Arithmetic Flashcards
When integers are added, subtracted, or multiplied, the result is always…
an integer.
When divided, the result may also be a fraction.
Subtracting a negative number is the same as…
adding it.
7– -1=8
The product of two positive integers is…
a positive integer.
(5)(6)=30
The product of 2 negative integers is…
a positive integer
(-5)(-6)=30
A positive integer times a negative integer is…
a negative integer
(-5)(6)=30
Factors
Positive or negative numbers that can be multiplied to get an integer.
1,2,3,4,6,12 are factors of 12
Multiple
A number divisible by integer x
60 is a multiple of 2
1 is a factor of…
Every integer
1 is a multiple of…
1 and -1
0 is a multiple of…
Every integer
0 is a factor of…
1
Lowest common multiple of c and d
C and D are non zéro integers and the least positive integer that is a multiple of c AND d
Greatest common factor (divisor) of C and D
The greatest positive integer divisible by both C AND D
If C and D are not both divisors of integer x, the result is represented as a…
Fraction, decimal, or quotient with remainder where both are integers
The sum of two even integers is…
an even integer
The sum of two odd integers is…
an even integer
The sum of an even and an odd integer is…
an odd integer
The product of two even integers is…
An even integer
The product of two odd integers is…
An odd integer
The product of an even integer and an odd integer is…
An even integer
Prime number
Integer greater than one divisible only by 1 and itself
The first ten prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Prime factorization for 12
(2)(2)(3) = (2^2)(3)
Fraction
c/d where c (numerator) and d (denominator) are integers and d is NOT equal to 0
Rational number
A number that can be turned into a fraction where d is NOT 0. All integers are rational numbers because all integers can be n/1.
Equivalent fractions
Any fraction can be multiplied by the same nonzero integer to create an equivalent fraction.
Any fraction can have the same common factor factored out to be reduced to an equivalent fraction.
Adding and subtracting fractions with same denominator
3/11 + 2/11
Add straight across. Keep the denominator the same.
3/11 + 2/11 = 5/11
Adding and subtracting fractions with different denominators
1/3 + -2/5
Find a common denominator. Convert both fractions to equivalent fractions with the same denominator. Add numerators and keep common denominator.
1/3 + -2/5
1/3 = 5/15 and -2/5 = -6/15
5/15 + -6/15 = -1/15
Multiplying fractions
8/7)(7/3
Multiply the two numerators. Multiply the two denominators.
(8/3)(7/3) = 56/9
Dividing fractions
17/8 \ 3/5
Invert the second fraction (find the reciprocal). Multiply the first fraction by the inverted fraction.
17/8 * 5/3 = 85/24
Mixed number
Integer plus a fraction
Converting a mixed number to a fraction
4 3/8
Convert the integer part to an equivalent fraction. Add the fraction.
4 = 32/8 + 3/8 = 35/8
Base and exponent
3ˆ4
3 = base 4 = exponent
A negative number raised to an even power is always…
(-3)ˆ2
Even
9
A negative number raised to an odd power is always…
(-3)ˆ3
Odd
-27
(-3)ˆ2=
-3ˆ2
(-3)ˆ2 = 9 -3ˆ2 = -(3)(3) = -9
aˆ0=
1
0ˆ0
undefined
aˆ-1=
1/a
aˆ-2=
1/aˆ2
(√a)ˆ2 =
(√3)ˆ2 =
a
3
√(aˆ2)=
√4=
a
√4 = √(2ˆ2) = 2
√a √b=
√3√10
√ab
√30
√a/√b =
√5/√15=
√ab
√5/√15 = √5/15 = √1/3
For odd order roots
There is exactly ONE root for EVERY number n, even when n is negative
For even order roots
There are exactly TWO roots for every POSITIVE number and NO roots for any NEGATIVE number
3√8 (cube root)=
4√8 (4th root)=
-4√8 (- 4th root)=
4√-8 (4th root of a negative number) =
1 cube root (2)
8 is positive so has two 4th roots– 4 and -4
-8 is negative, so no 4th root because the root is positive and the number is negative (imaginary numbers)
irrational number
a number whose decimal does not terminate or repeat; cannot be written as a fraction
The set of real numbers consists of…
all rational and irrational numbers
Organize the sets from smallest to largest:
Imaginary numbers
Rational numbers
Integers
Real numbers
Integers
Rational numbers
Real numbers
Imaginary numbers (separate set)
Absolute value
The distance between the number X and zero on the number line.
r(s + t) =
(rs) + (rt)
if rs = 0, then…
ex: -2s=0
either r is 0, s is 0, or both are 0
-2s=0, s=0
|r + s| is…
ex: |3+ -6|
less than or equal to |r| + |s|
|3+-6| = |-3| = 3
|3| + |-6| = 3 + 6 = 9
|r||s| =
|5||-2|
|rs|
|(5)(-2)| = |-10| = 10
If r is greater than 1, then rˆ2 is…
If r is between 0 and 1, then rˆ2 is…
greater than r (5, 5ˆ2=25)
less than r (1/25 is less than 1/5)
2 apples and 3 oranges in a basket
What is the ratio?
2 apples to 3 oranges or 2/3
Can be reduced like a fraction
Proportion and how to solve
An equation relating two ratios: 9/12 = X/4 Solve by cross multiplying: 9x = 36 x = 36/9 = 4
Solving for percents proportion
part / whole = percent / 100
Percent increase/decrease formula
amount of increase/decrease
over
base (original amount)
