First flash cards

Question 1

Natural Numbers/Whole Numbers/Integers

Answer 1

The numbers you use to count objects, beginning with 1, are called the natural numbers. In set notation, this is written as {1,2,3…} . The “…” indicates that the numbers go on and on in this pattern. Adding zero (0) to this set of numbers gives you the set of whole numbers: {0,1,2…} . Adding negative numbers gives you the set of integers: {…−2,−1,0,1,2…} . Zero is an integer, but it is neither positive nor negative.

Question 2

Absolute Value

Answer 2

The value of a number, whether it is positive or negative, is called the absolute value. For example, 3 and −3 both have an absolute value of 3. For negative numbers, the higher the absolute value, the further to the left they appear on the number line, and for positive numbers, the higher the absolute value, the further to the right they appear.

Question 3

Rounding Numbers

Answer 3

Multidigit numbers can be rounded to any place. To round an integer to a place, look at the digit to the right of that place. If that digit is 5 or higher, increase the digit in the place you’re rounding to by 1, and if it is less than 5, keep that digit the same. Either way, replace digits to the right of the rounded place with zero(s). For example, to round 15,789 to the nearest thousand, look to the hundreds place. The digit in the hundreds place is 7, which is greater than 5, so we’ll increase the 5 in the thousands place to 6 and change all digits to the right of that to 0, giving us 16,000.

Question 4

Denominator

Answer 4

The number on the bottom of a fraction is called the denominator. The denominator tells how many parts the whole is divided into. The denominator can never be 0.

Question 5

Numerator

Answer 5

The number on the top of a fraction is called the numerator. The numerator tells how many of all the parts the fraction represents. For example, the fraction 12 means 1 out of 2 parts.

Question 6

Simplifying or Reducing Fractions

Answer 6

You can simplify or reduce a fraction by dividing both the numerator and the denominator by a common factor. For example:

912=9÷312÷3=34

Note: Any time you multiply or divide both parts of a fraction by the same number, the value stays the same.

Question 7

Adding or Subtracting Fractions with the Same Denominator

Answer 7

To add or subtract fractions with the same denominator, you can simply add the numerators. For example:

38+28=58

Question 8

Adding or Subtracting Fractions with Different Denominators

Answer 8

To add or subtract fractions with different denominators, you must first convert the fractions to equivalent fractions with a common denominator. To do this, find the lowest common multiple (LCM) of the denominators, multiply each denominator as necessary to make it equal to that common multiple, and multiply each numerator by the same number. For example:

14+13=1×34×3+1×43×4=312+412=712

Question 9

Multiplying Fractions

Answer 9

To multiply fractions, simply multiply the numerators and multiply the denominators and then simplify or reduce as needed. For example:

23×34=612=12

Question 9

Dividing Fractions

Answer 9

To divide fractions, multiply the first fraction by the reciprocal of second fraction. The reciprocal of a number is the number that when multiplied by that number gives a product of 1. For example, the reciprocal of 34 is 43 because 34×43=1212=1 . Notice that for a fraction, the reciprocal is that fraction flipped, or with the numerator and denominator reversed. Since all numbers are equal to themselves over 1, you can also use this rule for integers. For example, the reciprocal of 2 is 12 . Therefore, for example:

23÷2=23×12=26=13

Question 10

Distributive Property

Answer 10

The distributive property states that when multiplying a number by a sum of two numbers in parentheses, you can distribute the outside number to both numbers in the parentheses. For example:

12(13+14)=12(13)+12(14)=16+18=424+324=724

Question 11

Percentages

Answer 11

A percentage represents an amount out of 100, so decimals can also be converted to percentages by simply multiplying the decimal by 100%. For example:

0.5 = 0.5 × 100% = 50%

Question 12

Exponents

Answer 12

An exponent or a number raised to a power indicates the number of factors of that number that need to be multiplied together, and squaring a number means raising it to a power of two. For example:

42=4×4=16

Squaring 4 means multiplying two factors of 4 together, which gives you 16.

Question 12

Converting a Fraction to a Percentage

Answer 12

To convert a fraction to a percentage, convert the fraction to a decimal first, and then multiply the decimal by 100 and add the % symbol to the answer. To convert a percentage to a fraction, we reverse the process. First, convert the percentage to a decimal by dividing it by 100 and removing the % symbol. Then convert the decimal to a fraction as shown previously.

Question 13

Order of Operations

Answer 13

When performing operations on real numbers of any kind, it is important to remember to use the order of operations. We often use the acronym PEMDAS to help remember the order of operations. It stands for “parentheses, exponents, multiplication and division, addition and subtraction,” and that is the order in which operations in an equation should be performed.

Question 14

Commutative Property

Answer 14

Both addition and multiplication have a commutative property, meaning you can add or multiply two numbers in either order and the sum or product will be the same:

5+73×6==7+5=126×3=18

This does NOT apply to subtraction or division, however. If you subtract numbers in the opposite order, you get the additive inverse of the difference, and if you divide numbers in the opposite order, you get the multiplicative inverse or reciprocal of the quotient. For example:

7−55−76÷33÷6====2−2212

Question 15

Associative Property

Answer 15

Addition and multiplication have an associative property, meaning you can group additions or multiplications any way and the final sums or products come out the same. This property does not apply to subtraction or division. For example:

(7+8)+3(3×5)×6==(7+3)+8=18(3×6)×5=90

Question 16

Identity Property

Answer 16

Addition and multiplication each have their own identity property. For addition, any number plus zero equals itself, and for multiplication, any number times 1 equals itself. Multiplication also has its own zero property, which means that any number times 0 equals 0.

a+0a×1a×0===aa0

Question 17

Inverse Property

Answer 17

Multiplication has an inverse property, which means that any number times its multiplicative inverse or reciprocal always equals one. Note in the first equation that follows that when a fraction has the same numerator and denominator, it always equals 1.

5×15=5523×32ab×ba===111

Question 18

Distributive Property of Multiplication and Addition

Answer 18

The distributive property of multiplication and addition means that when multiplying a number by a sum of two or more numbers in parentheses, you can distribute the outside number to the numbers in the parentheses:

5(4+3)a(b+c)==5×4+5×3=20+15=35a×b+a×c

Question 19

Divisibility

Answer 19

We say a number is divisible by a second number if dividing the first by the second gives you an integer quotient. For example, 10 is divisible by 5 because 10 divided by 5 is 2, which is an integer, but 10 is not divisible by 4 because 10 divided by 4 is 2.5, which is not an integer. We call the set of numbers by which a number is divisible the factors of that number.

Question 20

Roman Numerals

Answer 20

There are some number systems that don’t use place or position value at all. For example, Roman numerals use different characters or letters to represent different values. The values of Roman numerals are shown in the following table:

Roman Numeral Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1,000

Question 20

Prime and Composite Numbers

Answer 20

All numbers can be classified as either prime or composite. Prime numbers are divisible only by themselves and 1, while composite numbers are divisible by other numbers, all of which can be written as products of prime numbers. All the factors of a composite number can be broken down into prime numbers, and that set of prime numbers is called the prime factorization of that number.

Question 21

Rules of Exponents: Product Rule

Answer 21

The product rule says that to multiply two exponents with the same base, you can simply keep the base the same and add the exponents.

43×45ax×ay==43+5=48ax+y

Question 22

Rules of Exponents: Quotient Rule

Answer 22

Similarly, the quotient rule says that to divide two exponents with the same base, you can simply keep the base the same and subtract the exponents.

86÷84ax÷ay==86−4=82ax−y

Question 23

Rules of Exponents: Power to a Power Rule

Answer 23

The power to a power rule says that to raise an exponent to a power, you can keep the base the same and multiply the exponents.

(53)4(ax)y==53×4=512axy

Question 23

Binomials

Answer 23

A binomial is an expression with two terms that is in parentheses and raised to a power such as:

(x4+x5)2

Question 24

Scientific Notation

Answer 24

One of the most important applications of exponents is using scientific notation to express very large or very small numbers. Scientific notation is written in the form a×10n , where 1≤|a|<10 and n is an integer.

To express a large number such as 30,000,000,000 (30 billion) in scientific notation, we move the decimal point to the left until the number is between 1 and 10 and then multiply by 10 to the power of the number of spaces the decimal was moved. In this case, we would need to move the decimal point 10 places to the left to get 3, so this number in scientific notation would be 3×1010 .

To express a small number such as 0.000005 (5 millionths) in scientific notation, we move the decimal point to the right until the number is between 1 and 10 and then multiply by 10 to the negative power of the number of spaces the decimal was moved. In this case, you would need to move the decimal point 6 places to the right to get 5, so this number in scientific notation would be 5×10−6 .

Question 25

The Nature of Sets

Answer 25

A set can be any grouping of elements or objects that is well defined. For example, if set S is defined as “the first five letters of the alphabet,” then it would contain the elements a,b,c,d, and e . The notation a∈S means “a is an element of S,” while the notation z∉S means “z is NOT an element of S.”

When working with sets, we call the largest set that contains all of the elements we are using the universal set, which is represented by the letter U. A set that contains no elements at all is called the empty set, also known as the null set or the void set. It is represented by the Greek letter phi (φ), by a circle with a line through it ( ⊘ ), or by a set of curly braces with nothing inside: { }. Two sets that share all of the same elements are called equal sets.

Question 26

Writing Sets: Roster Form

Answer 26

One way to write sets is called roster form, and it involves simply listing the elements of the set separated by commas inside curly brackets. For example, the set S containing the factors of 10 would be written as S={1,2,5,10} . If there are too many elements to list them all, you can establish a pattern and then use an ellipsis (…) to indicate that the pattern continues. For example, the set E of even numbers would be written in roster form as E={2,4,6,8,10…}