Chapter 1 Notes
A ~= symbol is used to indicate an approximate or rounded value
Numbers that are less than one are often written as fractions instead of decimals
Arithmetic is the branch of mathematics that includes four basic operations: Addition, Subtraction, Multiplication, and Division
Addition is the process of combining two or more numbers together to get a total
The numbers being combined are called addends, and the total is called the sum
When adding decimals, align the decimal point and annex zeros when necessary
Subtraction is the process of taking away one number from another number
Subtraction is the inverse operation of addition
The first number in a subtraction problem is called the minuend, and the number to be subtracted is called the subtrahend, and the answer is called the difference
Multiplication is the process of adding a number to itself as many times as indicated by another number
2×3 can also be written as 3+3 or 2+2+2, since either 3 two’s or 2 threes are being added
The numbers being multiplied are called the factors, and the answer is called the product
If the second factor has more than one digit, the problem will have two or more partial products that are added to get the final product
Division is the process of finding how many times one number is contained in another number.
Division is the inverse operation of multiplication
The quotient is the answer to the division problem, and the dividend is divided by the divisor
When a number doesn’t divide evenly, the part of the dividend left is called the divisor
Natural numbers are the numbers used for counting
When zero is included with the natural numbers, they are called the whole numbers.
Integers include whole numbers as well as their opposites, but integers, natural and whole numbers do not include fractions and decimals
Rational numbers are numbers that can be expressed as either an integer or a fraction of integers
Irrational numbers cannot be expressed as a integer or fraction of integers, since when they are expressed as decimals, they neither terminate or repeat
All rational and irrational numbers are in a classification group called the real numbers
Integers, fractions, and decimals greater than zero are called positive numbers
Integers, fractions, and decimals less than zero are called negative numbers
The distance a number is from zero is called the absolute value. In other words, the absolute value is the value of a number without it’s sign
The law of trichotomy states that all comparisons can only have ONE sign:.>, < or =
When comparing decimals or mixed decimals, if the whole numbers are different, the mixed decimal with the greater whole number is greater decimal
Factors of a number are all the different numbers that divide into it evenly
Prime numbers are numbers that only have 2 factors: 1 and the number itself
Composite numbers have more than 2 factors
The numbers 0 and 1 are neither prime or composite
A common factor is a factor that is shared by two or more factors
Numbers that are relatively prime are numbers that share no common factors except for 1
The expression of the number as a product of prime factors is the number’s prime factorization
One method to find the prime factorization of a number is division by primes, a process of dividing by prime numbers until the final quotient is prime
A factor tree is where are number is divided into pairs of factors until only prime factors remain
The base of a exponential equation is the repeated factor: In the expression, 5², 5 is the base
The exponent of a exponential equation is how many times the base repeats: In the expression, 5², 2 is the exponent
The entire exponential expression including the base and the exponent is called a power
The reciprocal of a number is 1 divided by the number: The reciprocal of 4 is ¼
Any number to the power of 0 is equal to 1: 3^0 is 1
The greatest common factor (GCF) is the largest common factor of two or more numbers
If one number is a factor of the other, the smaller number is the GCF: The GCF of 15 and 30 is 15
The GCF of two numbers that are relatively prime is 1: The GCF of 10 and 21 is 1
A multiple of a number is the product of the number and any natural number: The multiples of 3 are 3 ( 3×1), 6 ( 3×2), 9 ( 3×3), 12 ( 3×4) etc..
The least common multiple (LCM) is the smallest multiple that is shared by two or more numbers
A useful method for memorizing the process of LCM is this: List ( List the prime factorization of each number using exponents), Contest ( Hold a contest between the powers of each different prime factor), Multiply ( Multiply the circled factors from the contest)
If one number is a multiple of the other, the larger number is the LCM: The LCM of 15 and 30 is 30
The LCM of two numbers that are relatively prime is their product: The LCM of 10 and 21 is 210
The numerator (top number of a fraction) is used to represent how many equal parts are being considered
The denominator (bottom number of a fraction) is used to represent the total number of equal parts the whole or group is being divided into
Proper fractions are fractions in which the numerator is less than the denominator: 4/5 or 1/25
Improper fractions are fractions in which the numerator is equal to or greater than the numerator: 13/13 or 19/9
Mixed numbers consist of a whole number and a fraction: 6 1/3 or 14 7/8
An improper fraction that has the same value for both the numerator and denominator is equal to 1: 5/5 or 20/20
Finding an equivalent fraction by dividing both the numerator and denominator by the same number is called reducing
The goal of reducing a fraction is to get it into lowest terms ( a fraction that can be reduced no more)
To compare fractions, first find the LCD ( least common denominator): the LCM of the denominators of all fractions
A ratio shows the relationship between two numbers or quantities
The first term in a ratio is called the antecedent
The second term in a ratio is called the consequent
A proportion is two equivalent ratios joined by an equal sign
In a proportion like 3/4=15/20, 4 and 15 are the means, and 3 and 20 are the extremes
The fundamental property of proportions states that for every proportion, the product of the means must equal the product of the extremes
Multiplying fractions can be made easier by using cancellation, which is the process of removing a common factor from a numerator and a denominator before multiplying. Do not use cancellation with mixed numbers until they are made improper
To divide when a division problem includes fractions, express the problem using multiplication: invert the divisor and change the division symbol to multiplication ( Keep it, Change it, Flip it, KCF)
A complex fraction is a fraction containing a fraction or mixed number as the numerator, denominator, or both
To change a fraction to a decimal, divide the numerator by the denominator
A terminating decimal is a fraction that divides evenly: ½ divides evenly into 0.5
A repeating decimal is a fraction that does not divide evenly: 1/3 divides into 0.333…
Percent means parts per hundred or hundredths
Percent to decimal: Move the decimal point two places to the left and drop the percent symbol
Fractional percent to decimal: First express the fractional percent as a decimal percent and keep the percent symbol, then move the decimal point two places to the left and drop the percent symbol
Decimal to percent: Move the decimal point two paces to the right and add the percent symbol
Fraction to percent: First, express the fraction as a decimal. Then, express the decimal as a percent
A percent is also the ratio of a part of a whole to the whole
Percentage: The PART of a whole
Base: The WHOLE
Percentage: percent x base
Percent= part/whole=percentage/base
Base= percentage/percent
Percent Of Change: A percent that measures the amount of change between two values
Discount: A percentage of the original price
Percent of change= amount of change/original value
Discount rate: The percent used to calculate the discount
Sale price: The price the customer pays