Number Properties Flashcards
Properties of 0
Integer
Neither positive nor negative
Even
Not prime
Properties of 1
Integer
Positive
Odd
Not prime
Properties of 2
Integer
Positive
Even
Prime
When do you reverse the sign in an inequality?
When you multiply or divide by a negative number across the inequality
-(x/5) > 1
x < -5
Multiplying and dividing fractions
Multiply: simply multiply across
Divide: flip second fraction and multiply
Since fraction bar indicates division, same rule applies to fractions within fractions
3/4 ÷ 9/10 = 3/4 * 10/9 = 5/6
Reducing LARGE fractions
Break large fraction down into primes and then cancel
10/21 * 7/8 = (25)(7)/(37)(222) = 5/12
Adding or subtracting fractions with common denominator
Only add or subtract numerator
3/4 + 2/4 = 5/4
8/11 - 3/11 = 5/11
Adding and subtracting fractions with DIFFERENT denominators
Method 1: multiply each fraction by another fraction that makes both bases common. Then add or subtract
Method 2: bowtie (criss cross)
First multiply denominators to form base. The cross multiply numerators and denominators and add or subtract the products.
2/7 + 3/4 = (42) + (37) / 28 = 29/28
2/3 - 1/2 ÷ 1/12
(2/3 - 1/2) = 4-3/6 = 1/6
12(1/6) = 2
2
Order of operations
PEMDAS
Work exponents/roots as well as m,d,a,s from left to right
Factors
numbers that can be multiplied to make another number (x is a factor of y if y is divisible by x, where x and y are integers)
1,6,2,3 are the factors of 6 (factor pairs)
Prime factors - factor tree
A number is divisible by 2 if
it is even
ex. 110
A number is divisible by 3 if
its digits sum to multiple of 3
ex. 18255 (1+8+2+5+5 = 21)
A number is divisible by 4 if
Its last 2 digits are multiple of 4
Ex. 59456 (56÷4 = 14)
A number is divisible by 5 if
it ends in a 0 or 5
Ex. 65, 70
A number is divisible by 6 if
Its a multiple of both 2 and 3
Ex. 588 (even and sums to 18)
A number is divisible by 7
Take LAST digit and double it. Subtract answer from remaining digits - divisible by 7 if ends in 7
A number is divisible by 8 if
Its last 3 digits are multiple of 8
Ex. 17336 (336÷8 = 42)
A number is divisible by 9 if
its digits add up to a multiple of 3 and 9
Ex. 6417 (6+4+1+7 = 18)
A number is divisible by 10 if
it ends in a 0
Ex. 2,984,150
A number is divisible by 12 if
its a multiple of both 3 and 4
Ex. 4932 (32÷4 = 8; 4+9+3+2 = 18)
Multiples
numbers that can be made by multiplying by a certain number (all the numbers of an integer, y, where +/-1y, +/-2y…)
Multiples of 6: 6, 12, 18…
Least Common Multiple
the least common multiple of two integers is the smallest integer that is a multiple of both
to find LCM: break down multiples into prime factors and multiply largest common primes
ex. 18 = 2 x 3^2 and 24 = 2^3 x 3
LCM: 2^3 x 3^2 = 72
Exponent
exponents are just shorthand to express multiplication, and exponents can be negative (a number raised to a negative exponents is the reciprocal of that number raised to a positive exponent)
ex. x^-2 = 1/x^2
To multiply powers with the SAME base
add exponents
ex. 2^3 * 2^2 = 2^5
To divide powers with the SAME base
subtract exponents
ex. y^9/y^4 = y^5
When exponents have different bases, you CANNOT just add or subtract
exs.
x^4 * y^6 ≠ (xy)^10
x^5 + X^4 ≠ X^9
A negative number raised to an EVEN power becomes positive
exs.
(-4)^2 = 16 (-1/2) = 1/4
A negative number raised to an ODD power remains negative
exs.
(-2)^3 = -8 (-1/3)^3 = -1/27
When raising a power to power, MULTIPLY the exponents
exs.
(x^3)^5 = x^15 (3^2)^3 = 3^6 = 729
When raising a product in PARENTHESES to a power, DISTRIBUTE the power over BOTH factors
exs.
(x^2*y^4)^3 = x^6*y^12 (2x^2)^4 = 2^4*x^8 = 16x^8
Any number raised to the exponent 1
equals itself
exs.
2^1 = 2
-7^1 = -7
1 raised to any exponent
equals 1
exs.
1^2 = 1
1^0 = 1
0 raised to any power OTHER than 0
equals 0
exs.
0^4 = 0
0^1 = 0
Any number raised to the 0
equals 1
exs.
3^0 = 1
(.34)^0 = 1
TT^0 = 1
0 raised to the 0 power is
UNDEFINED
Any positive number greater than 1 gets _____as it is raised to a higher power
LARGER
exs.
2^2 = 4
2^3 = 8
2^4 = 16
Any positive number less than 1 (fractions) gets _____ as it is raised to a higher power
SMALLER
exs.
(1/3)^2 = 1/9
(1/3)^3 = 1/27
Roots can also be expressed as factional exponents
exs.
√ x = x^1/2
4^1/2 = √4 = 2
^3√x = x^1/3
8^1/3 = ^3√8 = 2
Square roots with SAME base can be added or subtracted
exs.
4√3 + 7√3 = 11√3
5√x - 2√x = 3√x
BUT
3√6 + 3√3 ≠ 3√9 (the numbers inside radicals are NOT the same)
The product of square roots is EQUAL to the square root of the product
exs.
(√3)(√7) = √21 √721 = (√9)(√4)(√2) = 6√2
The quotient of square roots is EQUAL to the square root of the quotients
exs.
√4/√2 = 2 √63/√7 = √9 = 3
When taking the square root of a fraction, the result will always be _____ than the fraction itself
LARGER
ex.
√1/4 = 1/2
√1/4 > 1/4
E x E
E
E x O
E
O x O
O
E +/- E
E
E +/- O
O
O +/- O
E
Prime numbers between 1 - 10
2, 3, 5, 7
Prime numbers between 10 - 20
11, 13, 17, 19
Prime numbers between 20 - 30
23, 29
Prime numbers between 30 - 40
31, 37
Prime numbers between 40 - 50
41, 43, 47
Prime numbers between 50 - 60
53, 59
Prime numbers between 60 - 70
61, 67
Prime numbers between 70 - 80
71, 73, 79
Prime numbers between 80 - 90
83, 89
Prime numbers 90 - 100
97
Absolute value
the distance between a number and o on the number line
P x P
P
N x N
P
P x N
N
Common radical approximations
TT ≃ 3 √1 ≃ 1 √2 ≃ 1.4 √3 ≃ 1.7 √4 = 2
Radical rules
√a√b = √ab
√a/√b = √a/b
a√c + b√c = (a + b)√c
(√a)^2 = a
BUT
√a + √b ≠ √a+b
√a - √b ≠ √a-b
Exponent rules
x * x = x^2 x^-a = 1/x^a x^0 = 1 x^a*x^b = x^a+b (x^a)^b = x^ab x^a/x^b = x^a-b (negative)^odd = negative (negative)^even = positive
Greatest common factor
largest factor that 2 numbers have in common
GCF: find prime factors of both integers and combine common ones
ex. 36 (2^23^2) and 54 (23^3) both have 2 and 3^2 in common so 233 = 18 = GCF
Comparing fractions
need common denominator but if more than 2 fractions, compare 2 at a time
use bowtie (criss-cross) method to see which is larger
