Algebra/Problem Bank

Put potential problems for the math WikiBooks here:

Algebra

A.1 (Determining Properties of Real Numbers) Determine if the following statements are always, sometimes, or never true. If the statement is always true, explain your reasoning. If the statement is not always true, provide a counterexample.

$a.\ An\ integer\ is\ a\ whole\ number.$
$b.\ If\ a\ number\ is\ whole\ it\ is\ a\ natural\ number.$
$c.\ If\ a\ number\ contains\ a\ decimal\ it\ is\ an\ integer.$
$d.\ If\ a\ number\ is\ nautural,\ then\ it\ is\ a\ real\ number.$
$e.\ The\ product\ of\ two\ irrational\ numbers\ is\ an\ irrational\ number.$

A.2 (Identifying Properties of Real Numbers) Identify the following properties being expressed.

$a.\ 4(3x+4)=12x+16$
$b.\ 6+0=6$
$c.\ (2+7)+5=(2+5)+7$
$d.\ (3/4)(4/3)=1$
$e.\ To\ divide\ 3072\ by\ 512,\ you\ can\ divide\ 3072\ by\ 16,\ again\ by\ 8,\ and\ again\ by\ 4.$

A.3 (Product Pattern) Use the Associative Law to explain why the products in each rule are equal.

$2*2=1*4$

$4*3=2*6$

$6*4=3*8$

$8*5=4*10$

$10*6=5*12$

$12*7=6*14$

$14*8=7*16$

A.4 (Suare of Sum/Difference) For two numbers $a$ and $b$, find the following:

$a.\ (a+b)^{2}$
$b.\ (a-b)^{2}$

A.5 (Secret of 1001) A boy claims that he can figure out the product of any three digit number and 1001. A student in his arithmetic class challenges him to find the product of 1001 and 865, and he gets the correct answer immediately. Compute the answer, and determine the boy's secret.

A.6 (ABCD) Prove that the following expression can be written as a product between $a-d$ and $b+c$

$ab-cd+ac-bd$

A.7 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.

$a.\ (a+b)+(c+d)=(a+d)+(b+c)$
$b.\ (a+b)+(c+d)=(a+c)+(b+d)$
$c.\ (a-b)+(c-d)=(a+c)+(-b-c)$
$d.\ (a-b)+(c-d)=(a+d)-(b+c)$
$e.\ (a-b)+(c-d)=(a-d)+(c-b)$
$f.\ (a-b)+(c-d)=-(b+d)+(a+c)$
$g.\ ((a+b)+c)+d=(a+c)+(b+d)$
$h.\ (a-b)-(c-d)=(a-c)+(d-b)$

A.8 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If $a$, $b$, and $c$ are integers, then the number $ab+bc$ is an even number.
b. If $b$ and $c$ are odd integers, and $a$ is an integer, then the number $ab+bc$ is an even number.

A.9 (Using Properties of Numbers) We define an integer $a$ to be of

• Type I if $a=4n$ for some integer $n$
• Type II if $a=4n+1$ for some integer $n$
• Type III if $a=4n+2$ for some integer $n$
• Type IV if $a=4n+3$ for some integer $n$

a. Provide at least two examples of each of the four types of integers above.
b. Is it true that if $a$ is even, then it is of type I or III? Justify your answer.
c. Is it true that if $a*b$ is of type I, whenever $a$ or $b$ are of type III? Justify your answer.