of Addition
(R1) (a + b) + c = a + (b + c) ( the addition + is associative)
(R2) Zero + a = a (existence of identity element for addition)
6. (R3) a + b = b + a (+ is commutative)
(R4) for each 2-color number a there is a 2-color number −a such that a + (−a) =
(−a) + a = Zero
(−a is the additive inverse element of a)
Two Axioms of Multiplication
(R5) (a . b) . c = a . (b . c) (the multiplication . is associative)
(R6) 1 . a = a . 1 = a (existence of identity element for multiplication)
Two Axioms of Distribution
(R7) a . (b + c) = (a . b) + (a . c) (left distributivity of multiplication)
(R8) (a + b) . c = (a . c) + (b . c) (right distributivity of multiplication)
Ni Suiti: Wow. That's right but I lost the visual beauty of the colored. numbers.
Ki Algo: Yes, but now you gain the beauty of logical consistency.
7. Part 3 :
Strange Numbers
Ni Suiti was so bewildered by Ki Algo exposition of Ring as the arithmetic
structure of 2-color numbers. She thought there is nothing strange with that at all.
All the Ring axioms are also followed by real numbers. So real numbers
arithmetic is also a Ring.
Ni Suiti: I suspects that the 2-colored numbers has similar arithmetic as the real
numbers.
Ki Algo: Oh, no. There are duet numbers which is squared to themselves. z2 = z
Ni Suiti: I think that is not so. Real number arithmetic has those too. Zero and Unity is
such a number
Ki Algo: Well the 2-color numbers have other numbers squared to themselves beside
them.
Ni Suiti: What numbers?
Ki Algo: They are z1= 1/2 + 1/2 and
z2= 1/2 - 1/2
Ni Suiti: My goodness. There are two of them.
Ki Algo: Mathematicians called the number as Idempotent number. Idem means
equal, potent means power. Because if you power them with any number then the results
will be equal to themselves. zn = z with n any integer.
Ki Algo: OK you know now that there are two really duet numbers that square
themselves to themselves. Now try to multiply them to each other.
Ni Suiti:
z1.z2= (1/2 + 1/2)(1/2 - 1/2)=Zero
Oh! It is very strange. In 2-color arithmetic, zero is equal to multiplication of two non
zero 2-colored numbers. No nonzero real numbers will multiply themselves to zero.
Ki Algo: They called by mathematician as Zero Divisors. In fact there are infinity of
8. zero divisors. All multiple of z1 and z2 are zero divisors. (3 + 3)(5 -
5)=Zero for example. The existence of strange numbers, Idempotents and Zero
Divisors, shows us that 2-color arithmetic is not similar in structure to real number
arithmetic.
Ni Suiti: OK, I am wrong. The arithmetic of 2-Color Numbers is not similar to the
arithmetic of the real numbers. They have more idempotents and infinity of zero divisors.
Ki Algo: Actually, mathematicians called the arithmetic of real number as Field and the
arithmetic of 2-color number as commutative Ring with unity (which is Black 1 as
unity). A Field is a commutative Ring with unity containing no Zero Divisor.
Ni Suiti: So, the 2-ColorNumber algebra is unique because it has unique structure as the
ring with infinite zero divisor and a pair of idempotent.
Ki Algo: No, it's not unique. The Ring of 2-Color Numbers has similar arithmetic
structure to counter-complex numbers with two units 1 and ε where both units
are squared to one. Each of them equivalent to 1 and 1 . Other arithmetic similar in
structure to the 2-Color arithmetic is the Group Algebra based on the 2-element
reflection group.
Ni Suiti: Anyway, I think all 2-color Numbers has common arithmetic property.
Ki Algo: I do not think so. Please wait for Si Nessa after her travel to Bichromic Two
and beyond. See what she found there.
Ni Suiti: Ok. We will see who is right. You or me?
Note:
I am not a mathematician, just a retired physicist. Exploring new kinds of number
is just my hobby. Please correct me if I am wrong.
Thank you.
Arma
9. Part 4:
Arithmetic Similarity
Si Nessa had returned from Bichromic Two which is a province in Numberland.
Bichromic Two is populated by 2-color numbers consisting black and pink
numbers. Bichromic One, that she had visited before, is a province populated
by 2-color numbers consisting of black and pink numbers. Si Nessa found out
that both regions have similar rules of composition except for the multiplication
rules for the same colored numbers. The pink number times a pink number is a
negative black number, while she know before that the red number times a red
number is simply black number. That's why she called the Black-Pink number is
a twisted 2-color number. She told her grandma Ni Suiti about her findings in the
company of her brother Si Emo and her Grandpa Ki Algo:..
Ni Suiti: Nessa, your discovery interesting, but I'll let you know that your grandpa Ki
Algo reformulate my verbal rules for 2-color number multiplication with the following
table.
X c d
a ac (ad)
b cb (bd)
Si Nessa: Wow, it is difficult for me to memorize.
Ni Suiti: For you, if you remember distribution axiom, I will simplify your grandpa's
table by changing all letters with number 1, then the multiplication table can be
simplified into
X 1 1
1 1 1
1 1 1
in more simplified form
1 1
1 1
Si Nessa: Yes. That is a simpler table.
10. Ni Suiti: We can simplify the table more, by drawing just a 2 x 2 checker board with just
two colors.
For black-red number, the table will be represented by
Si Nessa: That's beautiful and very easy to memorize. Now what about the Black-Pink
numbers that I found in Bichromic Two.
Ni Suiti: The multiplication checker board for Black-PinkNumbers is
o
o
where the white ring is representing the minus sign.
Si Nessa: How can I use the wonderful table
Ni Suiti: We can replace the formula (a + b)(c + d) = (ac+bd)+(ad+bd)
in this simple steps
• Make a multiplication checkerboard
• Put column (a,b) on the left of the table
• Put row (c,d) on the top of the table
• Multiply the elements of row and the column and
multiply them with the sign in the suitable board little boxes
• Add up all the elements of the table using the rule of addition.
Ni Suiti: I think this algorithm is easier for people's mind
who is stronger in intuition, like me, rather in logic, like Ki Algo.
Your Grandpa's algebraic formula is suitable for left-brainer
my diagramatic algorithm is suitable for right-brainer.
Si Emo: OK my brain is like grandpa's. For me the algorithm is too complicated. How is
about that Grandpa?
Ki Algo: Good. Your grandmother Ni Suiti has make the 2-color number multiplication
more visual. I will reformulate your grandmother's checkerboard with numeral and
letters. Let us symbolized black box with 1, the red box with the symbol e and pink
box with the simbol i
11. Si Emo: Grandma's multiplication black-red checker board will be symbolized by the
following table
1 e
e 1
and Grandma's multiplication black-pink checker board will be symbolized with the
following table
1 i
i -1
Si Nessa: Oh! So simple table. That's really a very simple table.
Ki Algo: Formulated as such symbolic table, mathematicians will directly know that
Black-Red numbers is nothing but another form of hyper-complex numbers and Black-
Pink numbers is nothing but another form of complex numbers.
Ni Suiti: In my words. Hyper-complex numbers is nothing but another form of Black-
Red numbers and complex numbers is nothing but another form of Black-Pink
numbers.
Ki Algo: In other words the arithmetic system of black-red numbers is similar to the
arithmetic of hyper-complex number ring and the arithmetic system of black-pink
numbers is similar to the arithmetic of complex number field. Mathematicians found out
that all the field axioms for the real number arithmetic are also followed by complex
numbers arithmetic.
Ni Suiti: Why is that black pink numbers form a field arithmetics?
Ki Algo: No zero divisors exist in its arithmetic due to the presence of minus sign in its
unit multiplication table.
Ni Suiti: OK. Now, by using my two color checkerboard we can teach the complex and
hyper-complex arithmetic to primary school kids as 2-color number arithmetic.
Ki Algo: That's a great idea. Hopefully teachers will take your advice.