Quant Group 3

quotient

the whole number of times one number goes into another number (example for 8 divided by 4 → quotient is 2

remainder

when you divide a number by another number, this is colloquially known as what’s “left over” or what’s “remaining”

example: 10 / 2 → remainder is nothing

remainder when denominator > numerator

whenever the number being divided is less than the divisor, and both integers are positive, the remainder is simply the number being divided

example: 5/6 → the remainder is 5

remainder with negative numbers

example: -29 / 7 = correct example -5 × 7 + 6 (remainder) = -29

it is NOT -1 because you cannot have a negative remainder

remainder patterns

for an integer x, try writing the first powers of x and see if the remainders repeat when dividing that power with a certain integer

example: first four positive powers of 7 repeat in a 2-4-3-1 pattern when divided by 5

remainders and addition

to find the overall remainder in this type of problem, find the remainder of each term separately and then add them together

example: 17 + 9 / 5 = 17/ 5 + 9 / 5 = remainder of 2 + remainder of 4 = 6

*since 6 is bigger than 5, you need to do 6 - 5 = 1 → 1 is final remainder

fractions

numbers that adhere to a particular form, where one integer is on top called the numerator and one integer on bottom called the denominator

square root of 2 / 3

this cannot be a fraction because the numerator is not an integer

improper fractions

fractions of the form a / b where a > b and b does not equal zero are called

mixed numbers

these occur when you combine an integer and a fraction, where the numerator is less than denominator

adding and subtracting fractions

step 1: check to see if denominator of each fraction is the same

• if yes, just add the numerators and keep denominators

step 2: if denominators are different, multiply top and bottom of fraction until denominators are the same

multiplying fractions

simply multiply the numerators and denominators together, then simplify the fraction if possible

dividing fractions

step 1: flip the numerator and denominator of any fractions that are not the first one

step 2: multiply all the fractions together

dividing by 1/something

6 / (1/5) = 6 × 5

rational numbers

numbers that can be written as a fraction

irrational numbers

numbers that cannot be written as a fraction

example: pi / 3

decimals

positional number system, meaning that the location of a digit in a number influences its value, It uses only 10 characters to express an infinite number of numbers

writing as powers of 10

because the decimal number system uses a base of 10, you can write any number with powers of 10

example: 789.91 = (7 × 10²) + (8 × 10^1) + (9 × 10^0) + (9 × 10^-1) + (1 × 10^-2)

other number systems like binary

other number systems like binary (base 2, i.e., numbers are represented only in terms of 0 and 1), and hexadecimal (base 16, i.e., numbers represented as 1-8A-F)

terminating vs. non-terminating

a fraction when converted to a decimal will do one of two things:

1) terminate → the decimals stop at some point (example ½ → 0.5)

2) not terminate → the decimals go on forever (example 1/3 → 0.3333…)

irrational numbers

these are also non-terminating (examples: pi = 3.14159… or square root of 2 = 1.414213…)

does the fraction terminate?

step 1: simplify the fraction to its simplest form so there are no shared factors among numerator and denominator

step 2: if there are only powers of 2 in the denominator, it definitely terminates

step 3: if there are only powers of 5 in the denominator, it definitely terminates

step 4: if there is a combination of only powers of 2 and 5 in the denominator, it definitely terminates

repeating decimals

these are always from fractions, where the numerator and denominator are both integers

examples: 4/9 = 0.44444…; 5/11 = 0.454545…

non-repeating decimals

these are always irrational numbers and never fractions

repeating decimals

shortcut for writing this type of number is to put a line above one or more of the numbers behind the decimal

irrational numbers

any number that, when written as a decimal, is non-terminating and non-repeating

converting terminating decimals to fractions

to do this, just write the decimal as a fraction with the denominator as a power of 10 and then simplify

repeating decimal to fraction

step 1: notice number of digits in repeating pattern. One digit repeat? Two? Three?

step 2:

• if only one digit repeats, put it over 9

• if two digits repeat, put them over 99

  • if three digits repeat, put them over 999

exponent rules I

1) if a^n = a^m, then n = m (this works if a doesn’t equal zero or 1)

• 5^x = 5^y+2 this equals x = y + 2

2) if you have a negative exponent, you can make it positive by “flipping” the fraction

• 5^-3 = 1 / 5³

• 2 / 3^-3 = 2 × 1 / 3^-3 = 2 × 3³

3) If you multiply two numbers together with the same base, you can add the exponents

4) if you divide a number by another number with the same base, you can subtract the exponents

5) if you raise anything to the 0 power, it equals 1, except in the 0^0 case, which is undefined

exponent rules II

rule 6: if you multiply two different numbers together with the same exponent, you can do the following:

• 5³ x 3³ = (5 × 3)³ or (2 × 9)^4 = 2^4 × 9^4

rule 7: if you divide two different numbers with the same exponent, you can do the following: (a / b)^m = a^m / b^m

rule 8: if you raise a number with an exponent to another exponent

• example: (2^5)^7 = 2³5

• MUST HAVE PARENTHESES