Math 20
Variables -- letters used to represent numbers in algebra
Factors -- the numbers or variables that are multiplied in a multiplication problem. i.e.: a x b=c a and b are factors of c
Numerator -- top number of a fraction
Denominator -- bottom number of a fraction
Simplified (reduced to its lowest terms) -- the numerator and denominator have no common factors other than 1
Real number system – double R is symbol for real numbers
Whole numbers – 0, 1, 2, 3, 4, etc.
Counting numbers (natural numbers) – 1, 2, 3, etc.
Integers -- …, -3, -2, -1, 0, 1, 2, 3, …
Rational numbers – all numbers that can be expressed in a fraction (ratio) 2/3 , ½, -3/7, 4/1, etc.
Irrational numbers – cannot be expressed any other way.
Prime numbers -- can only be divided by themselves and 1 -- 2, 3, 5, 7, 11, 13, etc
1 is not a prime number. It is a unit.
All other numbers are composite numbers
To simplify a fraction:
Find the largest number that will divide without a remainder both the numerator and denominator. This number is called the greatest common factor or GCF
Then divide both the numerator and denominator by the GCF
6/18 = 6 divided by 6 over 8 divided by 6 = 1/3
To multiply a fraction:
Multiply their numerators together
Then multiply their denominators together
7/13 x 5/12 = 7x5 over 13x12 = 35/156
To avoid having to simplify the answer:
Divide both a numerator and a denominator by a common factor
6/17 x 5/12 6 and 12 divide by 6
6 divides 1 time and 12 2 times
1/17 x 5/2 = 1 x 5 over 17 x 2 = 5/34
To multiply a whole number by a fraction:
Write the whole number with a denominator of 1 and then multiply
12 x 5/64
12/1 x 5/64 12 and 64 both divide by GCF of 4
12 divides 3 times and 64 divides 16 times
3/1 x 5/16 = 3 x 5 over 1 x 16 = 15/16
To divide a fraction by another:
Invert the divisor -- the second fraction
Multiply
3/5 divided by 5/6
3/5 x 6/5 = 3 x 6 over 5 x 5 = 18/25
Evaluate an expression -- to obtain the answer to the problem using the operations given
Evaluate 3/8 divided by 9
3/8 divided by 9/1 = 3/8 x 1/9 9 is divisible by 3
1/8 x 1/3 - 1 x 1 over 8 x 3 = 1/24
Fractions with the same (common) denominator can be added or subtracted
To add (or subtract) fractions with the same denominator:
Add (or subtract) the numerators
Keep the common denominator
9/15 + 2/15 = 9 + 12 over 15 = 11/15
8/13 - 5/13 = 8 - 5 over 13 = 3/13
The smallest number that is divisible by 2 or more dominators is the least common denominator or LCD
To add (or subtract) fractions with unlike denominators:
Rewrite fractions with the same (common) denominator -- multiply numerator and denominator with the same number
Add (or subtract)
1/2 + 1/5 10 is the LCD
1/2 = 1 x 5 over 2 x 5 = 5/10
5/10 + 2/10 = 5 + 2 over 10 = 7/10
3/4 - 2/3 12 is the LCD
3/4 = 3 x 3 over 4 x 3 = 9/12
2/3 = 2 x 4 over 3 x 4 = 8/12
9/12 - 8/12 = 9 - 8 over 12 = 1/12
Mixed number consists of a whole number followed by a fraction: 5 2/3 = 5 + 2/3
To change a mixed number to a fraction (an improper fraction)
Multiply the denominator of the fraction by the whole number
Add the numerator to the product from step 1. This sum represents the numerator of the fraction you are seeking.
The denominator of the fraction you are seeking is the same as the denominator of the fraction in the mixed number
5 2/3 = 5 x 3 = 15
15 + 2 over 3 = 17/3
5 2/3 = 17/3
To change a fraction greater than 1 to a mixed number:
Divide the numerator by the denominator
Note the quotient and remainder
Write the mixed number. The quotient found in step 1 is the whole number of the mixed number.
The denominator in the fraction of the mixed number will be the same as the denominator in the original fraction
17/3
17 divided by 3 = 5 (whole number) and a remainder of 2
17/3 = 5 2/3
To add, subtract, divide, or multiply mixed numbers they are often changed into improper fractions
Cross divide to check answers:
13/26 = 1/2
Take 13 x 2 and 26 x 1. Both should give the same answer of 26. That tells you the answer is correct
Absolute value – the bars make the number inside positive.
│2 │ = 2
│-2│ =2
Negative symbol outside of absolute stays with number
-│-2│ = -2
For < or > solve the absolute first
│-7│ < or > │-8│
│-7│ = 7 and │-8│ = 8
7 < 8
│-7│ < │-8│
-│-3│ < or > │2│
-│-3│ = -3 and │2│ = 2
-3 < 2
-│-3│ < │2│
For decimals ignore the negative sign, find the greater than and do the opposite.
-1.83 < or > -1.82
1.83 > 1.82 = -1.83 < -1.82
DSS – if the signs of the numerals are different, subtract and take the sign of the larger number
-3 + 5 = 5 – 3 = 2
5 is larger than 3 so the number is positive
-12 + 7 = 12 – 7 = 5 = -5
12 is larger than 7 so the answer is negative
SSA – if the signs of the numerals are the same, add and keep the sign
-9 + -7 = 9 + 7 = 16 = -16
Two negatives make a positive
-5 - -7
-5 + 7 = 7 – 5 = 2 (DSS)
A negative and a positive double sign becomes a negative
3 + -7
3 -7 = 7 – 3 = 4 = -4 (DSS)
Same rules apply for multiplication.
Mixed signs are negative. Same signs are positive
Commutative property of addition and multiplication – states that the order in which any two real numbers are added does not matter.
A + B = B + A
4 + 3 = 3 + 4
7 = 7
Associative property of addition and multiplication – states that in addition of three or more numbers, parenthesis may be placed around any two adjacent numbers without changing the results.
(A+B) + C = A + (B+C)
(3+4) + 5 = 3 + (4+5)
7 + 5 = 3 + 9
12 = 12
Distributive property of multiplication over addition – anything on the outside multiplies by anything on the inside.
A(B+C) = AB + AC
Ex: A = 2, B = 3, C = 4
2(3+4) = (2x3) + (2x4)
2x7 = 6 + 8
14 = 14
Also
A(B-C) = AB – AC
Ex: A = 4, B = 3, C = 2
4(3 – 2) = (4x3) – (4x2)
4 x 1 = 12 – 8
4 = 4
Order of operations – what order the problem is done. Parentheses, exponents, multiplication, division, addition, subtraction
P
E
MD
AS
Negative numbers to the odd power will always give a negative number
Negative numbers to an even power will always give a positive number