Elementary Algebra

1. Geometric representation of any numberImagine a straight line. Choose a point and call it origin.Let origin serve as the geometrical image of the number 0.Select another point to the right, call it the unit point and let it correspond to number 1.The direction from origin to unit point is positive, the opposite direction is negative.

The distance between the origin and the unit point is the unit distance.We call such a line an axis.

1.1 Graph and coordinatesEvery point on an axis serves as the geometric image of a unique number.We call the number the coordinate of the corresponding point, and the point the graph of the number.

1.2 Correspondence between points and different kinds of numbersWhole numbers are simply the numbers without fractions: 0, 1, 2, 3, 4, 5, …

Counting numbers are whole numbers without the zero, because you can't count zero: 1, 2, 3, … Natural numbers can mean either counting numbers or whole numbers. Integers are whole numbers including negative numbers. Still no fractions.

A number that is the ratio of two integers is called rational. They include all the integers and fractions. Examples:

•2719

, ½, 1.25 and 3.

• The last two numbers are rational because 1.25 = 125100

and 3 = 62

-5 -4 -3 -2 -1 0

O

1

I

2 3 4 5

0 1 2 3 4 5

-5 -4 -3 -2 -1 0

O

1

I

2 3 4 5

• Since expression such as 50

is meaningless, a fraction with zero denominator does not

represent a number.

If you draw a square of size “1”, what is the distance across the diagonal? Pythagorean Theorem gives answer as: x2 = 12 + 12 = 2, so x = √2√2 = 1.4142135623730950...

The above question cannot be answered using the ratio of two integers:

√2 ≠ pq

So, √2 is not a rational number, it is irrational.

Real numbers include both rational and irrational numbers. A real number can be thought of as any point anywhere on the number line.

Now, is there √−1 ?In other words, what can we multiply by itself to get −1?Answer: Imagine that the √−1 exists, and it is an imaginary number i.

So, i = √−1Example:

• What is √−9 ?• Answer: √−9 = √9x−1 = √9 x √−1 = 3 x √−1 = 3i

The number i has interesting property that if you square it, you get -1, which is back to being a real number. This is hence its definition:“An imaginary number is a number whose square is negative real number.”

What if you put a real number and an imaginary number together? You get complex numbers.Example:

• 3 + 2i• 27.2 – 11.05i

-5 -4 -3 -2 -1 0 1 2 3 4 5

-½ ½-3/2 3/2

Since either the real part or the imaginary part of a complex number could be zero, we have:• a real number is also a complex number: 4 is a complex number, 4 + 0i• an imaginary number is also a complex number: 7i is a complex number, 0 + 7i

1.3 Number summary

Type of number Quick descriptionCounting numbers {1, 2, 3, … }Whole numbers {0, 1, 2, 3, … }Integers {… -2, -1, 0, 1, 2, … }Rational numbers {p/q : p and q are integers, q ≠ 0}Irrational numbers Not rational, e.g. sqrt(2)Real numbers Rationals and IrrationalsImaginary numbers Squaring them gives a negative Real NumberComplex numbers Combinations of Real and Imaginary Numbers

2. Constants and variables

2.1 Constant and valueA symbol that in a particular context is the name of just one specific thing is called a constant. In other words, a constant is a proper name. Grammarians call constants proper nouns. The thing that a constantnames is called its value.

The distinction between a constant and its value is simply the distinction between a name and the thing it names. A constant always names its value. We name a constant by enclosing it in quotation marks.

Example:• “FDR” is a constant for Franklin D. Roosevelt. • The man FDR is the value of the symbol “FDR”.• “Sri Aurobindo” is a constant. • “10” is a constant.

2.2 NumeralA symbol that stands for a number is called a numeral.Example:

• The numeral “2” stands for the number 2. • We also say that “2” has the value 2.

2.3 Variable

In “A chair has four legs”, the word “chair” does not name any particular chair. • Therefore, “chair” is not a constant.

In mathematics, when we wish to refer to an unspecified object of certain kind, we usually use an arbitrary letter or symbol, such as x. A symbol that, in a particular context is not a constant but for which any one of certain constants may be substituted is called a variable.

A variable is a symbol that holds a place for constants.Suppose a variable occurs in an equation or formula. What constants are permitted to replace it?

• With each variable is associated a set.• The names of the elements in the set are the permitted replacements for the given variable. • The associated set is the range of the variable.

A constant has just one value, whereas a variable has more than one value. This means, a single object is associated with a constant whereas a collection of several objects (set) is associated with a variable. The variable may be thought of as standing for some unspecified object in the set.

A handy method of creating/specifying variables:• Place a subscript to the right of the variable, say x.• So we have: x1, x2, x3, x4, … xk … xn.

Variables occur frequently together with certain expressions called quantifiers. • Quantifiers deal with “how many”.• Universal quantifier: for all or for each, ∀ x• Existential quantifier: there exists, ∃

Example:• If m, n are natural numbers, then “ ∀ x , ∃m m > n” means “For each natural n, there is a

natural m such that m > n.”

If an occurrence of a variable is accompanied by a quantifier, then the variable is bound, otherwise it is free.

• In mathematical discourse, variables frequently occur as free variables. • For example, we often find, “If x is a non-zero real number, then ...” or “Let x be a non-zero real

number. Then ...”

2.4 Avoiding confusion about variables and constantsThis can be done if a careful distinction is made between symbols and their values. Since a variable is aplaceholder for which constants are to be substituted, it appears in its context in the same way as if it were a constant. But a variable is not the name of any particular thing and has no definite value. It is merely a symbol used to indicate a place in which a constant may be substituted. It is a blank, a hole to be filled in a certain way.

• Thus in “3 + x”, “x” appears in the same way as does “2” in “3 + 2”, but “x” is not the name of any particular number.

3. Vectors

3.1 What is a vector?A vector quantity, or vector, provides information about not just the magnitude but also the direction ofthe quantity. When giving directions to a house, it isn't enough to say that it's x miles away, but the direction of those x miles must also be provided for the information to be useful.

If the house is 3 miles due west, the direction would be negative, if 5 miles due east, the direction would be positive. Corresponding to each such arrow there is a unique real number equal to the length of the arrow if it points to the right and the negative of this length if it points to the left.

Arrows such as those pictured above are sometimes called directed distances and are called vectors. A vector has magnitude (how long it is) and direction. In contrast, a scalar has only magnitude (e.g. mass or temperature). We see that every real number determines a unique vector and, conversely, that every horizontal vector determines a unique real number.

A vector can be thought of as a total shift. If an earthquake hits a town and people move 2 miles east and 1 mile up to get out of the town, we have a total shift of 2 + 1 = 3 miles.

In the above figure, the length of the line shows the magnitude and the arrowhead points in the direction.

From now on, we shall think of numerals as standing for numbers, points on a real axis or vectors, according to convenience. For example:

• we use “-1” to stand for the number -1, • the point -1 (the point one unit to the left of the origin), or• the vector -1 (the vector of length 1 pointing to the left)

-5 -4 -3 -2 -1 0

O

1 2 3 4 5

2

12 + 1 = 3