Arithmetic Flashcards
The product of a positive integer and a negative integer
Negative integer
The product of an even integer and an odd integer
Even integer
The product of two odd integers
Odd integer
The product of two even integers
Even integer
The sum of an even integer and an odd integer
Odd integer
The sum of two odd integers
Even integer
The sum of two even integers
Even integer
What is a prime number
an integer greater than 1 that has only two positive divisors: 1 and itself.
The first ten prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29
An integer greater than 1 that is not a prime number
Composite number
If both the numerator and denominator of a fraction have a common factor, then
the numerator and denominator can be factored and the fraction can be reduced to an equivalent or simplified fraction.
What is a fraction?
a number of the form c/d , where c and d are integers and d (the denominator) does NOT equal 0.
Every integer is a fraction or rational number; e.g. 3 = 3/1
What are the equivalent fractions for integers c and d?
c/d
-c/d
c/-d
-(c/d)
How do you add two fractions with the same denominator?
add the numerators and keep the same denominator.
3/11 + 2/11 = 5/11
-8/11 + 5/11 = -3/11 or -(3/11)
How do you add or subtract two fractions with different denominators?
First, find a common denominator. Second, convert both fractions so that they have the same denominator. Third, add or subtract the numerators.
Ex. 1/3 + 2/5
1. The common denominator is 15.
- 1/3 x 5/5 = 5/15 AND 2/5 x 3/3 = 6/15
- 5/15 + 6/15 = 11/15
How do you multiply two fractions?
multiply the two numerators and multiply the two denominators.
Ex. 10/7 x -1/3 = -10/21
How do you divide two fractions?
To divide one fraction by another, first invert the second fraction (that is, find its reciprocal), then multiply the first fraction by the inverted fraction.
Ex. (3/10) / (7/13) = 3/10 x 13/7 = 39/70
What is a mixed number?
It consists of an integer part/whole number and a fraction part, where the fraction part has a value between 0 and 1
Ex 4(1/3) means 4 + 1/3
How do you convert a mixed number to a fraction?
To convert a mixed number to a fraction, convert the integer part to an equivalent fraction with the same denominator as the fraction, and then add it to the fraction part.
Ex. 4 3/8 = 4/1 x 8/8 = 32/8;
32/8 + 3/8 =35/8;
Thus 4 3/8 is equivalent to 35/8
OR Multiply the denominator (8) times the whole number (4) then add the numerator (3), place the answer over the denominator to create the improper or equivalent fraction.
Ex to convert 4 3/8…
8 x 4 + 3 = 35;
Thus the new fraction is 35/8
What are fractional expressions?
Numbers of the form c/d, where either c or d is not an integer and d is NOT 0, are called fractional expressions. Fractional expressions can be manipulated just like fractions.
A negative number raised to an even power is always
Positive
a negative number raised to an odd power is always
Negative
What is the difference between (-3)^2 and -3^2?
(-3)^2 = 9 -3^2 = -9
For all nonzero numbers a, a^0 =
1
For all nonzero numbers a,
a^-1 =
a^-2 =
a^-3 =
….
a^-1 = 1/a a^-2 = 1/a^2 a^-3 = 1/a^3
(a)(a^-1) =
(a) (a^-1) =
(a) (1/a) = 1
A square root of a nonnegative number n is
a number r such that r^2 = n
Ex. 4 is a square root of 16 because 4^2 = 16.
Another square root of 16 is −4, since (-4)^2 = 16.
All positive numbers have two square roots, one positive and one negative.
What are the four important rules of square roots?
(✔️a)^2 = a; (✔️3)^2 = 3
✔️a^2 = a; ✔️2^2 = 2
✔️a✔️b = ✔️ab; ✔️3✔️10 = ✔️30
✔️a/✔️b = ✔️a/b; ✔️5/✔️15 = ✔️5/15 =✔️1/3
For odd order roots, there is/are________ root(s) for every number n, even when n is negative.
For odd order roots, there is exactly one root for every number n, even when n is negative.
For even order roots, there is/are _______ roots for every positive number n and _____ roots for any negative number n.
For even order roots, there are exactly two roots for every positive number n and no roots for any negative number n.
For example, 8 has exactly one cube root, 3✔️8 = 2, but 8 has two fourth roots, 4✔️8 and -4✔️8, whereas -8 has exactly one cube root, 3✔️-8 = -2, but -8 has no fourth root, since it is negative.
Convert 2.3 to an equivalent fraction
2.3 = 2/1 + 3/10 = (2/1 x 10/10) + 3/10 = 20/10 + 3/10 = 23/10
Convert 90.17 to an equivalent fraction
90.17 = 90/1+ 17/100 =
(90/1 x 100/100) + 17/100 =
9,000/100 + 17/100 =
9, 017/100
Convert 0.612 to an equivalent fraction
0.612 = 612/1000
What are irrational numbers?
Decimals that do not terminate (ex. 0.375) or repeat (0.1111…)
Ex. 1.41421356237
What are real numbers?
All rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals.
What does absolute number mean?
The distance between a number x and 0 on the number line.
Ex. |-3| = 3
|9| = 9
R(s + t) =
r(s + t) = Rs + rt
If rs = 0, then
If rs = 0, then r = 0 OR s = 0, OR both
Division by 0 is
Division by 0 is undefined
Ex. 5/0 is undefined
If both r and s are negative, then
If both r and s are negative, then
R + S is negative and RS is positive
If R is negative and S is positive, then
If R is negative and S is positive, then RS is negative
Triangle inequality
|r + s| is less than or equal to |r| + |s|
When expressing a ratio as a fraction, which number is the numerator and which is the denominator?
the first quantity is the numerator and the second quantity is the denominator.
For example, if there are 2 apples and 3 oranges in a basket, we can say that the ratio of the number of apples to the number of oranges is 2/3 , or that it is 2 to 3, or
that it is 2 : 3.
What is a proportion?
A proportion is an equation relating two ratios.
for example, 9/12 = 3/4
How do you solve an equation with ratios?
Write a proportion and cross multiply.
Example: To find a number x so that the ratio of x to 49 is the same as the ratio of 3 to 21, you can first write the following equation.
X/49 = 3/21
You can then cross multiply to get
21x = (3)(49) and finally you can solve for x to get x = 147/21 = 7
Define percent
The term percent means per hundred, or hundredths. Percents are ratios that are often used to represent parts of a whole, where the whole is considered as having 100 parts. Percents can be converted to fraction or decimal equivalents.
