algebra introduction

7/25/2019 Algebra Introduction

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FUN TIME WITH ALGEBRA

What is the missing number?

2 = 4

OK, the anser is !, right? Be"ause 6 2 = 4# Eas$ stu%%#

We&&, in A&gebra e '(n)t use b&an* b(+es, e use a letterusua&&$ an + (r $, but an$

&etter is %ine-# .( e rite/

x 2 = 4It is rea&&$ that sim0&e# The &etter in this "ase an +- 1ust means 2e '(n)t *n( this $et2,

an' is (%ten "a&&e' the unknown(r the variable#

An' hen e s(&3e it e rite/

x = 6

Algebra - Basic Definitions

It may help you to read Introduction to Algebra first

What is an E4uati(n

An e4uati(n sa$s that t( things are e4ua It i&& ha3e an e4ua&s sign 252 &i*e this/

x + 2 = 6

That e4uati(n sa$s/ what is on the left (x + 2 is e!ual to what is on the right (6

.( an e4uati(n is &i*e a state"ent2thise4ua&s that2

6arts (% an E4uati(n

.( 0e(0&e "an ta&* ab(ut e4uati(ns, there are na"es%(r 'i%%erent 0arts better than

sa$ing 2that thing$ there27-

Here e ha3e an e4uati(n that sa$s 8+ 9 :e4ua&s ;, an' a&& its 0arts/
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A #ariableis a s$mb(& %(r a number e '(n)t *n( $et# It is usua&&$ a &etter &i*e + (r $#

A number (n its (n is "a&&e' a $onstant#

A $oefficientis a number use' t( mu&ti0&$ a 3ariab&e 4xmeans 4times x, s( 4is a

"(e%%i"ient-

.(metimes a &etter stan's in %(r the number/

E+am0&e/ a+


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The e+0(nent su"h as the < in +


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Li*e Terms are ter"sh(se 3ariab&es an' their e+0(nents su"h as the < in +


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Wh$ 'i' e a'' < t( b(th si'es?

T( 2*ee0 the ba&an"e2###

In Balance

Add 2 to Left Side

Out of Balance!

Add 2 to i!ht Side Also

In Balance A!ain

Dust remember this/

To ee# the $alance, what we do to one sideof the "%"

we sho&ld also do to the other side'

See this in action at the Algebra Balance Animation .
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An(ther 6u&e.(&3e this (ne/

x + 3 = 02

Start with: ( ) * % +2

What we are aimin! for is an answer lie "( % ...", and the plus 5is in the wa of that'

We can cancel o&t theplus 5$ doin! asubtract 5$eca&se **%/0

So, let &s have a !o at s&$tractin! * from both sides: ()* 5% +2 5

A little arithmetic ** % / and +2* % 10 $ecomes: ()/ % 1

Which is &st: x = 7

Solved!

3&ic 4hec: 1)*%+20

Ha3e a Tr$ (urse&%

.(&3e the %(&&(ing1:

+= 5

2:

+ 5

3:

+=: 5 :

4:


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=+ 5 @ = < 5 20

*'ample:I% +5@ but e '(n)t *n( 2$2-, then hat is x2+ x)?

6ut 2@2 here 2+2 is/

@ 9


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T( un&i*e signs be"(me a negative

sign

:=9 9


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These are a&& e4uati(ns, but (n&$ s(me are %(rmu&as/

x 2y- # $o!%u&a (!e&atin' xand y)

a2 b2 c2 $o!%u&a (!e&atin' a band c)

x/2 # * +ot a $o!%u&a (,ust an euation)

With(ut the E4ua&s

.(metimes a %(rmu&a is ritten ith(ut the 252/

E+am0&e/ The %(rmu&a %(r the 3(&ume (% a b(+ is/

lwh

But in a a$ the 252 is sti&& there, be"ause e "an rite # = lwhi% e ant t(#

.ub1e"t (% a F(rmu&a

The 2sub1e"t2 (% a %(rmu&a is the sing&e 3ariab&e usua&&$ (n the &e%t (% the 252- that

e3er$thing e&se is e4ua& t(#

E+am0&e/ in the %(rmu&a

s 5 ut = at

; 5 23

! .4uare' 5 !< 5 ! > ! 5 ,6
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The s4uares are a&s(

(n the Mu&ti0&i"ati(n Tab&e/

Negati3e Numbers

We "an a&s( s4uare negative nu"bers#

E+am0&e/ What ha00ens hen e s4uare 9;- ?

Anser/

9;- > 9;- 5 23

be"ause a negati3e times a negati3e gi3es a 0(siti3e -

That as interesting7

When e s4uare a negativenumber e get a &ositiveresu&t#

Dust the same as s4uaring a 0(siti3e number/

F(r m(re 'etai& rea' .4uares an' .4uare R((ts in A&gebra -

.4uare R((ts

A s!uare rootg(es the (ther a$/
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@ s4uare' is , s( a s!uare root of is ,

A s4uare r((t (% a number is ###

### a 3a&ue that "an be"ulti&lie b) itselft( gi3e the (rigina& number#

A s4uare r((t (% is ###

### ,, be"ause when , is "ulti&lie b) itselfe get #

It is &i*e as*ing/

What "an e mu&ti0&$ b$ itse&% t( get this?

'o hel& )ou re"e"berthin* (% the r((t (% a tree/

&I (no, the tree- but ,hat is the root that made it2

In this "ase the tree is 22, an' the r((t is 2@2#

Here are s(me m(re s4uares an' s4uare r((ts/

Ce"ima& Numbers

It a&s( (r*s %(r 'e"ima& numbers#

Tr$ the s&i'ers be&(# /ote: the numbers here are only sho,n to 2 decimal places.

@ 2015 MathsIsFun.com v 0.81

Using the s&i'ers remembering it is (n&$ a""urate t( < 'e"ima& 0&a"es-/

What is the s4uare r((t (% *?


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What is the s4uare r((t (% ?

What is the s4uare r((t (% 0.?

What is 0s4uare'?

What is 010s4uare'?

What is 216s4uare'?

Negati3es

We %(un' (ut be%(re that e "an s4uare negati3e numbers/

E+am0&e/ 9@- s4uare'

9@- > 9@- 5

An' (% "(urse @ > @ 5 a&s(#

.( the s4uare r((t (% "(u&' be ,(r +,

E+am0&e/ What are the s4uare r((ts (% 9;- 5 ; 5


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We&&, e 1ust ha00en t( *n( that ;, s( hen e mu&ti0&$ ; b$ itse&% ; > ;- e

i&& get


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Getting "&(ser t( , but it i&& ta*e a &(ng time t( get a g((' anser7

At this #oint, I !et o&t m calc&lator and it sas:

3.1622776601683793319988935444327

B&t the di!its &st !o on and on, witho&t an #attern.

So even the calc&lator5s answer is only an approximation!

/ote: numbers li(e that are called Irrational /umbers- if you ,ant to (no, more.

The Easiest Wa$ t( a&"u&ate a .4uare R((t

Use $(ur "a&"u&at(r)s s4uare r((t butt(n7

An' a&s( use $(ur "(mm(n sense t( ma*e sure $(u ha3e the right anser#

A Fun Wa$ t( a&"u&ate a .4uare R((t

There is a %un meth(' %(r "a&"u&ating a s4uare r((t that gets m(re an' m(re a""urate

ea"h time ar(un'/

a- start ith a guess&et)s guess 8 is the s4uare r((t (% -

b- 'i3i'e b$ the guess8 5


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H( t( Guess

What i% e ha3e t( guess the s4uare r((t %(r a 'i%%i"u&t number su"h as 2 ; > ; 5


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! "ube' 5 !@ 5 ! > ! > ! 5


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111 so the answer is 3

The ube R((t .$mb(&

This is the s0e"ia& s$mb(& that means 2"ube r((t2, it is the"radical"s$mb(& use'

%(r s4uare r((ts- ith a &itt&e three t( mean cuber((t#

(u "an use it &i*e this/ e sa$ 2the "ube r((t (% 9; 5 9


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3.1%"232%$3###!!#""!!242"224 ###

### but the 'igits 1ust g( (n an' (n, ith(ut an$ 0attern# .( e3en the "a&"u&at(r)s anser

is onl) an approximationI

urther reading: these (ind of numbers are called surds ,hich are a special type

ofirrational number

nth 8oot

he &nth 5oot& used n timesin a multiplicationgives the original value

2 nth ? 2

0st, 2n', ,r', 4th, 3th, ### nth ###

Instea' (% ta&*ing ab(ut the 28th2, 2!th2, et", i% e ant t( ta&* genera&&$ e sa$ the

2nth2#

The nth R((t

The 2


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Using it

We "(u&' use the nth r((t in a 4uesti(n &i*e this/

uesti(n/ What is 2n2 in this e4uati(n?

Anser/ I 1ust ha00en t( *n( that 623 = 34, s( the 4th r((t (% !


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E+am0&e/

A''iti(n an' .ubtra"ti(n

But e cannot'( that *in' (% thing %(r a''iti(ns (r subtra"ti(ns 7

E+am0&e/ 6$thag(ras) The(rem sa$s

a2 b2 c2

.( e "an "a&"u&ate " &i*e this/

" 5 Qa


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When a 3a&ue has an ex&onent of nan' e ta*e the nth roote get the value back

again###

### hen a is &ositive(r er(-/ i.e. for a 0

E+am0&e/

### (r hen the ex&onent is o/ i.e. ,hen n is odd

E+am0&e/

### but hen a is negativean' the ex&onent is evene get this/

Ci' $(u see that @ be"ame =@ ?

### s( e ha3e/ ,hen n is even

N(te/ JaJmeans the abs(&ute 3a&ue (% a, in (ther (r's an$ negati3e be"(mes a

0(siti3e-

E+am0&e/

.( that is s(mething t( be "are%u& (%7 Rea' m(re at E+0(nents (% Negati3e Numbers #

Here it is in a &itt&e tab&e/

" " ee

a 0

a 0

nth R((t (% at(themth6(er
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N( &et)s see hat ha00ens hen the e+0(nent an' r((t are 'i%%erent 3a&ues "an' n-#

E+am0&e/

.( ### e "an m(3e the e+0(nent 2(ut %r(m un'er2 the nth r((t, hi"h ma$ s(metimes be

he&0%u

But there is an e3en "ore &owerful "etho### e "an "(mbine the e+0(nent an' r((t

t( ma*e a ne e+0(nent, &i*e this/

E+am0&e/

That is be"ause the nth rootis the same as an ex&onent of (05n/

E+am0&e/


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An e+0(nent (% 5

.( is the same as Q

Tr$ An(ther Fra"ti(n

Let us tr$ that again, but ith an e+0(nent (% (ne4uarter 8-/

E+am0&e/ +S
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+S> +S> +S> +S5 +S=S=S=S-5 +-5 +

.( +S, hen use' 8 times in a mu&ti0&i"ati(n gi3es +, an' s(xLis the 4th root of x#

Genera& Ru&e

It (r*e' %(r M, it (r*e' ith L, in %a"t it (r*s genera&&$/

+n5 The n-th R((t (% +

.( e "an "(me u0 ith this/

A !actiona& exonent &ike 1/ %eans toae he -h

!:

E+am0&e/ What is


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A fractional e(#onent lie m/n means:

Do the m-th power, then tae the n-th root

OR Tae then-th rootand then do the m-th power

.(me e+am0&es/

E+am0&e/ What is 8@8- 5 Q!8- 5 *

(r

8@


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.tart ith m5 an' n5, then s&(&$ in"rease n s( that $(u "an see


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We&& ar(un' Q< 5


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.( Q< is sim0&er as @ > @ > ;- 5 Q@ > @- > Q< > ;- 5 @Q

Fra"ti(ns

There is a simi&ar ru&e %(r %ra"ti(ns/

E+am0&e/ sim0&i%$ Q@ Q

First e "an "(mbine the t( numbers/

Q@ Q 5 Q@ -

Then sim0&i%$/

Q@ - 5 Q@

.(me Har'er E+am0&es

E+am0&e/ sim0&i%$ Q Q;- Q

algebra introduction

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