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Founding Definitions of The Real Number System
Founding Definitions of The Real Number System
To sum it up in 1 definition, the real number system, as it name suggests, is a system of categorizing real numbers into more specific categories, as shown in the diagram. A real number in math is any value that can be found on a number line, and/or basically any number you usually use in math, depending on the way you think about it. Additionally, there are imaginary numbers that you may encounter further on in your math career, but no need to worry about that now. When looking at the diagram, imagine it like a sort of categorization system such as one used for determining a species in biology, further specifying it as it goes along in the system. For example, since 1 is a natural number, it must also a whole number, integer, rational number, and real number. My apologies, because although I did say that I was teaching concepts and techniques rather than stuff to memorize, many definitions, theorems, postulates, and more used throughout all of mathematics are usually have to be memorized overtime, but you will most like subconsciously memorize them rather than having to use flashcards. Now, going into definitions:
Rational number: Any number that can be written as a fraction (though the denominator cannot equal 0).
For examples: 1/2, 8, -5, 0.333 (Note: while not all of these examples are written as fractions, they all have the ability to be, such as 8 being equal to 8/1 and 0.333 being equal to 1/3)
Irrational number: Any number that CANNOT be written as a fraction, meaning the decimals of the number do not repeat in patterns.
For examples: π, √2, √3 (Note: an irrational number will most commonly be found as the square root of a number and pi, as shown in the examples)
Integers: Both positive and negative whole numbers, and 0.
For examples: -1, 0, 1
Whole number: Any positive whole number, and 0.
For examples: 0, 1, 2, 3
Natural number: Any positive whole number, also known as counting numbers.
For examples: 1, 2, 3 (Note: 0 is not included)
Examples
1-What are the classifications of -4
Let's go and break this down step-by-step
Firstly, it is a rational number since -4 can be written as -4/1, which also indicates it isn't an irrational number.
Secondly, it is an integer due to -4 being a negative whole number, and an integer's definition being both positive and negative whole numbers, and 0.
Finally, since we can conclude that -4 is an integer and a rational number, we can also conclude it is a real number due to its being in the real number system. Additionally, we can conclude -4 is not a whole number or natural number, due to those sets not including negative numbers.
Answers: Integer, rational number, real number.
2-What are the classifications of π
Firstly, π is an irrational number because it cannot be written as a fraction due to pi going on forever, having infinitely many decimal places, and therefore, π cannot be a rational number, integer, whole number, or natural number, and only an irrational number and real number. While you see irrational numbers less commonly, they are still very prominent in some topics and ideas of math, so it is still good to know what it means to be an irrational number.
Answers: Irrational number, real number.
3-What are the classifications of 0
Firstly, it is a rational number due to its being able to be written as 0/1, which then also, of course, makes it a real number due to all rational numbers being real numbers.
Secondly, it is also an integer due to an integer being both positive and negative whole numbers, and 0.
Finally, we can also say it is a whole number because it includes any positive whole number and 0. Additionally, it cannot be a natural number due to its being only positive numbers.
Answers: whole number, integer, rational number, real number.
BONUS-What are the classifications of √-1
You may be wondering how √-1 works due to there not being an answer to it, because the two factors of -1 are 1 and -1. While this will not be talked about for most/all of high school math, it is still good to get a slight idea behind it. For cases like this, mathematicians created imaginary numbers, and the answer to √-1 is simply written as i, meaning i²=-1. Due to it being an imaginary number, it isn't even in the real number system, so there really is no classification other than imaginary.
Answer: Imaginary
Founding Properties of Real Numbers
Founding Properties of Real Numbers
As shown as the diagram to the right, real numbers have many different properties, and throughout Algebra I, we will use these to simplify and solve equations. Additionally, these properties do not work for subtraction and division simply due to the nature of how they work in mathematics. Additionally, specific examples are not needed in my opinion, due to them all being rather self-explanatory and intuitive.
Commutative property of addition/multiplication
For examples: 2+3=3+2, 5•2=2•5
Associative property of addition/multiplication
For examples: 1+(2+3)=(1+2)+3, 2•(5•4)=(2•5)•4
Distributive property.
For examples: 2•(3+4) = 2•3+2•4
Identity property of addition/multiplication
For examples: 5+0=5, 5•1=5
Inverse property of addition/multiplications
For examples: 4+(-4)=0, 5•1/5=1
Problem set
1: What is the most specific classification of -4
2: What are all the classifications of √49
3: What is the most specific classification of 8.9
4: What is the most specific classification of 5
5: What are all the classifications of √22
6: Name the property used in 5•18=18•5
7: Name the property used in 4+(8+2)=(4+8)+2
8: Rewrite 6•-4 using the commutative property
9: Rewrite 4•(5+9) using the distributive property
10: What property justifies that 12+0=12
11: Explain why √19 is irrational
12: Give an example of the distributive property
Answer Key (Real number is optional when classifying)
1: Integer (Negative whole number)
2: Real number, rational number, integer, whole number, natural number (√49=7)
3: Rational number (89/10)
4-Natural number (positive whole number and isn't 0)
5: Real Number, irrational number
6: Commutative property (of multiplication)
7: Associative property (of addition)
8: -4•6
9: 4•5+4•9
10: Identity property (of addition)
11: √19 is irrational due to it's decimals not repeating in a pattern and therefore not being able to be written as a fraction
12: X•(Y+Z) (X, Y, and Z can be any number as long as it follows that formula)
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