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**,

In the English statement the **verb** " ** is **" becomes the

**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 = **2**J 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.

intro |
nouns |
division |
minus x minus |

verbs |
exponentiation |
translating |
collecting terms |

**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

- We use

- So the English sentence: "If

- The

- The

- We know that 2 is

- We know that 7 is

- But a

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 |

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:

- +
**sum**as in x + y, the**sum**of x and y. - -
**difference**as in x - y, the**difference**of x and y. - ×
**product**as in 5 × a, the**product**of 5 and a.****** - ÷
**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****(**you owe me $10.00 and you have nothing -- right?__negative__ten dollars);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 $10The

**1st minus**sign instructs us to(an operator) and__take away__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**, orthe

**negative****quality**of loss or debt or retreat (down or left)**a plus sign can also mean 2 things**:the

**operation**to**add**, orthe

**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.

**intro****nouns****division****minus x minus****verbs****exponentiation****translating****collecting terms**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__twoand 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**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.__horizontal__fraction lineThis 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

or__square__**raise to the**is to__2nd power__**multiply**something**by itself**.So

**5 squared = 5****× 5**--**2**nd 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

or__cube__**raise to the**is to__3rd power__**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 timesSo, 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.or fractional exponentiation__Roots and Radicals__: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}= 81Get the idea??

Did you notice that

**anything squared is positive**?If then*a*^{ 2}= b,is a*a*of*square root*.*b*The

of*Square Root*is written .*b*Say we're working on a question about the area of rectangle which leads us to the statement:

**x**^{ 2}= 36It'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}= 27the 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**button.^{ x}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.**intro****nouns****division****minus x minus****verbs****exponentiation****translating****collecting terms**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 and*a**b*+*a**b*the square of *x**x*^{ 2}three times the cube of *b*3 3*b*^{ }the sum of the squares of and*a**b**a*^{ 2}+*b*^{ 2}the square of the sum of and*a**b*******()*a + b*^{ 2}three times the product of and*a*squared*x*3( *ax*^{ 2}) or 3*ax*^{ 2}the difference of 4 and 7*a**b*4 - 7*a**b*the square root of is*m***3**the fifth root of is -*m - 2**9*the sixth root of raised to the*m - 2***8th**powerthe sum of twice the cube of and its square root*b*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 "" are -- otherwise how can anyone expect you to collect them?*like terms*Again -- as always in math -- the words say it all.

**in***like terms***the****variable****and its****powers****are****identical****.**So,

**x**and^{ 3}, 17x^{ 3}, -9x^{ 3}, 2ax^{ 3}**4267x**are all^{ 3}*like terms*When I collect them I will get

**(4276 + 2a )x**^{ 3}Whereas

**x**and^{ 3}, 17^{ 2}, -9x^{ 7}, 2ax^{ - 8}**4267x**are^{ 0. 357}__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}**3x**and**7x**are like terms =**10x****-5y**and -**2y**are like terms =**- 7y****x**has no buddies, so he stands alone.^{ 2}**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**-- always!!__perfect sense__**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 - ).*