Math Terms - Subset II

how to reduce fractions to simplest form

divide both numerator and denominator by any number by which they both evenly divisible (ex:

mentally compute

use understanding of place value (round to tens) - estimate ex: what would be the best way to mentally compute 31 x 4? 30 x 4 = 120

To find percent increase or decrease:

what percent of the starting point is the change? (ex: 8 people then 4 people come, now 12 people. what % increase? what percent of 8 is 4?

what percent of 8 is 2?

x/100 · 8 = 2 Percent (by the hundred) - x/100 is - equals what fraction of 8 is 4? 2/8 OR 2/8 = x/100 = 2 ÷ 8 = .25 = 25%

Quotient rule with same exponent a^n/b^n

(a/b)^n

Decimals to Percents

move decimal 2 places right ex: 0.14 = 14/100 (reduce) = 7/50

0^n =

0 , for n>0

b^0 =

1

x^-2

1/(x · x)

Estimating

Finding a number that is close enough to the right answer. You are not trying to get the exact right answer. What you want is something that is close enough. Also, involves the concept of predicting, or making an educated guess.

algorithm

how to compute something

divisibility rule for 10

if 0 is in the ones place (ends in zero)

divisibility rule for 3

if the sum of the digits is divisible by 3

what percent of 16 is 4?

what fraction of 16 is 4? percent - per 100 4/16 = 1/4 = ?/100 multiply top and bottom by 25 to get it to 100 25 /100 = 25% something over 100 4/16 = 4 ÷ 16 = 0.25 (25 hundredths) = 25/100 = 25%

product rule with same exponent a^n ⋅ b^n =

(a ⋅ b)^n

percent

"by the hundred;" with a preceding numeral expressing a proportion of the whole amount per centum "by the hundred" cent - hundred per - by means of, through A percentage is a ratio whose second term is 100. Percent means parts per hundred. p% is equivalent to: p:100 p/100 the sum of all parts of the whole is 100 percent. 18

multiple

A product of a given whole number and any other whole number.

divisibility rule for 9

If the sum of the digits is divisible by 9

Power rule II b^n^m

b^(n^m)

addition or subtraction under a radical

numbers must be combined under radical before any computation of square roots may be done ex: √10+6 = √16 = 4 √10+6 ≠ √10 + √6 √93-12 = √81 = 9

power rule with radicals b^ 1/n =

n√b

percent increase

(original value-new value)/original value x 100

Convert improper fraction to mixed number

1. find how many times the denominator divides into the number. Make that your whole number. 2. then, multiply the whole number times the denominator and subtract that number from the numerator to get the numerator of the remainder (ex: 9/4 = 2 1/4 4 goes into 9 twice Subtract 8 from 9 Remainder is 1 Put remainder over denominator)

ordering fractions

2 ways: -convert all fractions into fractions with a common denominator Ex: 2/3 ; 3/4 ; 1/2 16/24 ; 18/24 ; 12/24 12/24 ; 16/24; 18/24 or -convert all the fraction into decimals by dividing the numerator by the denominator

real number

All rational and irrational numbers can exist in the read world (ex. 12, 5/8, √3)

rational number

Any number that can be written as a fraction/ratio of two integers (ex. 1, 5/8, 0.5, 10%, ...)

prime factorization

Breaking down a composite number until all of the factors are prime (Ex: 24 = 2*2*2*3)

Associative Property of Addition

Changing the grouping of three or more addends does not change the sum. grouping does not matter A + (B + C) = (A + B) + C

Percent increase formula

New Value - Original Value ---------------------------------- x 100 Original Value

signed numbers

Numbers that are either positive or negative.

order of operations

PE(R)MDAS Parentheses, (Square) Roots and Exponents, Multiplication, Division, Addition, Subtraction "Please (really) excuse my dear aunt sally" Work from left to right

Percents to Fractions

Place the number over 100 (as % means) ex: 12% 12/100 (reduce) = 6/50= 3/25

remainder

The amount left over when one number is divided by another.

divisibility rule for 4

The last 2 digits are divisible by 4

Divisibility Rule for 5

The last digit is 0 or 5

divisibility rule for 6

The number is divisible by both 2 and 3

How do we calculate percent changes?

When calculating a percent change from an initial value to a final value: 1. Find the difference between the initial and final values. 2. Divide the difference by the initial value. 3. Convert the quotient to a percentage. % change = ((final - initial) / (initial)) x 100 (ex: the percent discount on jeans, the percent increase in the amount of potato chips in a bag, etc.

Perfect Squares

a square of a whole number

Power rule I (b^n)^m=

b^n*m

Power rule with radicals m√(b^n) =

b^n/m

whole number line

can be useful when working with addition and subtraction of negative numbers

percentage change

change/original x 100 applies to increase or decrease

binary number system

computers use A base 2 positional numbering system. only 2 values - 0, 1

Fractions to Percents

convert fraction to decimal then multiply by 100

multiplicative inverse property

the product of a nonzero number and its reciprocal is 1 a * 1/a = 1 any number times its multiplicative inverse equals 1 ex: 4 · (1/4) = 1

what is p% of a?

x = p/100 * a

x²

x · x

common factor

a number by which two other numbers are both divisible

mixed number

a number made up of a whole number and a fraction

irrational number

a number that can not be expressed as a ratio of two integers or a fraction Ex: pi or any square root of an imperfect square

factor

a number that is multiplied by another number to find a product; the different numbers that can be multiplied together to arrive at the original number

prime number

a number which can only be divided by itself and 1; all are positive (ex: 2, 3, 5, 7, 11, 13, 17, 19, etc.)

product rule with same base a^n ⋅ a^m =

a^(n+m)

quotient rule with same base a^n/a^m

a^n-m

divisibility rule for 2

ones digit must be even (0, 2, 4, 6, 8)

If a is p% of b, then:

p = a/b x 100 can be rearranged to show a or b in terms of other value a = p/100 x b b = a/ (p/100) = (100xa) / p

two numbers multiplied under a radical (square root) sign =

the product of the square roots ex: √(4)(25) = √4 x √25 = 2x5=10 or √100 = 10 √64/4 = √64/√4=8/2 =4 or √16=4

√2

~1.4

when multiplying or dividing: (+) x (+) =

(+) number

when multiplying or dividing: (-) x (-) =

(+) number

when multiplying or dividing: (-) x (+) =

(-) number

Negative exponents rule b^-n

1/b^n

Commutative Property of Addition

Changing the order of the addends does not change the sum. a + b = b + a order does not matter

whole numbers

Natural numbers ( counting numbers) and (ex. zero; 0, 1, 2, 3 etc.) positive integers

Percent decrease formula

[(original value-new value)/original value] x 100

a is p% of what number?

a = p/100 * (x) turn p into decimal and divide number by p

If a is what percent of b?

a = x(b) a/b = x divide and multiply by 100 p% = x

common denominator

a common multiple of the denominators of two or more fractions (you can just multiply all the denominators, then multiply the numerators too)

factor tree

a diagram that is used to break down a composite number into its factors until all the numbers left are prime to find its down into its prime factors don't rely on these when seeking common factors unless you are seeking only common prime factors

to find a square root which will not be a whole number

you should approximate ex: √57 falls somewhere between √49 and √64, it will fall between 7 and 8; just about half way b/w 49 and 64; aprroximately 7.5 √83 √81 < √83 < √100 9 < √83 < 10 since √83 is closer to 81 than it is to √100, √83 will be closer to 9 than 10. decimal value will then be b/w 9 and 9.5. since √83 is closer to √81, a reasonable estimate is 9.1

√3

~1.7

Which numbers can be represented on a number line?

all real numbers

Identity Property of Addition

any number plus 0 equals itself a + 1 = a

Additive Inverse Property

any number plus its additive inverse (opposite) equals 0 a + (-a) = 0 ex: 4 + -4 = 0

Identity Property of Multiplication

any number times 1 equals itself (a · 1 = a)

percent decrease

(new value-original value)/original value x 100

Which numbers are neither prime nor composite?

0 and 1

Decimal Places and their names

0.1 - tenth place 0.01 - hundredths 0;

Decimals to Fractions

1. move decimal point to the right of the last digit 2. place this number over 1 followed by one 0 for every place you moved the decimal point.

Number System

A collection of symbols and the rules for ordering them.

composite number

A positive number with more than two factors; can divide them into equal groups (if greater than one and not _____, then it's automatically this type of number).

Associative Property of Multiplication

Changing the grouping of three or more factors does not change the product. grouping does not matter A · (B · C) = (A · B) · C

Commutative Property of Multiplication

Changing the order of the factors does not change the product. a · b = b · a order does not matter

Fractions to Decimals

Divide the numerator by the denominator. 1/4 = 1 divided by 4 = 0.25

Rules for rounding

Drop 4 or less Round up 5 or greater Be mindful of what place the problem is asking you to round to

greatest common factor (GCF)

The largest factor that two or more numbers have in common. a common factor is a number by which two other numbers are both divisible.

counting numbers

The numbers used to count; the members of the set {1,2,3,4,5,...}. Also called natural numbers.

Integer

The set of whole numbers and their opposites can be written without a fraction or a decimal (ex: 0,1,2,3,-1,-2,-3)

improper fraction

a fraction whose numerator is larger than the denominator

Distributive Property

a property indicating a special way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac A(B+C) = AB + AC (the opposite of distributing is factoring).

base 10 number system

each movement left or right changes the place value by a unit of 10

when working with a fractional number line:

first determine the unit value represented by each increment

Adding and Subtracting Fractions

first find a common denominator, then add or subtract the numerators

Dividing Fractions

flip(invert) the second fraction (take its reciprocal) then multiply ex: 7/6 ÷ 2/8 = ? 7 x 8 = 56 6 x 2 = 12 reduce to simplest form and/or convert to mixed number =14/3 = 4 2/3

Percents to Decimals

move the decimal point to the left two places ex: 65% → 0.65; 4% → 0.04

what are the algorithmic opposites?

multiply / divide add / subtract square roots / exponents

multiplying fractions

multiply numerator times numerator denominator x denominator you do not need to find a common denominator ex: 3/4 x 2/5 = ? 3 x 2 = 6 4 x 5 = 20 reduce to simplest form. =3/10

whenever there is a perfect square within a radical:

the root of that square may be removed from under the radical (ex. √4 = √(2X2) = 2) simplify : 2 √(21 +4) + √(6)(6) 2 √25 + √36 2(5) + 6 =10 +6 =16

least common multiple (LCM)

the smallest number that can be divided evenly by those original two numbers; smallest multiple in common For ex: of 5 and 6 it's 30, because it is the smallest number that both 5 and 6 go into Used often for finding a common denominator with fractions list out all the multiples of a number and find those in common with the other number or if no factors in common, multiple two orginal numbers together