The Language of Algebra

Algebra is a language which we use to describe reality in its most basic form. However, we first describe that reality in another language -- in my case -- English or French. Problems can arise when we translate from one language to another -- so this lesson is designed to teach the definitions of Algebraic terms and how to translate English statements into Algebra statements.

When we first study grammar in any language, we learn that it is composed of parts of speech like nouns, verbs, adjectives, adverbs and a few more things. Well, Algebra works exactly the same way.

For example:

John is 7 years old to an Algebran (my term for a native Algebra speaker) is:

J = 7; J in years

Notice the semi-colon? Algebra, like all languages even has punctuation.

But let's discuss the nouns and verbs before we get to punctuation.

In the English statement John is 7 years old, John and 7 are nouns.

In the Algebra statement J = 7; J in years,

J -- known as the variable -- is a noun.
7 -- known as a constant -- is also a noun.

In the English statement the verb " is " becomes the equal sign " = " in the Algebra statement.

Verbs in Algebra are called operators.

Adjectives -- used to modify nouns in English -- are called coefficients in Algebra --

The English sentence: Mary is twice as tall as John is written M = 2J in Algebra.

2 -- the Algebra equivalent of twice is a coefficient.

There are adverbs too in Algebra. Advanced courses define new types of addition and subtraction -- denoted with symbols like /, 0 -- which modify the operators or verbs of the language but we'll ignore those for the time being.

This lesson, then, will enable us to speak Algebra well enough to continue with the lessons that follow.

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1. Variables & Constants: algebraic nouns

The nouns in the language of Algebra are known as variables and constants.

Variables are the letters.

Constants are the numbers.

Variables, as their name declares -- are things that can change in value -- the x's and y's.

Constants as their name declares -- are things that can't change in value -- 7, -3, o, 2153...

We use variables to name things -- such as:
the length of a rectangle, Sally's age, the first integer, a certain number , etc.
We use constants to symbolize the numbers they represent.
So the English sentence: "If two be added to a certain number, the result is seven." becomes, when translated into Algebra:
2 + x = 7
The constants are 2 and 7 -- they don't change -- they remain constant.
The variable is x -- a certain number -- not sure what it is -- could be anything.
We know that 2 is always 2 -- that's constant.
We know that 7 is always 7 -- that's constant too!
But a certain number ? -- that's definitely variable!

Now, on to the verbs or operators -- there's one in the sentence above -- the +

 intro nouns division minus x minus verbs exponentiation translating collecting terms

2. Operators: algebraic verbs

The signs and symbols of mathematics tell us what to do to the variables and constants in our expressions and equations.They represent operators.

For instance, + means add, ÷ means divide, etc.

We assign words to or define the results of these operations. They are:

1. + sum as in x + y, the sum of x and y.
2. - difference as in x - y, the difference of x and y.
3. × product as in 5 × a, the product of 5 and a. **
4. ÷ quotient as in 5 ÷ a, the quotient of 5 and a

** Because we often use x as a variable, we don't often use × to denote multiplication in our algebra statements.

We either write the terms all together like this: 2ax, 3mn

or

We use brackets or a dot ( · ) instead.

So, for five times x plus 7 we write 5(x + 7) or 5 · (x + 7) instead of 5 × (x + 7).

As you can see, we could get our x's mixed up -- so it's usually safer to use brackets.

These are considered the 4 basic operations in math but, really they're all the same operation -- and that operation is addition. "How come?" you ask. Well, subtraction is reverse or backward addition, multiplication is repeated addition, and division is repeated subtraction which is repeated backward addition. Even the other operators such as exponentiation and taking roots are addition in some convoluted form.

 intro nouns division minus x minus verbs exponentiation translating collecting terms

Now let's discuss a major bone of contention for all new "Algebrans".

 Why in the world is a minus times a minus equal to a plus???

The easy answer is: because it's backward, backward addition -- and when you're going backwards and you reverse direction -- where's the only place you can go???

Forward!

Think about it this way:

The first minus sign means the operation of subtraction -- take it away!

The second minus sign assigns a negative quality to the number or variable -- a loss or a debt.

When you take away a loss or a debt, there has to be a gain.

To take away 3 negative degrees, we must add 3 positive degrees.

To take away(1st minus) 7 steps left(2nd minus) -- you must go 7 steps right -- yes?

Say you owe me \$10.00 but you have no money in your pocket now to pay your debt.

You have -\$10.00 (negative ten dollars); you owe me \$10.00 and you have nothing -- right?

I'm feeling generous for some strange reason and I say "Forget about the \$10 you owe me."

Now, you have \$0 , which is \$10 more than you had before I took away a debt of \$10.

The 1st minus sign instructs us to take away (an operator) and

the 2nd minus sign is the negative quality of the \$10.00, because it is a debt.

So - (- 10) = + 10 and 5x - (- 3x) = 5x + 3x = 8x.

and 3a - 2( -5a ) = 3a + 10a = 13a

So, just as a minus sign can mean 2 things:

the operation to subtract, or

the negative quality of loss or debt or retreat (down or left)

a plus sign can also mean 2 things:

the operation to add, or

the positive quality of a gain or advance (up or right).

When we work with positive constants or variables, we don't usually show the + sign.

We don't generally write 6 - +2 since 6 - 2 will do. When you first start working with integers, you will see expressions like (+ 6) - (+ 3) - (- 7) because you're just learning to work with operators and signed numbers. However, once you're on your way, you'll write 6 - 3 - (-7), since the first two plus signs are unnecessary.

If a negative is not indicated before a number or variable we know it is positive.

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Now let's inspect the division sign, for it too is often replaced by an alternate symbol.

Look at this symbol

÷

What does it look like? Something over something else -- yes?

That's exactly what it means -- that's why we use a fraction to represent division.

When using the number keypad on our computers,

how do we instruct the computer to divide?

with the slash /

we enter 8 / 4 to divide 8 by 4 -- don't we?

Another symbol used to indicate division is the colon ( : )

We use it in proportion statements such as

6 : 12 as 1 : 2 , which is read:

six is to twelve as one is to two.

and could be written .

Look at this symbol ( : ) -- it's the division sign without the horizontal line.

the / slash is really the division sign without the colon

so both parts of the division sign can be used independently to indicate division.

We use the colon almost exclusively in proportion statements.

As for the slash, it is best to use a horizontal fraction line rather than the slanted slash. Otherwise you'll get the denominators mixed up with the numerators when you've got a whole mess of fractions to deal with.

This is much clearer than (1/(2x-3)) - (5/3(x - 2)) = 9

 intro nouns division minus x minus verbs exponentiation translating collecting terms

Exponentiation or raising to powers is repeated multiplication of something by itself.

Since multiplication is just repeated addition, exponentiation is repeated repeated addition.

To square or raise to the 2nd power is to multiply something by itself.

So 5 squared = 5 × 5 -- 2nd power -- we see 2 of them.

x squared = x · x = x 2

the square of the sum of a and 7 = ( a + 7 )(a + 7) = (a + 7) 2 .

To cube or raise to the 3rd power is to multiply something by itself 3 times.

So 5 cubed = 5 × 5 × 5 -- 3rd power -- we see 3 of them.

x cubed = x · x · x = x 3

the cube of the sum of a and 7 = ( a + 7 )(a + 7) (a + 7) = (a + 7) 3 .

To raise a variable or constant to the nth power, multiply it by itself n times.

So, to raise 5 to the nth power I write 5 n .

If n is assigned a value -- say 3 -- then 5 n = 5 3 or 125.

Roots and Radicals: or fractional exponentiation

So far, all our exponents have been positive integers -- whole numbers greater than zero.

Negative numbers and fractions are also used as exponents.

The Concept of a Square Root:

The most common mistake my students make when taking a square root is to take half the number rather than its root. Since it's tough to describe the concept well with words, we'll use examples.

3 is the square root of 9 because 3 2 = 9

- 3 is the square root of 9 because ( - 3 ) 2 = 9

9 is the square root of 81 because 9 2 = 81

- 9 is the square root of 81 because ( - 9 ) 2 = 81

Get the idea??

Did you notice that anything squared is positive?

 If a 2 = b, then a is a square root of b.The Square Root of b is written .

Say we're working on a question about the area of rectangle which leads us to the statement:

x 2 = 36

It's pretty obvious that x has to be 6 or - 6

the next operation in our solution then is to

take the square root of both sides of the equation. But since we're discussing area, x must be a the length or width of the rectangle so it can't = - 6.

Our solution would be x = 6 As in the example, sometimes we must omit the negative root

since it doesn't fit the reality of the situation.

A rectangle can't be - 6 centimetres wide or long -- not yet, anyway!

If our question leads us to the statement:

x 3 = 27

the next step is to take the CUBE ROOT of both sides of the equation. ROOTS CAN BE EXPRESSED AS FRACTIONAL EXPONENTS.

In Lesson 1: Simplifying Expressions, (in the Algebra MathRoom) we learn the laws of exponents which explain that fractional exponents are the same as roots.

So, instead of , I could have written .

and is written This fact is very useful for calculations on your calculator. Usually, there is no special button for an nth root -- you have to go searching for the format in one of the lists.

The easy way to enter something like using the ( ^ ) button or the y x button.

In a case where n = , you could enter 32 0. 2 since one fifth = 0.2. However, not all fractions can be expressed as decimals (irrational numbers), so it is best to put brackets around the fraction in order to enter the exact value.

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In the table, you'll see English phrases translated into Algebra phrases.

If you see symbols you don't yet know -- check out the Symbols List on this site.

 English phrase Algebra phrase the sum of a and b a + b the square of x x 2 three times the cube of b 3b 3 the sum of the squares of a and b a 2 + b 2 the square of the sum of a and b ** (a + b) 2 three times the product of a and x squared 3( ax 2 ) or 3ax 2 the difference of 4a and 7b 4a - 7b the square root of m is 3 the fifth root of m - 2 is -9 the sixth root of m - 2 raised to the 8th power the sum of twice the cube of b and its square root the quotient of x and the square of its cube root ** Spoken: a plus b ALL squared.

Now one more topic and we're ready to speak Algebra.

 intro nouns division minus x minus verbs exponentiation translating collecting terms

Throughout the lessons in the Algebra MathRoom, you'll see the phrase collect like terms as a step in a question's solution, so it would be nice if you knew what "like terms" are -- otherwise how can anyone expect you to collect them?

Again -- as always in math -- the words say it all.

in like terms the variable and its powers are identical.

So, x 3 , 17x 3 , -9x 3 , 2ax 3 and 4267x 3 are all like terms

When I collect them I will get (4276 + 2a )x 3

Whereas x 3 , 17 2 , -9x 7 , 2ax - 8 and 4267x 0. 357 are not like terms

because though the variable, x, is the same, the powers or exponents are different.

If we have 3x - 5y + 7x - 2y + x 2 , we would collect like terms as follows:

3x and 7x are like terms = 10x

-5y and - 2y are like terms = - 7y

x 2 has no buddies, so he stands alone.

3x - 5y + 7x - 2y + x 2 = 10x - 7y + x 2 .

One of the greatest sources of confusion for new students of Algebra when they start collecting terms is the fact that we don't write a one (1) when we talk about one x, or one y.

We simply write x or y not 1x, or 1y.

In math, it's pretty obvious -- if we say x, we mean one x. If we meant five x, that's what we'd say.

So, if we have 4x + x , we have 5x :

and x - 7x = - 6x

and since we divide x by itself and we get 1 (in the numerator).

We also don't write a one (1) when we talk about x to the first power. We just write x.

When you learn the laws of exponents in Lesson 1, you'll learn that to multiply variables with exponents you add the powers.

So don't forget that x = x 1

which means that x 2 · x = x 2 + 1 = x 3

And now you're ready to talk math. You should find your time in the Algebra MathRoom pretty easy since I've set the examples up in a 2-column format. If you don't understand what was done in the Algebra -- look to your right -- you'll find the explanation in English. This way, you don't have to read 2 or 3 paragraphs of text explaining the steps you understood to get to the explanation you need. It's right there -- on the same line as the math text.

Have fun -- and remember

-- everything about math makes perfect sense -- always!!

If it makes no sense -- it ain't math!

TtT

 intro nouns division minus x minus verbs exponentiation translating collecting terms (all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).