Glory to God in the highest; and on earth, peace to people on whom His favor rests! - Luke 2:14

The Joy of a Teacher is the Success of his Students. - Samuel Chukwuemeka

Numbers and Notations

Samdom For Peace

I greet you this day,

First, read the notes. Second, view the videos. Third, solve the questions. Fourth, check your answers with the calculators.

I wrote the codes for the calculators using JavaScript, a client-side scripting language and AJAX, a JavaScript library. Please use the latest Internet browsers. The calculators should work.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me. If you are my student, please do not contact me here. Contact me via the school's system.

Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng., A.A.T, M.Ed., M.S

Objectives


Students will:

(1.) Define natural numbers.

(2.) Define whole numbers.

(3.) Define integers.

(4.) Define fractions.

(5.) Define decimals.

(6.) Define rational numbers.

(7.) Define irrational numbers.

(8.) Define real numbers.

(9.) Define complex numbers.

(10.) Define even numbers.

(11.) Define odd numbers.

(12.) Define prime numbers.

(13.) Define composite numbers.

(14.) Define perfect squares.

(15.) Define perfect cubes.

(16.) Define perfect numbers.

(17.) Define abundant numbers.

(18.) Write the symbols for some set of numbers.

(19.) Discuss the notations for representing real numbers.

(20.) Write whole numbers as rational numbers.

(21.) Write mixed numbers as rational numbers.

(22.) Determine if a given rational number is a repeating decimal.

(23.) Determine if a given rational number is a terminating decimal.

(24.) Order real numbers from least to greatest.

(25.) Order real numbers from greatest to least.

(26.) Add, subtract, multiply, and divide numbers.

(27.) Perform the order of operations involving numbers.

(28.) Solve real-world problems involving real numbers.


Skills Measured/Acquired


(1.) Use of prior knowledge

Ask students to list all the types of numbers they have ever used.
In what ways have they used these numbers? What did they use the numbers to do?

(2.) Critical Thinking

Ask students to list any other types of numbers they think that exists.
What other ways do they think they might use these numbers?
Why have they not used those numbers yet?
Some students may ask you to explain complex numbers.
Of what use are complex numbers? Are they "ever gonna" ever going to use complex numbers?
What disciplines of life may use complex numbers (the imaginary part of complex numbers)?
Can all real numbers be written as complex numbers?

(3.) Interdisciplinary connections/applications

Ask students to list their disciplines.
What kinds of numbers would they use primarily in their respective disciplines?
Why would they use those kinds of numbers?
In what way would those numbers be used?

(4.) Technology

Ask students if they could write programs that can list the different kinds of numbers.
Based on their responses, determine the next questions to ask. Would they be interested in writing such programs as independent class projects?
Another use of technology would be to write a program that tells the kind of number when a user inputs a number.

(5.) Active participation through direct questioning

Encourage students to ask questions. If they do not ask, ask them.
Please answer any questions students ask. If you do not know the answer, inform them you will get back with them after finding the answers.

(6.) Student collaboration in Final Project

Depending on the "student population", the final project may be independent project, or group project.
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Vocabulary Words


natural number, counting number, whole number, fraction, numerator, denominator, proper fraction, improper fraction, rational number, integer, mixed number, decimal, terminating decimal, exact decimal, repeating decimal, recurring decimal, irrational number, real number, complex number, arithmetic, arithmetic operators, sum, difference, product, quotient, augend, addend, minuend, subtrahend, multiplier, multiplicand, factor, dividend, divisor, positive, negative, nonpositive, nonnegative, constant, number, variable, term, add, subtract, multiply, divide, expression, equation, equal, equality, inequality, left hand side (LHS), right hand side (RHS), end points, set notation, interval notation, braces, brackets, parenthesis, even number, odd number, prime number, perfect sqaures, perfect cubes, perfect number, abundant number, excessive number

Symbols and Meanings


  • $\mathbb{N}$ = set of natural numbers
  • $\mathbb{W}$ = set of whole numbers
  • $\mathbb{Z}$ = set of integers
  • $\mathbb{Z}_+$ = set of positive integers
  • $\mathbb{Z}_-$ = set of negative integers
  • $\mathbb{Q}$ = set of rational numbers
  • $\mathbb{I}$ = set of irrational numbers
  • $\mathbb{R}$ = set of real numbers
  • $\mathbb{C}$ = set of complex numbers
  • $\mathbb{P}$ = set of prime numbers
  • { } (braces) = used in set notation
  • [ ] (brackets) and ( ) (parenthesis) = used in interval notation
  • [ ] = closed interval (closed at both ends)
  • ( ) = open interval (open at both ends)
  • [ ) = half-closed half-open interval (closed at $1^{st}$ end, open at $2^{nd}$ end)
  • ( ] = half-open half-closed interval (open at $1^{st}$ end, closed at $2^{nd}$ end)
  • [c, d] = closed interval - includes $c$ and $d$
  • (c, d) = open interval - excludes $c$ and $d$
  • [c, d) = half-closed half-open interval - includes $c$, excludes $d$
  • (c, d] = half-open half-closed interval - excludes $c$, includes $d$
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Definitions


A natural number is any positive integer.
It is also known as a counting number. It is a number you can count.
It does not include zero.
It does not include the negative integers.
It is not a fraction.
It is not a decimal.

A whole number is any nonnegative integer.
It includes zero and the positive integers.
It does not include the negative integers.
It is not a fraction.
It is not a decimal.

An integer is any whole number or its opposite.
Integers include the whole numbers and the negative values of the whole numbers.

The basic arithmetic operators are the addition symbol, $+$, the subtraction symbol, $-$, the multiplication symbol, $*$, and the division symbol, $\div$

Augend is the term that is being added to. It is the first term.

Addend is the term that is added. It is the second term.

Sum is the result of the addition.

$$3 + 7 = 10$$ $$3 = augend$$ $$7 = addend$$ $$10 = sum$$

Minuend is the term that is being subtracted from. It is the first term.

Subtrahend is the term that is subtracted. It is the second term.

Difference is the result of the subtraction.

$$3 - 7 = -4$$ $$3 = minuend$$ $$7 = subtrahend$$ $$-4 = difference$$

Multiplier is the term that is multiplied by. It is the first term.

Multiplicand is the term that is multiplied. It is the second term.

Product is the result of the multiplication.

$$3 * 10 = 30$$ $$3 = multiplier$$ $$10 = multiplicand$$ $$30 = product$$

Dividend is the term that is being divided. It is the numerator.

Divisor is the term that is dividing. It is the denominator.

Quotient is the result of the division.

Remainder is the term remaining after the division.

$$12 \div 7 = 1 \:R\: 5$$ $$12 = dividend$$ $$10 = divisor$$ $$1 = quotient$$ $$5 = remainder$$

An even number is a whole number that is divisible by $2$ without a remainder.
Examples include: $2, 4, 6, 8, 10, 12, 70, 84$, etc.

An odd number is any integer that is not a multiple of $2$.
Examples include: $1, 3, 5, 7, 9, 75$, etc.

A prime number is a whole number greater than $1$, which is divisible by only $1$ and itself without a remainder.
Examples include: $2, 3, 5, 7, 11, 13, 17, 19$, etc.

A perfect square is the square of a rational number.
Examples include: $1, 4, 9, 16, 25, 36$, etc.
$ 1^2 = 1 \\[3ex] 2^2 = 4 \\[3ex] 5^2 = 25 $

A perfect cube is the cube of a rational number.
Examples include: $1, 8, 27, 64, 125$, etc.
$ 1^3 = 1 \\[3ex] 2^3 = 8 \\[3ex] 5^3 = 125 $

A perfect number is a positive integer that is equal to the sum of its "proper positive divisors".
"Proper positive divisors" refers to all divisors excluding that number.
Examples include: $12, 18, 20, 24$, etc.
$ 6 = 1 + 2 + 3 \\[3ex] 28 = 1 + 2 + 4 + 7 + 14 $

An abundant number is a positive integer in which the sum of its proper positive divisors is greater than the number.
"Proper positive divisors" refers to all divisors excluding that number.
An abundant number is also called an excessive number.
Examples include: $6, 28$, etc.
$ For\:\: 12 \\[3ex] Proper\:\: divisors\:\: of\:\: 12 = 1, 2, 3, 4, 6 \\[3ex] Sum = 1 + 2 + 3 + 4 + 6 = 16 \\[3ex] 16 \gt 12 \\[5ex] For\:\: 18 \\[3ex] Proper\:\: divisors\:\: of\:\: 18 = 1, 2, 3, 6, 9 \\[3ex] Sum = 1 + 2 + 3 + 6 + 9 = 21 \\[3ex] 21 \gt 18 $

A fraction is a part of a whole.
It is the part of something out of a whole thing.
It is also seen as a ratio.
It is also seen as a quotient.
The numerator is the part.
It is the "top" part of the fraction.
The denominator is the whole.
It is the "bottom" part of the fraction.
A proper fraction is a fraction whose numerator is less than the denominator.
An improper fraction is a fraction whose numerator is greater than or equal to the denominator.

A mixed number is a combination of an integer and a proper fraction.


A decimal is a linear array of digits that represent a real number, expressed in a decimal system with a decimal point; and in which every decimal place indicates a multiple of negative power of 10.
A terminating decimal is a decimal with a finite number of digits.
A terminating decimal is also known as an exact decimal.
A repeating decimal is a decimal in which one or more digits is repeated indefinitely in a pattern or sequence.
A repeating decimal is also known as a recurring decimal.
A non-repeating decimal is a decimal in which there is no sequence of repeated digits indefinitely.
A non-repeating decimal is also known as a non-recurring decimal.

A rational number is any number that can be written as a fraction where the denominator is not equal to zero.
You can also say that a rational number is a ratio of two integers where the denominator is not equal to zero.
A rational number is a number that can be written as: $${c\over d}$$ where $c, d$ are integers and $d \neq 0$
A rational number can be an integer.
It can be a terminating decimal. Why?
It can be a repeating decimal. Why?
It cannot be a non-repeating decimal. Why?
Ask students to tell you what happens if the denominator is zero.

An irrational number is a number that cannot be expressed as a fraction, terminating decimal, or repeating decimal.
When you compute irrational numbers, they are non-repeating decimals.

A real number is any rational or irrational number.
It includes all numbers that can be found on the real number line.

A complex number is a number that can be expressed in the form of $a + bi$ where $a \:and\: b$ are real numbers, and $i$ is an imaginary number equal to the square root of $-1$.

A constant is something that does not change. In mathematics, numbers are usually the constants.

A variable is something that varies (changes). In Mathematics, alphabets are usually the variables.

A mathematical expression is a combination of variables and/or constants using arithmetic operators.

A mathematical equation is an equality of two terms - the term or expression on the LHS (Left Hand Side) and the term or expression on the RHS (Right Hand Side).
This implies that we should always check the solution of any equation that we solve to make sure the LHS is equal to the RHS.
Whenever we solve for the variable in "any" equation, how do we know we are correct? CHECK!

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Classify Numbers


Classify these numbers according to the "standard types" of numbers.

Questions Answers (Click "Answers:" to show/hide answers)
$$30$$
Answers:
$${2\over5}$$
Answers:
$${5\over2}$$
Answers:
$$3{2\over5}$$
Answers:
$$-30$$
Answers:
$$0$$
Answers:
$$2.25$$
Answers:
$$1.3333333$$
Answers:
$$1.\overline{3}$$
Answers:
$$0.4527272727$$
Answers:
$$0.45\overline{27}$$
Answers:
$$0.23759456287$$
Answers:
$$π$$
Answers:
$$3 + 7i$$
Answers:



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Add and Subtract Integers


NOTE:
$(1.)$ positive, add, plus mean the same thing. There is no difference.
$(2.)$ negative, subtract, minus mean the same thing. There is no difference.

Clear all confusions.
$2 + 3$ is read as: two plus three, two added to three, the sum of two and three

$2 + -3$ = $2 + (-3)$ is read as: two plus negative three, two plus minus three (not "two plus-minus three" $2\pm3$), two added to negative three, two added to minus three, the sum of two and minus three, the sum of two and negative three

$2 - 3$ is read as: two minus three, two subtract three, three subtracted from two, the difference between two and three

$2 - - 3$ = $2 - (-3)$ is read as: two minus minus three, two minus negative three, negative three subtracted from two, two subtract negative three, two subtract minus three, the difference between two and negative three, the difference between two and minus three

$-2 - 3$ is read as: minus two minus three, negative two negative three, negative two minus three, minus two negative three, three subtracted from negative two, three subtracted from minus two, minus two subtract three, negative two subtract three, the difference between negative two and three, the difference between minus two and positive three

$-2 - -3$ = $-2 - (-3)$ is read as: minus two minus minus three, negative two minus negative three, negative two minus minus three, negative three subtracted from negative two, negative two subtract negative three, the difference between minus two and minus three, the difference between negative two and negative three, the difference between minus two and negative three

First Method: Basic Rules
Single Sign Between Integers
First Case: For two integers with "same/common" signs (positive-positive or negative-negative);
Add the integers
Put the common sign.

Example 1
$(1.)\:\: 2 + 3$
This is the same as saying $+2 + 3$
Add the two integers. $2 + 3 = 5$
Put their common sign. Their common sign is $+$
So, the answer is $+5$ = $5$

Example 2
$(2.)\:\: -2 - 3$
Add the two integers. $2 + 3 = 5$
Put their common sign. Their common sign is $-$
So, the answer is $-5$

Second Case: For two integers with "different" signs (positive-negative or negative-positive);
Subtract the smaller integer from the bigger integer
Put the sign of the bigger integer.

Example 3
$(3.)\:\: 2 - 3$
This is the same as saying $+2 - 3$
Smaller integer = $2$
Bigger integer = $3$
Subtract $2$ from $3$ means $3 - 2= 1$
Put the sign of $3$. The sign of $3$ is $-$. That is the sign before $3$
So, the answer is $-1$

Example 4
$(4.)\:\: -2 + 3$
Smaller integer = $2$
Bigger integer = $3$
Subtract $2$ from $3$ means $3 - 2= 1$
Put the sign of $3$. The sign of $3$ is $+$. That is the sign before $3$
So, the answer is $+1$ = $1$


Second Method: Samuel Chukwuemeka's (SamDom For Peace) Method
Method of "have" and "owe"

Single Sign Between Integers
$+$ = "have"
$-$ = "owe"

Examples
$(1.)\:\: 2 + 3 = +2 + 3$
Say you had $2 yesterday and you have $3 today.
So, altogether; you have $5
$2 + 3 = 5$

$(2.)\:\: 2 - 3 = +2 - 3$
Say you have $2, but you owe someone $3
The person asks you to pay back. You gave the person the $2 and said that was all you had.
Are you still owing the person, or do you have anything with you?
You still owe! That means the answer must be $-$ because you still "owe".
How much are you still owing? $1
$2 - 3 = -1$

$(3.)\:\: -2 + 3 = -2 + 3$
Say you owe someone $2, but you have $3
The person asks you to pay back. You gave the person the $2.
Are you still owing the person, or do you have anything with you?
You still have something left - a dollar! That means the answer must be $+$ because you still "have".
How much do you have? $1
$-2 + 3 = 1$

$(4.)\:\: -2 - 3$
Say you owe Mr. $A$ $2 yesterday and you owe Mr. $B$ $3.
So, altogether; you owe both of them. That means the answer must be $-$ because you still "owe".
How much do you owe altogether?
$5
$-2 - 3 = -5$

NOTE: These rules apply to the Addition and Subtraction of Numbers (not just Integers).

Ask students their preferred method. What are the reason(s) for that preferred method?
Ask them if they have any other method that could be used to add and subtract numbers.

What about Double Signs Between Integers?

Whenever you see double signs between integers, change it to a single sign first.
Then, solve it as we solved single signs between integers.
Change Double Signs Between Integers to a Single Sign:
(Please excuse me to use the word, "hate" to explain this.)
$+$ = "love"
$-$ = "hate"

$+(+) = ++ = love-love$: When you love to love someone, you love the person

$+(-) = +- = love-hate$: When you love to hate someone, you hate the person

$-(+) = -+ = hate-love$: When you hate to love someone, you hate the person

$-(-) = -- = hate-hate$: When you hate to hate someone, you love the person

This means that:

$ +(+) = ++ = + \\[3ex] +(-) = +- = - \\[3ex] -(+) = -+ = - \\[3ex] -(-) = -- = + \\[3ex] $ NOTE: These same rules apply to the Multiplication of Numbers.

$ + * + = + \\[3ex] + * - = - \\[3ex] - * + = - \\[3ex] - * - = + \\[3ex] $ NOTE: These same rules apply to the Division of Numbers.

$ + / + = + \div + = + \\[3ex] + / - = + \div - = - \\[3ex] - / + = - \div + = - \\[3ex] - / - = - \div - = + \\[3ex] $ Examples
$ (1.)\:\: -2 + (-3) = -2 - 3 = -5 \\[3ex] (2.)\:\: -2 - (-3) = -2 + 3 = 1 \\[3ex] (3.)\:\: -3 -(-2) = -3 + 2 = -1 $

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Numbers Calculator (Answers are in Integers or Fractions)


  • All outputs/answers are in integers or the simplified forms of fractions.
  • When you input fractions, please type only proper fractions or improper fractions.
  • For mixed numbers, please put the "plus" sign between the integer and the proper fraction.
  • Example: $2{3\over4}$ should be typed as 2+3/4.
  • Convert a decimal to a fraction: Just type the decimal and click the "Calculate" button
  • Example: 0.25 should give ${1\over4}$.
  • Add, Subtract, Multiply fractions, integers, and decimals
  • It follows the order of operations
  • Example: 3 + ${5\over9}$ - 0.7 * ${2\over5}$ (written as 3 + 5/9 - 0.7 * 2/5) should give ${737\over225}$
  • Divide fractions, integers, and decimals
  • Use parenthesis appropriately
  • Example: ${(2 + 3)\over4}$ written as ((2 + 3) / 4) should give ${5\over4}$.
  • Example: ${3\over5}$ ÷ ${5\over3}$ (written as (3/5) / (5/3)) should give ${9\over25}$
  • Example: 3 ÷ 0.3 (written as 3 / 0.3) should give $10$
  • Example: ${3\over5}$ ÷ 0.3 (written as (3/5) / 0.3) should give $2$
  • Add, Subtract, Multiply, and Divide fractions, integers, and decimals
  • Use parenthesis appropriately for the division
  • Example: $-{5\over12} + ({3\over5}$ ÷ ${7\over10})$ written as (-5/12) + ((3/5) / (7/10)) should give ${37\over84}$
  • Example: ${3\over4} * 4(2 + 7)$ ÷ $-{1\over2} * -{1\over3}$ written as (3/4) - 4*(2 + 7) / (-1/2)*(-1/3) should give $-{93\over4}$

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Numbers Calculator (Answers are in Integers or Decimals)


  • All outputs/answers are in integers or decimals.
  • When you input fractions, please type only proper fractions or improper fractions.
  • For mixed numbers, please put the "plus" sign between the integer and the proper fraction.
  • Example: $2{3\over4}$ should be typed as 2+3/4.
  • Convert a fraction to a decimal: Just type the fraction and click the "Calculate" button
  • Example: ${1\over4}$ (written as 1/4) should give 0.25
  • Add, Subtract, Multiply fractions, integers, and decimals
  • It follows the order of operations
  • Example: 3 + ${5\over9}$ - 0.7 * ${2\over5}$ (written as 3 + 5/9 - 0.7 * 2/5) should give $3.2755555555555556$
  • Divide fractions, integers, and decimals
  • Use parenthesis appropriately
  • Example: ${(2 + 3)\over4}$ written as ((2 + 3) / 4) should give $1.25$.
  • Example: ${3\over5}$ ÷ ${5\over3}$ (written as (3/5) / (5/3)) should give $0.36$
  • Example: 3 ÷ 0.3 (written as 3 / 0.3) should give $10$
  • Example: ${3\over5}$ ÷ 0.3 (written as (3/5) / 0.3) should give $2$
  • Add, Subtract, Multiply, and Divide fractions, integers, and decimals
  • Use parenthesis appropriately for the division
  • Example: $-{5\over12} + ({3\over5}$ ÷ ${7\over10})$ written as (-5/12) + ((3/5) / (7/10)) should give $0.4404761904761905$
  • Example: ${3\over4} * 4(2 + 7)$ ÷ $-{1\over2} * -{1\over3}$ written as (3/4) - 4*(2 + 7) / (-1/2)*(-1/3) should give $-23.25$

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