“Doing arithmetic with symbols instead of numbers is the basis of algebra.”

—

Harold R. Jacobs, Elementary Algebra

The purpose of algebra is to substitute symbols for numbers. Symbols that substitute for numbers are called variables.

For example, consider harvesting and selling fruit.

Figure 1: Fruit harvest. [Dayton Art Institute]

Say you are growing crops, and you get paid 5 currency units (dollars, yen, etc.) for each kilogram of a crop you harvest.

You could denote the number of kilograms harvested as a symbol, such as a letter of the alphabet, any letter, for example the letter H (to help remind you that it symbolizes harvest). That would yield the following algebraic formula for calculating total number of dollars (yen, etc.) earned for that harvest:

5 × H

where H is a variable symbol (denoting harvest in kilograms), and × is the multiplication (times) symbol. Other multiplication times symbols include a dot (·) or no symbol at all. The following are equivalent to the preceding formula:

5 · H

5 H

The variable for harvest of a particular crop could be any symbol, not necessarily H. It could be A, or P, or any other symbol you choose (for example to denote apples, pears, etc., in this case again letters to remind you of what the variables symbolize).

Figure 2: Pears

Figure 3: Apples

Say your apples and pears are both earning 5 dollars per kilogram. Denoting the apple harvest as A and the pear harvest as P, the total harvest H, of apples and pears, is A plus P:

H = A + P

and the dollars earned are:

5 H = 5 (A + P)

When parenthesis are used, as in 5(A+P), it has higher precedence (is evaluated before multiplication or addition).

Precedence specifies the order of evaluating mathematical operators like multiplication and addition. Multiplication and division have higher precedence than (are evaluated before) addition and subtraction, and parenthesis have even higher precedence than multiplication and division.

For example, in the formula A + B × C, first B and C are multiplied together (before adding A) because multiplication has higher precedence than addition.

However, if you enclose A + B in parenthesis:

( A + B ) × C

then A + B are added together before that sum is multiplied with C, because parenthesis have higher precedence than multiplication.

Returning to the example of harvesting fruit for 5 dollars per kilogram, denoting the total harvest as H, the dollars earned are:

5 H

The harvest in kilograms (H) consists of apples (A) and pears (P):

H = A + P

Multiplying (scaling) both sides of the equation by 5 yields the dollars earned for the harvest:

5 H = 5 (A + P)

(Note: In algebra, to maintain the same equation, you can multiply one side of the equation by a number if you multiply the other side of the equation with the same number, as we have just done).

According to the rules of precedence, parenthesis are evaluated first, so for the right hand side (RHS) of this equation, which is 5(A+P), first A and P are added together since they are enclosed in parenthesis, then that sum (A+P) is multiplied with 5.

If the parenthesis were left out as follows:

5 A + P

Wrong!

then A would be multiplied by 5 but P would not be. That would be incorrect. The accurate formula without parenthesis for this example would be as follows:

5 A + 5 P

Multiplication has higher precedence than addition. Thus, in this formula without parenthesis, A is multiplied by 5, and P is multiplied by 5, before addition is performed.

Numbers could be whole numbers (integers) and real numbers.

Whole numbers could be used when considering a harvest of individual fruits, like delivering 20 pears to a work kitchen, for example if they are ordering one pear for each of 20 meals.

Whole numbers are called integers since they are “integral” units (each unit “in and of itself”). Each pear is a whole (integral) pear in this type of number system.

Other uses are possible. Consider you had 6 pears, and you eat half a pear and put the rest of it in the refrigerator for later consumption. Then you would have 5.5 pears, a real number (not an integer).

Likewise, when measuring harvest by mass (e.g., kilograms) the value is a real number, since mass units are by defnition not based on a regularly occurring indivisible object being measured.

A number (whether integer or real), that is multiplied into another value (or variable), is called a scalar, because it “scales” (multiplies) the other value by the amount of the scalar integer or real number.

Figure 4: Caramel apples. The number of apples on a tray of caramel apples is an integer.

Figure 5: Apple pie. The amount of apples in an apple pie is a real number.

A number with an exponent is multiplied with itself. The exponent specifies how many copies of the number are included in the multiplication. The exponent is usually a superscript after the number:

X2 = X · X

X3 = X · X · X

The exponent is called the power of the number. Note that a number to the first power is the number itself (if no exponent is given, the exponent assumed to be 1):

X1 = X

X to the power of two ( X2 ) is also referred to as X squared, and X to the third power ( X3 ) is also referred to as X cubed.

For systems that do not support superscript, an upward caret (^) preceding the exponent may be used:

X^2 = X · X

X^3 = X · X · X

For exponents that are positive integers, multiplying copies of the same variable that have different exponents sums the exponents:

Xa · Xb = Xa+b

and division subtracts the exponents:

Xa ╱ Xb = Xa−b

A number raised to a power results in a number that can also be raised to a power, according to the following formula:

(Xa)b = Xab

For example:

(23)4 = (2·2·2) 4 = (8)4 = 4096

(23)4 = 23×4 = 212 = 4096

More formulas for integer exponents:

(XY)a = XaYa

(X ⁄ Y)a = Xa ⁄ Ya

The following definitions are for non-zero x:

x0 = 1

x−a = 1 ⁄ xa

A polynomial of a variable is an equation, in which terms of the variable are added or subtracted, each term defined as follows:

𝐚𝒙n

where 𝒙 is a variable, 𝐚 is a scalar, and n is a non-negative integer exponent.

Here are examples of polynomials (from Jacobs):

x3 + 8

2x5 − x4 − 1

(1⁄3)x7

−9

The degree of a polynomial is the highest power of a term, which for those examples are 3, 5, 7 and 0 respectively.

The last example is of degree zero because:

−9 = −9x0

Subtraction applies to the immediately following term. It is best to think of the subtraction symbol as a minus sign (negation) for the term that immediately follows (negating that term):

x2 − 3x + 1 = x2 + (−3x) + 1

To multiply two polynomials together, each term in one of the polynomials is multiplied with each term of the other polynomial.

When done with pen and paper, it is handy to draw a line connecting the terms. That way, you are not done until each term in one polynomial has a line to each term in the other polynomial.

Polynomials may have more than one variable. Here are polynomials with two variables (x and y):

x + y

x − y

Squaring those polynomials produces results that will come in handy when factoring polynomials (in another lesson later):

(x + y)2 = (x + y)(x + y) = x2 + 2xy + y2

(x − y)2 = (x − y)(x − y) = x2 − 2xy + y2

Problem: Find the square of x+y

Solution: The square of x+y is (x+y)2, which is (x+y)(x+y).

In this example, we begin by drawing a line, from the first term of the first polynomial, to the first term of the second polynomial, giving us the first term of the solution:

Next we draw a line, from the same term in the first polynomial, to the other term in the second polynomial, providing the next term of the solution:

Then a line, from the second term of the first polynomial, to the first term of the second polynomial, providing the third term of the solution:

Finally, a line from the second term to the second term, providing the fourth term of the solution, and then add the two middle terms of that solution together, to reduce the solution to three terms:

If you have pen and paper, repeat this exercise to get the hang of multiplying polynomials (try calculating the square of x−y). Note that in this example, it was also possible to draw the second line below the equation and the last line above, to make a “smiley face” (try doing that when you practice).

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2021–Sep–22 21:05 UTC