Ch. 28 ASVAB Numbers

Which of the following statements about natural numbers is FALSE? A.) Natural numbers are never negative. B.) If you add two natural numbers, you always get a natural number. C.) If you multiply two natural numbers, you always get a natural number. D.) If you subtract two natural numbers, you always get a natural number.

D.) If you subtract two natural numbers, you always get a natural number.

Are the example 'coins' considered natural numbers or non-natural numbers?

Coins are not natural numbers. Any amount of coins less than 100 ends up being a decimal and therefore NOT a natural number.

Which one of the following is a natural number? A.) - 5 B.) 2.112 C.) 3/7 D.) 1,000,000

D.) 1,000,000

Identify all of the counting numbers in the following list: 1/2, 5/3, 15/5 A.) None of the numbers listed are Counting Numbers B.) 1/2 C.) 1/2 and 15/5 D.) 15/5

D.) 15/5 x x x x x 5 o o o o o o o o o o o o o o o 15 = 3

What is the golden ratio? A.) A fraction printed in a yellow font B.) The circumference of a circle divided by its diameter C.) An ancient Greek hoax D.) An irrational number, often called phi

D.) An irrational number, often called phi

Why is 10 a rational number? A.) Because it never ends. B.) Because it is a positive number. C.) Because it is an even number. D.) Because it can be written as a fraction, e.g., 10/1.

D.) Because it can be written as a fraction, e.g., 10/1.

Which of the following are natural numbers? A.) Numbers with several places after the decimal point. B.) Negative numbers C.) Fractions that can't be simplified D.) Positive integers

D.) Positive integers

What classification of numbers contains both integers and non-integers? A.) The number of pets you have. B.) Whole numbers C.) Natural numbers D.) Rational numbers

D.) Rational numbers Explanation Just a side note - some classifications of numbers contain both integers and non-integers. Rational numbers are numbers that can be written as fractions. As a result, many rational numbers like 1/2 or 1/4 are not integers, because they have a fractional piece that can't be simplified. However, integers like 1 or 2 are both rational numbers and integers.

golden ratio

Phi (also called golden mean or divine proportion) Another transcendental irrational number is derived from the ratios of the sides of certain geometric shapes

Is √(7)2 a rational number or not? Why?

This is a rational number. The square of a square root is the number inside the square root. So this would be 7, a rational number.

Can two irrational numbers be combined using multiplication or division to get a rational number?

This is a trick question. Two radicals, in some cases, can be multiplied to get a rational number. For example, √(8) * √(2) = √(16) = 4.

Can two irrational numbers be combined using addition or subtraction to get a rational number?

Two irrational numbers cannot be combined with addition or subtraction to get a rational number unless the irrationals cancel each other out as in π + -π.

Can two rational numbers be combined using addition or subtraction to get an irrational number?

Two rational numbers cannot be combined using addition or subtraction to get an irrational number.

Can two rational numbers be combined using multiplication or division to get an irrational number?

Two rational numbers cannot be combined using multiplication or division to get an irrational number.

Whole numbers

Whole numbers is a subset of an integer and includes all positive numbers and 0. {0, 1, 2, 3, ...}

Is zero a rational number or not? Why?

Zero is a rational number. It can be written as 0/1.

irrational numbers

a kind of number that cannot be written as a fraction (ratio) using only positive and negative counting numbers (integers) Examples: - can write the rational number 2.11 as 211/100 - cannot turn the irrational number 'square root of 2' into an exact fraction of any kind

negative number

a number with a value less than zero

Rational numbers

are numbers that can be represented as the ratio of two integers. - Consists of positive numbers, negative numbers - Can be written as fraction - can be positive, negative or 0

Natural numbers

do not include zero

N 1

do not want the 0 included

e = ?

e = 2.718281828...

What are the different subsets of integers?

-whole numbers - natural numbers - negative numbers - prime numbers - even numbers - odd numbers i

Is 0 a counting number?

0 is not a counting number. When counting, the first number is always a 1. Counting numbers are the positive integers beginning with 1.

whole number

0, 1, 2, 3, 4, 5, 6, 7, 8 Are all positive numbers, including zero, which do not include any fractional or decimal parts (1/2 and 0.5)

Turn the terminating decimal 0.123 into a rational number

0.123 = 123/10^3 10 * 10 *10 = 1000 123/1000 = cannot be simplified anymore.

Turn the terminal decimal 0.2 into a rational number

0.2 = 2/10 2/10 = 1/5 2 * 1 = 2 (no) 2 * 2 = 4 (no) 2 * 3 = 6 (no) 2 * 4 = 8 (no) 2 * 5 = 10 (yes) Using a power of 10 as the denominator allows us to quickly find an appropriate fraction representation of each of these numbers, which can then be reduced to lowest terms.

Write a repeating decimal 0.333.... as a fraction.

0.333... = 1/3

Turn the rational number 1/3 into the terminating decimal

1/3 = 0.33333... = 0.3

Turn the rational number 17/99 into the terminating decimal

17/99 = 0.17171717... = 0.17

Is a natural number squared also a natural number? How about the square root of a natural number?

2) A natural number squared is a natural number multiplied by itself. So, a natural number squared is still a natural number, since a natural number times a natural number is a natural number. For example, 3^2 = 3*3 = 9 is still a natural number. The square root of a natural number can be a natural number, but usually is not. For example, the square root of 4 is 2, which is a natural number, but the square root of 5 is approximately 2.236 which is not a natural number since it is not a whole number. The square root of a natural number is not guaranteed to be a natural number.

Which of the following numbers are natural numbers? 3, 19, -9, 27.5, 1, -3. How do you know?

3, 19, 1 The numbers 3, 19, and 1 are natural numbers because they are positive whole numbers. -9 and -3 are not natural numbers because they are negative and 27.5 is not a natural number because it is not a whole number.

Repeating decimals

A decimal in which one or more digits repeat infinitely and does not end. Example: 0.3333333333333333

Subscript

A number written slightly below and to the right of a chemical symbol that shows how many atoms of an element are in a compound.

Is a piece of pie an example of natural numbers or non-natural numbers?

A piece of a pie. A piece or slice of pie is always a part of a whole pie, so it always ends up being a fraction or decimal. For example, taking a slice equal to a quarter of the pie, the fraction is 1/4, which is, therefore, NOT a natural number.

Which of the following numbers is not irrational? A.) 2.11 B.) e C.) Φ D.) Square root of 7

A.) 2.11

Odd numbers

Odd numbers are not divisible by two and are both positive and negative.{..., -5, -3, -1, 1, 3, 5, ...}

π = ?

(pi) 3.14 or 3.1415926535...

e

(Also called Napier's constant but is commonly called Euler's number) the result of adding a tiny bit to 1 and then raising that to a really big power

Nonnegative integers

(Whole numbers) 0, 1, 2, 3, 4...

Counting Numbers

(also called natural numbers & positive integers) are the numbers people use to count with. When counting, the first number said is one, then two, then three, and so forth. 1, 2, 3, 4, ...

ϕ = ?

(phi) 1 + √521 + 52 or approximately 1.6180339887...

What are the different properties of whole numbers?

- The associative property - The closure property - The commutative property - The distributive property

A decimal number is rational if either:

- The decimal expansion terminates - The decimal expansion doesn't terminate, but takes on a repeating sequence of digits that goes on forever past its decimal point.

Ratio

- a comparison of two or more numbers - often written as a fraction

Terminating decimal

- a decimal that ends - all terminating decimal numbers are rational numbers because they can be converted to fractions. Examples: 1.2 --> 12/10 or --> 6/5 3.25 --> 325/100

Turn the terminating decimal -0.45 into a rational number

-0.45 = -45/10^2 10 * 10 = 100 = -45/100 = -9/20 5 * 1 = 5 (no) 5 * 2 = 10 (no) 5 * 3 = 15 (no) 5 * 4 = 20 (no) 5 * 5 = 25 (no) 5 * 6 = 30 (no) 5 * 7 = 35 (no) 5 * 8 = 40 (no) 5 * 9 = 45 (yes) 5 * 10 = 50 (no) 5 * 11 = 55 (no) 5 * 12 = 60 (no) 5 * 13 = 65 (no) 5 * 14 = 70 (no) 5 * 15 = 75 (no) 5 * 16 = 80 (no) 5 * 17 = 85 (no) 5 * 18 = 90 (no) 5 * 19 = 95 (no) 5 * 20 = 100 (yes) Using a power of 10 as the denominator allows us to quickly find an appropriate fraction representation of each of these numbers, which can then be reduced to lowest terms.

Which item below would NOT necessarily be measured in integers? A.) Bank account balance B.) The number of coins you have in a jar C.) How many friends you have D.) The number of socks you have

A.) Bank account balance Explanation The monetary value of a bank account is an example of a non-integer. Whether someone has a positive or negative balance, that balance is likely to include some decimal part of a dollar: $542.35; or maybe -$12.25 if over drawn. These would be examples of non-integers, because integers do not include any decimal portion.

Is zero a natural number? A.) It depends on who you ask. B.) Yes! C.) Only on Sundays. D.) No!

A.) It depends on who you ask.

Rational numbers are numbers that _____. A.) can be written as a fraction B.) have radicals in them C.) are only positive D.) never repeat and never end

A.) can be written as a fraction

positive integers

All numbers greater than zero. 1, 2, 3, 4, 5, ...

Integer

All whole numbers (both positive and negative) and zero. Does not include any fractional or decimals.

Identify all of the counting numbers in the following list: -1, 0, 1, 1.1 A.) 0, 1 B.) 1 C.) -1, 1 D.) -1, 0, 1

B.) 1

How do you get the number pi? A.) 3.14 B.) The circumference of a circle divided by its diameter C.) e D.) 22/7

B.) The circumference of a circle divided by its diameter

Find the TRUE statement. A.) The natural numbers include fractions. B.) The natural numbers are the counting numbers. C.) The natural numbers go up to 100. D.) The natural numbers can be positive or negative.

B.) The natural numbers are the counting numbers.

All positive numbers, including zero, which do not include fractional or decimal parts are called: A.) Natural numbers B.) Whole numbers C.) Counting numbers D.) Real numbers

B.) Whole numbers Explanation Whole numbers are all positive numbers, including zero, which do not include fractional or decimal parts.

Is the number .313131313131... rational? Why? A.) No, because it is a repeating decimal. B.) Yes, because it is a repeating decimal. C.) Yes, because it never repeats and never ends. D.) No, because it never ends.

B.) Yes, because it is a repeating decimal.

Which of the following types of numbers is NOT rational? A.) Fractions B.) pi C.) Repeating decimals D.) Mixed numbers

B.) pi

Identify all of the counting numbers in the following list: 0, 1, 2, 3, 4, 5 A.) 0 B.) 0, 1, 2, 3, 4, 5 C.) 1, 2, 3, 4, 5 D.) 1, 2, 3

C.) 1, 2, 3, 4, 5

Which of these problems has a whole number for an answer? A.) 10 / 4 B.) 10 / 8 C.) 10 / 5 D.) 10 / 6

C.) 10 / 5 x x x x x 5 o o o o o o o o o o 10 = 2 Explanation When you divide 10 by 5 you get 2. Two is a whole number because it has no decimal or fractional part. If you divide any of the other numbers into ten they do not go in evenly. As a result, you will end up with a fraction or decimal answer. For example: 10 / 4 = 2.5 The decimal part of the 2.5 means that it is not a whole number.

Which of the following numbers is a whole number? A.) 12 1/2 B.) 1/2 C.) 12 D.) 1.2

C.) 12 Explanation Whole numbers are numbers that do not have any fractional or decimal parts

Identify all of the counting numbers in the following list: 6.2, 3, 80, -7 A.) 6.2, 80, -7 B.) 3 C.) 3, 80 D.) 6.2, 3, 80, -7

C.) 3, 80

Identify all of the counting numbers in the following list: 356, 98, 11, 2, 101 A.) 2, 11, 98 B.) 2 C.) All of the numbers in the list are counting numbers. D.) 2, 11

C.) All of the numbers in the list are counting numbers.

The set of all integers includes which of the following: A.) Only negative numbers B.) Positive and negative numbers C.) Positive, negative numbers and zero D.P) Only positive numbers

C.) Positive, negative numbers and zero

What is the difference between whole numbers and natural numbers? A.) There is no difference; they are the same. B.) Whole numbers include positive numbers and natural numbers do not. C.) Whole numbers include zero, but natural numbers do not. D.) Whole numbers include fractions and natural numbers do not.

C.) Whole numbers include zero, but natural numbers do not. Explanation The difference between whole numbers and natural numbers is that the former includes zero, while the latter does not.

Which of the following might result in an answer that is NOT a whole number? A.) The number of shoes in your closet B.) The number of cars in a parking lot C.) The number of fish in a tank D.) The distance someone drives to work

D.) The distance someone drives to work Explanation While all of the other answers can only be represented as whole pieces, the distance someone drives to work might be 3.5 miles. It is rare that someone will have a commute of exactly a whole number of miles, so that answer will likely give you a value that is not a whole number.

Which of the following could be measured using only integers? A.) The amount of taxes owed at the end of a year B.) The mileage on a car C.) The degree to which a cup is full D.) The number of pennies put into or taken out of a jar

D.) The number of pennies put into or taken out of a jar Explanation Only the amount of pennies taken from or added to a jar will always be a whole value with no fractional or decimal parts. All the other possibilities are likely to include fractions or decimals.

Which of the following is a rational number? A.) The square root of 20 B.) The square root of 15 C.) The square root of 2 D.) The square root of 9

D.) The square root of 9 1 * 1 = 1 2 * 2 = 4 3 * 3 = 9 square root of 9 = 3

Which of the following types of numbers is NOT rational? A.) Repeating decimals B.) Mixed numbers C.) Fractions D.) pi

D.) pi

The complete set of integers is represented by which of the following: A.) {...-1 1/2, -1, -1/2, 0, 1/2, 1, 1 1/2...} B.) {0, 1, 1.5, 2, 2.5, 3, 3.5, 4...} C.) {-2, -1, .1, .2} D.) {...-2, -1, 0, 1, 2...}

D.) {...-2, -1, 0, 1, 2...}

Even numbers

Even numbers is a subset containing only numbers divisible by two and include positive and negative.{..., -4, -2, 0, 2, 4, ...}

When is a decimal number considered a counting number?

For example, the decimal 41.00 is a counting number even though it is a decimal with digits after. As long as the numbers after the decimal are zero, and the decimal itself is positive, then it is actually a counting number. These types of decimals with trailing zeros after the decimal point are most often found in engineering and statistics books and manuals.

How tall people are is considered to be a natural number. True or false?

How tall people are can be a natural number, but most of the time are not. For example, measuring a friend's height, it's usually several feet followed by some inches. When a measurement of feet has inches, it ends up as either a fraction or a decimal for the height in feet. Remember, fractions and decimals and NOT natural numbers.

Decimals which don't terminate and don't repeat represent?

Irrational numbers

Do you think e + e is a rational number or not? Why?

It is not a rational number, since e added to itself is irrational.

Do you think π2 is a rational number or not? Why?

It is not a rational number. Every multiple of π is irrational.

Are money bills an example of natural numbers or non-natural numbers?

Money bills are examples of natural numbers because they are all positive integers. For example

Natural numbers

Natural numbers is similar to the subset of the whole numbers; however, it does not include 0. {1, 2, 3, ...}

Is 4 an irrational number?

No, 4 is not an irrational number. It can be written as the fraction 4/1, which means that it is rational, not irrational.

Is 4.789 an irrational number?

No, 4.789 is not an irrational number. It is a terminating decimal, and terminating decimals are never irrational. In addition, it can be written as a fraction of two integers: 4.789 = 4,789/1,000.

Is 7/9 an irrational number?

No, 7/9 is not an irrational number. It is the ratio of two integers, so it is rational, not irrational. In addition, it is equal to 0.7777777777..., which clearly has a repeating pattern.

Are speed limits an example of natural numbers or non-natural numbers?

Speed limits are natural numbers because they are all positive and integers as well. For example, you'll see 25 mph, 35 mph, 65 mph, 75 mph, etc.

Is the example of 'the number of people who like chocolates' considered natural numbers or non-natural numbers?

The number of people who like chocolates is also a natural number. For example, numbers such as 1, 10, 275, etc., are all natural numbers. There won't be negative numbers, fractions, or decimals here.

What is not included in the counting numbers?

fractions, decimals, negative numbers, and 0 are not included

Distributive Property of Whole Numbers

governs how to remove parentheses from an expression including variables and multiplication. For example: 9(x + 3) 9(x + 3) 9 ⋅ x + 9 ⋅ 39 ⋅ x + 9 ⋅ 3 9x + 27

Set notation

group of items that you have carefully listed or described

integers

include all whole numbers as well as their negative opposites.

N 0

means for 0 to be included

negative integers

negative integers is a subset within the integer set that only includes negative integers. A related set called negative numbers includes all negative numbers, including fractions and decimals. {..., -3, -2, -1}

Irrational numbers

numbers that cannot be written as one integer divided by another integer - cannot be written as fractions - cannot be written as terminating decimals - cannot be written as repeating decimals

pi

part of a group of special irrational numbers that are sometimes called transcendental numbers cannot be written as roots

What are the special numbers of irrational numbers?

pi and e

prime numbers

prime numbers only prime integers; numbers can only divide by the number one and themselves. {1, 3, 5, 7, 13, ...}

What are the two types of numbers that are irrational numbers?

roots and radicals

Turn the rational number 5/7 into the terminating decimal

this can only get with calculator (won't be using calculator on ASVAB) answer: 0.714285714285...= 0.714285

Closure Property of Whole Numbers

when two whole numbers are added or multiplied, the product is always a whole number. This property does not apply to subtraction or division. Some examples are: 3 + 10 = 133 + 10 = 13 2 ⋅ 0 = 0

Associative Property of Whole Numbers

when whole numbers are added or multiplied, how they are grouped does not matter. Here's an example: 3 + 8 + (7+2) = 203 + 8 + (7+2) = 20 (3+8) +7 + 2 = 20 (3+8) + 7 + 2 = 20 3 + (8+7) +2 = 20

Commutative Property of Whole Numbers

when whole numbers are added or multiplied, how they are ordered does not matter. An example is: 9 + 2 = 119 + 2 = 11 2 + 9 = 112 + 9 = 11 Like the associative property, the commutative property always applies to problems where either addition or multiplication is used. It does not always apply to problems including both addition and multiplication.

Simplify the fraction 10/2

x x 2 o o o o o o o o o o 10 = 5

π will always equal?

π = 3.14 or 3.141592.... When making calculations with π, it is often approximated as 3.14 or by the fraction 22/7