Solutions for Lesson 1 Practice Questions

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Terminating Decimals

Which of the following can be written as terminating decimals?

a) \frac {2}{7}

Written as a decimal, \frac {2}{7} is 0.\overline{285714}. This is a repeating decimal, not a terminating one. 

b) \frac {3}{5}

Written as a decimal, \frac {3}{5} is 0.6. This is a terminating decimal. 

c) \frac {1}{9}

Written as a decimal, \frac{1}{9} is 0.\overline{1}. This is a repeating decimal, not a terminating one. 

d) \frac{3}{16}

Written as a decimal, \frac{3}{16} is 0.1875. This is a terminating decimal. 

e) 0.345

This number is already written as a decimal. Since there is no line over any of the numbers in the decimal, this decimal does not repeat. It is therefore a terminating decimal. 

f) 0.\overline{8}

This number is already written as a decimal. The line over the number 8 in the decimal indicates that this digit repeats forever, out to an infinite number of decimal places. Since this decimal repeats, it is not a terminating decimal. 

Integers

Which of the following are integers?
a) 0

0 is an integer, since it does not have a decimal. 

b) 0.5

0.5 is not an integer, since it has a decimal. 

c) 2

2 is an integer, since it does not have a decimal. 

d) \frac {6}{3}

\frac{6}{3} can be simplified to 2, which does not have a decimal, so this is an integer. 

e) \frac{3}{6}

\frac{3}{6} can be simplified to \frac{1}{2}, which can be written in decimal form as 0.5. \frac{3}{6} is therefore not an integer.

f) -6

-6 is an integer, since it does not have a decimal. 

g) -\frac{4}{2}

-\frac{4}{2} can be simplified to -2, which does not have a decimal, so this is an integer. 

Zero

Is zero... 

a) rational?

Yes! We can rewrite 0 as the ratio of two integers, like \frac{0}{1}, for example. 

b) irrational?

No! Since 0 is rational, we know that it can't be irrational. 

c) a whole number?

Yes! Whole numbers are 0, 1, 2, 3, 4 and so on. 0 is thus the smallest whole number, according to the definition we are using in this course. 

d) an integer?

Yes! 0 does not have a decimal, so it is an integer. 

e) a natural number?

No! In this course, we define natural numbers as 1, 2, 3, 4 and so on. The natural numbers are the whole numbers except for 0

f) a positive integer?

No! Zero is not a positive number, so although it is an integer, it is not a positive integer. 

g) a negative integer?

No! Zero is not a negative number, so although it is an integer, it is not a negative integer. 

Rational Numbers

Show that each of the following numbers are rational numbers by writing them in the form \frac{a}{b}, where a and b are integers and b\neq 0

a) 12

12 can be written as \frac{12}{1}, but there are many other ways to write it as a rational number as well. We could also say 12=\frac{24}{2}=\frac{36}{3}=\frac{48}{4}=\frac{156}{13}, just to name a few. 

b) 3\frac{2}{5}

3\frac{2}{5}=\frac{15}{5}+\frac{2}{5}=\frac{17}{5}

c) 0.7

0.7=7/10

Types of Number

a) Give an example of an integer which is neither negative nor positive. 

0 is the only integer that is neither negative nor positive. All of the other integers are either natural numbers (positive integers) or the negatives of the natural numbers (negative integers). 

b) What is the smallest natural number?

The smallest natural number is 1.  The natural numbers are 1, 2, 3, 4, 5 and so on. 

c) Give an example of a positive integer. 

Here you can put any natural number. I'll pick 11, since that is my favorite number :)

d) Give an example of a positive integer less than 20 whose square root is irrational. 

The only positive integers less than 20 that have rational square roots are 1, 4, 9, and 16, so for this question we can pick any positive integer less than 20 besides those numbers.  I could pick 11 again, or maybe I'd pick something different, like 8

e) Give an example of a positive integer whose square root is not irrational. 

Real numbers that are not irrational are rational, so we just need to pick a positive integer with a rational square root.  Examples of this would be 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and so on. 

Sets

What is the subset of the integers...
a) between 0 and 5 inclusive?

{0,1,2,3,4,5}

b) between -1 and 4 exclusive?

{0, 1, 2, 3}

Coefficients

  1. What is the coefficient of x in in the following expressions?
    a) 3x+5
    >! 3

b) 17-6x^2+2x

2

c) 12xy-3x+4

-3

  1. What is the coefficient of x^2 in the polynomial 5x^4+7x^3-11x^2+3?
    >! -11

Vocabulary - Algebra

Check the boxes corresponding to the correct statements. 

a) The sum of terms is called an equation. 

Incorrect. The sum of terms is an expression. An equation relates expressions that have the same value using an equals sign. 

b) Terms are sums of variables and constants. 

Incorrect. A term is the product of variables and/or constants. 

c) Coefficients are constants. 

Correct. The word "constant" can refer to either a constant term or a constant factor. Coefficients are constant factors, written at the beginning of terms. 

d) Any term which is just a number is a constant term and is sometimes just called a constant. 

Correct. 

e) An expression can be a single term. 

Correct. An expression can be one or more terms. 

Spotting Variables

The formula for the circumference C of a circle is C=2\pi r, where r is the radius of the circle.  What are the variables in the equation?

C,r

Investigating Integers

1) If the product of two integers is positive, then... 

c) either both are negative or both are positive. 

2) If the product of two integers is negative, then... 

d) one is positive and the other is negative. 

3) If the product of two integers is zero, then... 

e) one or both are zero. 

Sums & Products

For each row in the table, find the two integers whose sum and product are given.
a) sum = 12, product = 20

2, 10

b) sum = -12, product = 20

-2,-10

c) sum = 6, product = -7

-1, 7

d) sum = -6, product = -7

-7, 1

Inverses Part 1

a) What number can you add to 9 to get 0?

-9

b) If a is an integer, what number can you add to a to get 0?

-a

c) The answer to part b is _________ an integer. 

always

d) If a is instead any real number, what do you add to it to get 0?

-a

Inverses Part 2

a) What can you multiply by 9 to get 1?

\frac{1}{9}

b) There is one number that can't be multiplied by any other number to get 1. What number is it?

0

c) If a is an integer not equal to your answer to part b, what can you multiply a by to get 1?

\frac{1}{a}

d) The answer to part c is _________ an integer. 

sometimes

e) Fill in the gaps.
\frac{1}{2}\times ? = 1

2

\frac{3}{4}\times? = 1

\frac{4}{3}

\frac{a}{b}\times? = 1

\frac{b}{a}

Multiplying Integers

a) What is 7\times 8?

56

b) Is your answer an integer?

yes

c) When you multiply to integers together you _________ get an integer. 

always

Multiplying Rationals

a) What is \frac{2}{5}\times \frac{3}{7}?

\frac{6}{35}

b) Is your answer rational?

yes

c) Let's see if that's true in general by considering two rational numbers \frac{a}{b} and \frac{c}{d} (a, b, c, d are integers, b\neq 0, d\neq 0).  We need to find out if their product satisfies the definition of a rational number; that is, can it be written as a fraction where the numerator and the denominator are both integers and the denominator is not 0?
i) What is the product of \frac{a}{b} and \frac{c}{d}?

\frac{ac}{bd}

ii) Is the numerator an integer? Is the denominator an integer?

yes and yes

iii) Does it satisfy the definition of a rational number?

yes

So the product of rational numbers is ________ rational. 

always

Multiplying Irrationals

Round any irrational answers to 3 decimal places. (Hint: If your answer is not an integer, it's irrational.)
a) What is the product of \sqrt2 and \sqrt 8?

\sqrt 2\sqrt8= \sqrt{16} = 4

b) Is this irrational?

no

c) What is the product of \sqrt2 and \sqrt3?

\sqrt2\sqrt3 = \sqrt6\approx 2.449

d) Is this irrational?

yes

e) The product of irrational numbers is _________ irrational. 

sometimes