Algebra/Real Numbers

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In algebra, real numbers are used to measure continuous quantities - such as time, age, speed, area and temperature. Mathematicians use the symbol R (or alternatively, \Bbb{R} , the letter "R" in blackboard bold, Unicode ℝ) to represent the set of all real numbers.

[edit] Classification of Number

[edit] Natural or counting numbers

The set of natural numbers is typically represented by the letter \mathbb{N}. A natural number is any whole number greater than 0, and are thus countable, using an abacus, or fingers. The number of natural numbers extends to infinity, that is, no matter how large a number you can think of, there is a natural number that is one larger.

\mathbb{N} = \lbrace 1, 2, 3, 4, . . . \rbrace and so on.

[edit] Whole numbers

Whole numbers are generally represented as \mathbb{W}. The whole numbers are the set of natural numbers plus 0. It follows, therefore, that the natural numbers (\mathbb{N}) are included in the whole numbers--and as a result it is also infinitely large.

\mathbb{W} = \lbrace 0, 1, 2, 3, . . . \rbrace

[edit] Integers

Integers are often represented through the letter \mathbb{Z}. Integers, as you might expect, are an extension of the whole numbers, and include the negative of every natural number. The word "integer" comes from Latin, and means "whole, complete". The letter Z used to denote this set comes from the German for the word "numbers" - zahlen.

\mathbb{Z} = \lbrace . . . , -2, -1, 0, 1, 2, . . . \rbrace

[edit] Rational Numbers

Rational numbers are usually represented by \mathbb{Q}, the symbol of which derives from the German word quotient. The rational numbers, or rationals, are numbers that can be represented as a fraction of two integers, or equivalently, as the ratio of two integers. The word rational originated from the word ratio.

It follows that the integers are a subset of the rational numbers, because they can be expressed as fractions or ratios (e.g. 3 / 1 = 3).

\mathbb{Q} = \lbrace . . . , -3/1, 2/5, 3/6, 5/2, . . . \rbrace

or in closed form:

\mathbb{Q} = \lbrace \frac{a}{b} \vert a,b \in \mathbb{Z}, b \ne 0 \rbrace

[edit] Irrational Numbers

Irrational numbers are normally represented through the letter \mathbb{I}. Irrational numbers are any numbers that cannot be represented as fractions (rational numbers). These include infinite, non-repeating decimals, such as \pi \,, and \sqrt{2}.


[edit] Transcendental Numbers

Transcendental numbers are a subset of the irrational numbers for example, \ \pi is both irrational and transcendental. \ \pi has all the properties of being irrational and the additional property of not being the root of any polynomial equation with rational coefficients.

For example the equation:

\begin{matrix} x^2 - 2 && = && 0 \end{matrix}

has solutions \pm \sqrt{2} , which are irrational, but the coefficients in the equation were only integers. The claim is that even after exploring an infinite amount of these polynomials one will never obtain a transcendental solution. Proving that a number is transcendental can be very difficult.

[edit] Real Numbers

Real numbers are represented as \mathbb{R}. Real numbers represent the entire set of numbers we have mentioned before: Naturals, Whole, Integers, Rationals and Irrationals. All these families are subsets of each following set; that is, the naturals are a subset of the whole numbers, the whole numbers are a subset of the integers, and so on.

At this point one can think of the real numbers as:

\mathbb{R} = \mathbb{Q} \cup \mathbb{I}

[edit] Practice Problems

Name which sets of real numbers these numbers belong in:

1. \sqrt{49}

2. 0.0110211 \,

3. -32 \,

4. 0 \,

5. \sqrt{21} \,

[edit] Answers

1. The square root of 49 is 7, which is a natural number, whole number, integer, rational number, and a real number.

2. The decimal expansion ends, meaning it can be represented as fraction of two integers - a rational number thus also a real number.

3. Since the number is negative, it is an integer, rational number, and a real number.

4. 0 is not a natural number, but it is a whole number, integer, rational number, and real number.

5. The square root of 21 is an infinite, non-repeating decimal, thus it is an irrational number and therefore a real number.

[edit] Properties Of All Real Numbers

Although this has already been briefly discussed in Arithmetic and the Arithmetic Review, I'd like to go over some of the properties once again.

Property Name Addition Multiplication
Commutative a + b = b + a c(d) = d(c)
Associative (e + f) + g = e + (f + g) (h)[(i)(j)] = [(h)(i)](j)
Identity x + 0 = x (x)(1) = x
Inverse x + (-x) = 0 x(1/x) = 1
Distributive - a(b - c) = (a)(b) - (a)(c)


Out of all of those properties, the Distributive Property of Multiplication is most likely the most useful.

You can simplify many operations using the Distributive Property of Multiplication if you are operating on numbers which are hard to multiply.

An example of this would be 4(8.75x - 27y). 4(8.75x) may be difficult to calculate. You can actually perform a multiplication or division on the groups of numbers in parenthesis, if you perform the inverse operation on the number outside the parenthesis. We can simplify the expression by multiplying the numbers inside parenthesis by two, then dividing the one outside parenthesis by 2. The new resulting expression is 2(17.5x - 54y), which might be easier to figure out the answer of.


[edit] Practice Problems

Identify the following properties being expressed.


1. 4(3x + 4) = 12x + 16

2. 6 + 0 = 6

3. (2 + 7) + 5 = (2 + 5) + 7

4. (3/4)(4/3) = 1


Use the Distributive Property to simplify these expressions.


1. 2(14x - 26)

2. (2/3)(3x + 9)

3. 3(12x + 4y)

4. 2(5x - 6) + 3(3x + 2)

[edit] Answers

1. Distributive Property of Multiplication

2. Identity Property of Addition

3. Associative Property of Addition

4. Inverse Property of Multiplication (You multiply by the reciprocal. The reciprocal of 4 is 1/4, of -17 is -1/17, and of 2/5 is 5/2.)


1. 28x - 52

2. 2x + 6

3. 36x + 12y

4. 10x - 12 + 9x + 6 = 19x - 6 (Combine like-terms, as we learned in Arithmetic!)

[edit] Lesson Review

All numbers that we will be working with for the majority of Algebra are called Real Numbers. They consist of Rational and Irrational Numbers. Irrational Numbers are numbers that have infinite, non-repeating decimals, such as pi. Rational Numbers are all numbers that can be expressed as a fraction of integers, which include Natural Numbers, Whole Numbers, Integers, and Rational Numbers. For all Real Numbers, there are a few properties of addition and multiplication: Commutative, Associative, Identity, Inverse, and Distribution. The Distribution will come in handy for the rest of the course.


[edit] Lesson Quiz

Identify the set of numbers each number is in.


1. \sqrt{7}

2. -\sqrt{36}

3. ( − 4)2


Identify the property being expressed.

1. 2 \cdot\! 3 = 3 \cdot\! 2

2. x\left(\frac{1}{x}\right) = 1

3. x + (-x) = 0 \,


Perform the Distributive Property of Multiplication to simplify each of these expressions.

1. 3(2x + 7) \,

2. 15(6x - 22) \,

3. 3(20x + 42y) - 2(7x + 20y) \,

Challenge Questions.

1. When two rational numbers are multiplied, does it always result in a rational number? Why?

2. When two irrational numbers are multiplied, does it always result in an irrational number? Why?

3. When two irrational numbers are added, does it always result in an irrational number? Why?

4. When the square root of an irrational number is taken, does it have to be irrational? Why?

5. If x(x + 1) is irrational, does x have to be irrational? Why?

[edit] Quiz Answers

1. Irrational and Real

2. Integer, Rational, Real (Note that the negative comes after the square root!)

3. Natural, Whole, Integer, Rational, and Real (Note that the negative comes before squaring, because it's in parentheses!)


1. Commutative Property of Multiplication

2. Inverse Property of Multiplication

3. Inverse Property of Addition


1. 6x + 21

2. 90x - 330 (Remember that we could have simplified the expression to 30(3x - 11))

3. 46x + 86y (Don't forget to combine like-terms. Also, don't forget that you're subtracting 40y!)

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