Subject Area Exam - Math

integers (#)

(-5,-4,-3,-2,-1,0,1,2,3,4,5,6)

Why is it true that 3(4 + x) = 3(x + 4)?

All they did was move stuff around: Commutative Property

distributive property

Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.

True or False: A rational is an integer.

Not necessarily; 4/1 is an integer, but 2/3 is not! So this is false.

associative property

The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property. Copyright

Why is 12 - 3x = 3(4 - x)?

They factored: Distributive Property

Use the Commutative Property to restate "3×4×x" in at least two ways.

They want you to move stuff around, not simplify. In other words, the answer is not "12x"; the answer is any two of the following: 4 × 3 × x, 4 × x × 3, 3 × x × 4, x × 3 × 4, and x × 4 × 3

True or False: A number is either a rational or an irrational, but not both.

True! In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types!

negative exponents vs negative numbers

Warning: A negative on an exponent and a negative on a number mean two very different things! For instance: -0.00036 = -3.6 × 10-4 0.00036 = 3.6 × 10-4 36,000 = 3.6 × 104 -36,000 = -3.6 × 104

composite numbers

When a number has more than two factors it is called a composite number. (everything but 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.)

sum estimation

round each number and then add Ex: 3429 + 4983= 3400 + 5000

-sqrt(81)

Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: -sqrt(81) = -9, which is just an integer. The answer is: integer, rational, real

Write in decimal notation: 3.6 × 1012

3,600,000,000,000, or 3.6 trillion

prime numbers

A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.)

Why is 3(4x) = (3×4)x?

All they did was regroup: Associative Property

number sequences

An Arithmetic Sequence is made by adding the same value each time. A Geometric Sequence is made by multiplying by the same value each time. This Triangular Number Sequence is generated from a pattern of dots which form a triangle. The Fibonacci Sequence is found by adding the two numbers before it together.

x^0

Anything to the power zero is just "1".

3.14159

Don't let this fool you! Yes, you often use something like this as an approximation of pi, but it isn't pi! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike pi which is irrational! The answer is: rational, real

The Sieve of Eratosthenes

Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind. (5-7-6-5) (Since you skipped from 5 to 7, go back in order and give 2 quarters (1 to 2 and 1 to 5) (everything else gets zero) To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the first one hundred positive integers (1-100): Cross out 1, because it is not prime. Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number. Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2. Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number. Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100!

factors

Factors are what we can multiple to get the number. The reverse of a multiple. The factors of 6 are 1, 2, and 3. The multiples of 6 are 12, 18, 24, etc.

Simplify 2(3x), and justify your steps.

In this case, they do want you to simplify, but you have to tell why it's okay to do... just exactly what you've always done. Here's how this works: 2(3x) - original(given) statement (2x3)x - by the Associative Property 6x - simplification (2x3 = 6)

multiples

Multiples are what we get after multiplying the number by an integer (not a fraction). The reverse of a factor. The factors of 6 are 1, 2, and 3. The multiples of 6 are 12, 18, 24, etc.

Simplify (x3)(x4)

Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents: (x^3)(x^4) = (xxx)(xxxx) = xxxxxxx = x7 ( x^m ) ( x^n ) = x(^m+n )

Simplify (x2)^4

Note that x8 also equals x( 2×4 ). This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power: (x2)^4 = (x^2)(x^2)(x^2)(x^2) = (xx)(xx)(xx)(xx) = xxxxxxxx = x^8

10

Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real

Convert 4.2 × 10-7 to decimal notation.

Since I need to move the point to get a small number, I'll be moving it to the left. The answer is 0.000 000 42

Why is it true that 3(4x) = (4x)(3)?

Since all they did was move stuff around (they didn't regroup), this is true by the Commutative Property.

Why is it true that 2(3x) = (2×3)x?

Since all they did was regroup things, this is true by the Associative Property.

True or False: An integer is a rational number.

Since any integer can be formatted as a fraction by putting it over 1, then this is true.

Why is the following true? 2(x + y) = 2x + 2y

Since they distributed through the parentheses, this is true by the Distributive Property.

Use the Distributive Property to rearrange: 4x - 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is "By the Distributive Property, 4x - 8 = 4(x - 2)"

commutative property

The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

Rearrange, using the Associative Property: 2(3x)

They want you to regroup things, not simplify things. In other words, they do not want you to say "6x". They want to see the following regrouping: (2×3)x

1 2/3

This can also be written as 5/3, which is the same as the previous problem. The answer is: rational, real

-9/3

This is a fraction, but notice that it reduces to -3, so this may also count as an integer. The answer is: integer (possibly), rational, real

5/3

This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real

Classify according to number type; some numbers may be of more than one type.

This is a terminating decimal, so it can be written as a fraction: 45/100 = 9/20. Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real

Write 124 in scientific notation.

This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)(100) = 124 is, and 100 = 102. Then, in scientific notation, 124 is written as 1.24 × 102.

Simplify [(3x4y7z12)5 (-5x9y3z4)2]^0

Who cares about that stuff inside the square brackets? I don't, because the zero power on the outside means that the value of the entire thing is just 1.

3.14159265358979323846264338327950288419716939937510...

You probably recognize this as being pi, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so pi is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real

rational numbers

a number that can be expressed as a ratio, where p and q are integers and q/= 0. these can be written as a fraction (all rational numbers are also real numbers) (3/4, 0.75, -5, 4, 10)

irrational numbers

a number that cannot be expressed as a repeating or terminating decimal (all irrational numbers are also real numbers) (pi, e, square root, the golden ratio)

whole numbers

all positive whole numbers including zero (all whole numbers are also integers) (0,1,2,3,4,5,6...)

natural numbers (counting numbers)

all positive whole numbers not including zero (all natural numbers are also whole numbers) (1,2,3,4,5,6...)

integers

all whole positive and negative whole numbers (all integers are also rational numbers) (-5,-4,-3,-2,-1,0,1,2,3,4,5,6)

arithmetic sequence

always adding or subtracting the same value

geometric sequence

always multiplying or dividing by the same value

number properties

associative, commutative and distributive

common factors

factors that are the same for two numbers. Ex. take all the factors of 18 (1,2,3,6,9, and 18) and compare them with all the factors of 24 (1,3,4,6,8,12,and 24) and find the similar factors: 1,2,3 and 6. These are the common factors. 6 is the largest factor that divides evenly into both 18 and 24.

real numbers

the set of numbers that includes all rational and irrational numbers. it is measurable and concrete. (3, -6, 0, 10.5, 3/4, pi, square roots, NOT 3+2i, 2i)

front end sum estimation

use just the first 2 numbers and change the rest to zeroes Ex: 3429 + 4983= 3400 + 4900

whole numbers (#)

(0,1,2,3,4,5,6...)

natural numbers (#)

(1,2,3,4,5,6...)

real numbers (#)

(3, -6, 0, 10.5, 3/4, pi, square roots, NOT 3+2i, 2i)

rational numbers (#)

(3/4, 0.75, -5, 4, 10)

irrational numbers (#)

(pi, e, square root of 2, the golden ratio)