Week 1, Day 2 (Arithmetic) and Week 2, Day 1 (Arithmetic Continued)

Start Arithmetic Review

Some basic terminology:

Natural/counting numbers: 1, 2, 3, 4, …

Whole numbers: 0, 1, 2, 3, …

Integers (positive & negative): … -3, -2, -1, 0, 1, 2, 3, …

Rational numbers: numbers which can be expressed as integers or fractions

Irrational numbers: cannot be expressed as integers or fractions, i.e. √3 or π

Real numbers: all rational and irrational numbers

Imaginary numbers: √-1 or i   e.g., √-9 = 3i

Prime numbers: divisible only by 1 and itself (2, 3, 5, 7, 11, 13, 17, 19, 23…)

Composite numbers: divisible by more than 1 or itself (4, 6, 8, 9, 10, 12…)

Odd numbers: not divisible by 2 (3, 5, 7, 9,…)

Even numbers: divisible by 2 (2, 4, 6, 8, 10…)

Square numbers: product of a number times itself, i.e., 3 * 3 = 3² = 9 ß 9 is the square

Cubed numbers: product of a number times itself times itself, i.e., 3 * 3 * 3 = 3³ = 27 ß27 is the cube

START HERE MONDAY

Different ways to represent multiplication

a x b    a * b    (a)(b)   a(b)     (a)b     3ab ß only for variables/letters!

Math Symbols

Axioms/Laws/Properties of Addition & Multiplication

None of these rules applies to division or subtraction!

Commutation: a + b = b + a                2 + 3 = 3 + 2               a * b = b * a                2 * 3 = 3* 2          

            For commutation, order does not matter.

Association:    (a + b) + c = a + (b + c)                      (2 + 3) + 4 = 2 + (3 + 4)

                        (a * b) * c = a * (b * c)                       (2 * 3) * 4 = 2 * (3 * 4)

            For association, grouping does not matter.

Identity:          a + 0 = a          1 + 0 = 1         a * 1 = a          2 * 1 = 2

Additive inverse:        a + -a = 0         1 + -1 = 0

Multiplicative inverse:           a * 1/a = 1       2 * ½ = 1

Distribution:   a * (b + c) = ab + ac                2 * (3 + 4) = (2 * 3) + (2 * 4)

                        a * (b – c) = ab – ac                2 * (4 – 3) = (2 * 4) – (2 * 3)

Distribution will not work when the parenthetical operator is the same as the external operator (e.g.: 2 * (3 * 4) ≠ (2 * 3) * (2 * 4)).

Also note: even & odd numbers have some interesting properties to memorize as well.

Addition                                  vs        Multiplication

e + e = e          2 + 2 = 4                     e * e = e          2 * 2 = 4

o + o = e          3 + 3 = 6                     o * o = o          3 * 3 = 9

e + o = o          2 + 3 = 5                     e * o = e          2 * 3 = 6

Place Value

Ones (ones, tens, hundreds), e.g. 103 – one hundred and three

Thousands (thousands, ten thousands, hundred thousands), e.g., 182,976 – one hundred eighty-two thousand and nine hundred seventy-six

Millions (millions, ten millions, hundred millions), e.g. 543,235,695 – five hundred forty-three million, two hundred thirty-five thousand, six hundred ninety-five

So the number 3,240,586,917 has a 3 in the billions place, a 2 in the hundred millions place, a 4 in the ten millions place, and a 0 in the millions place, etc.

1,000,000,000,000 = 1 trillion;           1,000,000,000 = 1 billion;      1,000,000 = 1 million;           1,000 = 1 thousand and so forth.

For decimals, the number 3.01257896 has a 3 in the units or ones place, a 0 in the tenths place, a 1 in the hundredths place, a 2 in the thousandths place, etc.

0.1 = 1 tenth; 0.01 = 1 hundredth; 0.001 = 1 thousandth and so forth.

Expanded Notation p. 12

This is practice for Scientific Notation (p. 59-60).

Basically the idea is to separate each digit into its place.      345 = 300 + 40 + 5

Expanded Notation: (3 * 10²) + (4 * 10¹) + (5 * 10⁰)

Scientific Notation: (3.45 * 10²)

43.25 = 40 + 3 + 0.2 + 0.05

Expanded Notation: (4 * 10¹) + (3 * 10⁰) + (2 * 10^-1) + (5 * 10^-2)

Scientific Notation: (4.325 * 10¹) 

Grouping Symbols p. 13

{[()]} – Braces, brackets, and parentheses are usually given in this order: () parentheses are innermost, followed by [] square brackets, and then {} curly braces; the pattern repeats again as necessary: for example, a + b * 3c + 4 ¸ 5 – 7 + 2d * 3 = [(a + {[(b * 3c) + 4] ¸ 5}) – 7] + (2d * 3) etc.

Order of Operations p. 14

PEMDAS (Please Excuse My Dear Aunt Sally) is how we remember

1) Parentheses

2) Exponents (including square roots or fractional exponents)

3) Multiplication and/or

4) Division (go left to right), then

5) Addition and/or

6) Subtraction (go left to right)

Rounding Off p. 15

1. Find the place value to which you are rounding

2. Look to the right

3. If that number is five or larger, the digit in the place to which you are rounding goes up 1; if that number is four or smaller, the digit in the place to which you are rounding remains the same.

The Number Line p. 16

Operations (with and without Signed Numbers) p. 17

Addition with the same sign: keep the sign, add the numbers as usual

Addition with different signs: subtract the numbers, use the sign of the number farthest from zero

Subtraction: change the sign of the number being subtracted and then add

Minus before parenthesis: change the signs of each number inside the parentheses

Multiplication or Division: multiply or divide as usual, but an odd number of negative signs

produces a negative result

Multiplying with zero: always equals zero

Dividing zero by any number: always equals zero

Dividing by zero: always equals undefined

Other divisibility rules MEMORIZE! p. 20

A number is divisible by...

2 if it is even (ends in 0, 2, 4, 6, or 8)

3 if the sum of its digits is divisible by 3

4 if the number formed by the last 2 digits is divisible by 4

5 if it ends in 0 or 5

6 if it is divisible by both 2 and 3

7 (no simple rule, do long division)

8 if the number formed by the last 3 digits is divisible by 8

9 if the sum of its digits is divisible by 9

Fractions p. 21

Numerator: number on top = how many pieces are considered relevant

Denominator: number on bottom = how many pieces into which the whole is divided

e.g.: ½  ß In this fraction (one half), the 1 is the numerator and the 2 is the denominator

All rules for signed numbers apply to fractions.

Negative fractions: where the negative sign goes does not matter.

Proper and Improper Fractions vs. Mixed Numbers p. 22

Proper = numerator is smaller than the denominator, e.g. ¼ = 0.25 = 25%

Improper = numerator is larger than the denominator, e.g. 4/3 = 1.33 = 133%

Mixed = one integer with a proper fraction, e.g. 1½ = 1.5 = 150%

Turning an improper fraction into a mixed number requires division; turning a mixed number into an improper fraction requires multiplication.

Equivalent Fractions p. 23

The GRE takes all equivalent fractions in numeric entry. However, for multiple-choice answers and for some computations it is best to reduce fractions to their lowest terms. To do so, divide both the numerator and the denominator by the same number until they can be reduced no further.

Enlarging the denominator may be required for some computations as well. To do so, multiply both the numerator and the denominator by the same number until you reach the desired denominator.

Factors p. 25

The integers which are multiplied together to produce a number are known as factors.

Common factors are the factors which are the same for two such numbers.

The greatest common factor is the largest factor common to two numbers.

Multiples p. 27

Multiples are found by multiplying a number by increasingly larger numbers.

Common multiples are the multiples which are the same for two numbers.

The least common multiple is the smallest multiple common to two numbers. ß This will be important for finding the least common denominator for two fractions when performing operations between fractions.

Number Sequences p. 29

Also known as patterns!

The key: look for the changes between each number in the sequence. This may be the same change (add 4 each time) or a change from one number to the next (add increasing values, 1, 2, 3, etc.); alternatively, the pattern may be a combination of numbers in a sequence (e.g., n₁ + n₂ = n₃, for all n > n₂) or between separated numbers (e.g., n₁ + n₃ = n₅ and n₂ + n₄ = n₆)

Operations with Fractions  p. 31

Addition and Subtraction

To add and subtract fractions, the fractions must have a common denominator; find the least common multiple for the denominators and that becomes the least common denominator. Then you can combine the numerators accordingly. ONLY add or subtract the numerators.

To add and subtract mixed numbers, the same rule applies for the fraction part of the number. In many cases it will be easier to convert mixed numbers into improper fractions, find the least common denominator, and then add or subtract the numerators before converting the answer back into a mixed number. Alternatively, you can just add or subtract the whole numbers and fractions separately, but this can involve borrowing from the wholes to perform the operation with the fractions.

Multiplication and Division p. 38

To multiply fractions, you multiply across from numerator to numerator and then denominator to denominator. Then you can reduce to lowest terms by canceling common factors from both the numerator and denominator. Multiplying mixed numbers is easiest if you convert to improper fractions, multiply, and then convert back into a mixed number.

For dividing fractions, you put the first number (the dividend) down as it is, then invert the second number (the divisor), and finally multiply the result. The song:

“Dividing fraction’s so easy to do, it’s oh so easy to do. You just take one down, turn the other around, and multiply the both of them.”

A complex fraction is just a fraction division problem presented vertically instead of horizontally. We treat this just like a regular division problem. The GRE really likes to use very complex fractions, so we start with the lowest and divide until we have finished the problem.

Dividing mixed numbers is easiest when we convert to improper fractions and back again to mixed numbers once we have the result.

“Simplifying” just means “solving” for our purposes.

STOP HERE; START HERE WEDNESDAY

Decimals p. 45

Fractions can be converted to decimals simply by dividing the numerator by the denominator. To Turn a decimal back into a fraction, we choose the place value of the last significant digit (i.e., non-zero numeral) and use that as a denominator. Then we place the decimal digits on top and reduce as far as possible.

Add and subtract decimals by lining up the decimals and inserting zeros where necessary.

To multiply, count the number of decimal places and then be sure the answer has this number of decimal places.

To divide decimals, move the decimals the same direction and number of spaces so you can divide as if they were whole numbers.

Percentage p. 48

A percent is a fraction out of one hundred, or a decimal to the hundredths place. Move the decimal two places to the right and add on a % (percent sign). If you are given a fraction, you will convert to a decimal before doing this. To turn a percent into a decimal, you move the decimal two places to the left and remove the %. To turn a percent into a fraction, you put the percent number over 100 and reduce as necessary. 

Important equivalents to memorize are on p. 51-52.

Find the Percent of a Number p. 52

To determine the percent of a number, use a fraction or decimal (whichever you prefer) and multiply. For our purposes “of” is a keyword meaning “multiply.”

Here are the other keywords for word problems:

For example: 10% of 100 = 10/100 * 100 = 10; 10% of $56.70 = 10/100 * 56.70 = $5.67 etc.

With our keywords, we can turn any percent question into an equation. For example:

5 is what percent of 20? ß “is” becomes “=” and “what” becomes “x.” Our equation is now:

5 = x% * 20    which we can solve as            5 = x/100 * 20            

and then divide both sides by 20 to get x by itself     5/20 = x/100  

and then multiply both sides by 100 to further isolate x        5/20 * 100 = x

and then reduce the fraction   ¼ * 100 = x     and then divide 100 by 4 to get          25 = x

so 5 is 25% of 20, or 5 = 25% * 20

This method of isolating x can be used to solve any of the following types of percent problems:

What is x% of y?        X% of what number is y?       X is what percent of y?           etc.

Proportion Method p. 54

Using percent as a fraction of 100 allows us to use the percent-proportion method where cross-multiplication reveals the missing value. Proportions are statements that two values expressed as fractions are equal. Remember: given any two values in a percent problem, we can always find the missing third value.

Percent Change  p. 56

Roots & Exponents + Operations with Roots & Exponents p. 57-58

Measurements

Measurement conversions to memorize p. 67

How to convert measurements:

A. Convert one unit into its larger unit through multiplication.

            e.g. 1 foot = 12 inches; 2 feet = ?? inches; multiply 12 inches by 2 to get 24 inches

B. Given an equality between two units, set up a proportion and solve for the missing measure

through cross-multiplication.

            e.g. 36 inches equals 1 yard, so how many inches are in 3 yards?

                        1/36 = 3/x       36 * 3 = 108, so x = 108 or:

                        Therefore, x = 108, or in other words, 3 yards are 108 inches.

C. Given an equality between two units, convert through division.

e.g. 2.2 pounds approximately equals 1 kilogram (2 lb. ≈ 1 kg.) so how many kilograms are in 10 pounds?

            10 lb. ¸ 2.2 kg. ≈ 4.5 kg.

Glossary of arithmetic terms; important terms to include are:

sum – result of addition

difference – result of subtraction

product – result of multiplication

quotient – result of division (between a dividend [number divided] and a divisor [number

dividing]; may or may not have a remainder if not divided evenly)

 Review & practice set