Week 6, Day 2: Arithmetic Review

Week 6, Day 2

 

Chapter 10: Math Foundations – Arithmetic and Number Properties Review

 

I will not include all the terms listed at the beginning of the chapter here, but suffice it to say that this is math vocabulary you need to memorize. Note important distinctions between kinds of numbers (integers, fractions/decimals, etc.) and especially take note of the different names of place value in a given number. U.S. English gives different place values from U.K. English when it comes to large values like millions (1,000,000) and billions (1,000,000,000). The symbols commonly used on the GRE are listed right after the terms.

 

 

Rules of Operation

If you do not follow the correct order of operations, you will get incorrect answer. PEMDAS is the acronym we use to memorize these rules. The order goes: first do what is inside Parentheses, and next do any Exponents (or roots), and third you can do either or both Multiplication and Division from left to right, followed finally by Addition and/or Subtraction from left to right. U.S. students learn this as “Please Excuse My Dear Aunt Sally.”

The commutative laws of addition and multiplication entail the ability to switch the order of any values being added or multiplied. You CANNOT do this with division or subtraction.

E.g.: a + b = b + a

The associative laws of addition and multiplication entail the ability to change the grouping of any values being added or multiplied. You CANNOT do this with division or subtraction.

E.g.: (a + b) + c = a + (b + c)

The distributive law entails the ability to distribute or share a value being multiplied to a parenthetical statement over each value.

E.g.: a(b + c) = ab + ac

 

 

Number Properties

Adding and Subtracting

When we add two numbers that have the same sign (positive or negative for both), we simply add the numerals and give the result the sign of both numbers.

When we add two numbers with different signs (one is positive and the other is negative), we subtract the numerals and give the sign of the larger number.

Subtraction is the opposite of addition: we add a negative second number to the first number – this is more useful when we are dealing with unusual circumstances. To subtract a negative number, remember that the negatives cancel out, so we simply add. But when we subtract a positive number from a negative number, OR when we subtract a positive number from a smaller positive number, we change the sign of the number we subtract and treat it as addition.

 

Multiplication and Division of Positive and Negative Numbers

Multiplying or dividing with the same sign on both produces a positive result, but multiplying or dividing with different signs will produce a negative result.

 

Absolute Value

Absolute value is the distance of any number from zero.

E.g., |-3| and |3| are both exactly three spaces from zero, and so |-3| = 3.

When operations happen with absolute value, we treat them like parentheses; when the absolute value is set equal to some number, we have to solve for both the positive and the negative number.

E.g., |x – 3| = 10 must be solved as (x – 3) = 10 and as (x – 3) = -10.

 

Properties of Zero

Adding or subtracting zero does not change a number; this is known as additive identity.

However, multiplying by zero changes the result to zero. Dividing zero by any number results in zero, but you CANNOT divide by zero. 0/3 = 0 but 3/0 = undefined.

 

Properties of 1 and -1

Multiplying or dividing by 1 does not change the number; this is known as multiplicative identity. However, multiplying or dividing by -1 changes the sign of the number.

 

Factors, Multiples and Remainders

Multiples and Divisibility

A multiple is the product of a number and an integer. Factors are the integers that produce a given product.

E.g., the multiples of 2 include 2, 4, 6, 8, 10…

The factors of 10 are 1, 2, 5, and 10.

 

Divisibility is based on what integers evenly divide a given number without any remainders or decimals. The rules are listed under this heading and should be memorized; they are also included in Appendix D.

 

 

Odds and Evens

Even numbers are evenly divisible by 2, and end in 0, 2, 4, 6, or 8. Odd numbers are not evenly divisible by 2 and end in 1, 3, 5, 7, or 9.

Odd + odd always gives an even result, e.g., 3 + 3 = 6.

Even + even always gives an even result, e.g., 2 + 2 = 4.

However, odd + even produces an odd result, e.g., 2 + 3 = 5.

Odd x even always gives an even result, e.g., 2 x 3 = 6.

Even x even always gives an even result, e.g., 2 x 8 = 16.

However, odd x odd produces an odd result, e.g., 3 x 3 = 9.

 

Factors, Primes, and Prime Factors

Factors or divisors include both positive and negative integers.

The greatest common factor is the largest factor that is the same for two numbers.

Prime factors are the prime numbers that are the factors for a number. There are two methods to do the prime factorization of a number. Method 1: work your way up through prime numbers starting with 2; divide by 3 next, then 5, then 7, then 11, etc. Method 2: figure out one pair of factors and then work your way down until you have only prime factors left.

 

Least Common Multiple

The least common multiple is the smallest multiple that is the same for two numbers. This is useful for combining fractions. To find it, you can always do a prime factorization; each prime factor should appear the maximum number of times it appears in either number, and then all of the primes get multiplied together. The least common multiple of two integers should be smaller than their product if they have any factors in common.

 

Remainders

Remainders are the number “left over” or remaining when you have divided a number as many times possible without creating a decimal but you still have not completely divided the number. For example, 7 goes into 22 three times, but there is still 1 leftover.

 

 

Exponents and Roots

Rules of Operations with Exponents

The rules of operations are on our memorization chart; generally the rule for multiplication is, when the base is the same, you add the exponents. For division, when the bases is the same, you subtract the exponents. When a power is raised to a power, you multiply the exponents. When the bases are different but the exponent is the same, you multiply the bases together and raise the result to the power. A negative exponent means you give the reciprocal of the base and raise it to the indicated power. A fractional exponent means you take the root of the base.

 

Commonly Tested Properties of Powers

Fractions raised to a power produce a smaller result; e.g., ½ squared produces ¼.

Negative numbers raised to even powers produce positive results, but negative numbers raised to odd powers produce negative results; e.g., -2 squared is 4, but -2 cubed is -8.

Raising even numbers to positive exponents gives even numbers, while raising odd numbers to positive exponents produces odd results.

 

Powers of 10

Increasing powers of 10 produces additional zeros after a number. For example, 10 squared has two zeroes, 100, while 10 cubed has three zeros, 1,000, and so forth. Multiplying a number by 10 raised to a power moves the decimal to the right when the exponent is positive, but it moves to the left when the exponent is negative. For example, 0.029 times ten cubed produces 29, but 416.03 times ten to the negative 4^th power produces 0.041603.

 

Scientific Notation

Scientific notation is based on multiplying numbers by different powers of ten. We ALWAYS have only one digit to the left of the decimal when representing any number in scientific notation; all other non-zero (i.e. significant) digits must be to the right of the decimal. For example, 123,000,000,000 is 1.23 x 10¹¹.

 

 

 

 

Rules of Operations with Roots and Radicals

A number squared is equal to that number multiplied by itself. The product can then have a square root equal to that number. E.g., 5 square equals 25, so the square root of 25 equals five. If the radical symbol has another digit on it, then you take that root of the number, so the cube root of 81 would equal 3. We do not look for the square root or any other even root of negative numbers, because these are imaginary numbers and the GRE only uses real numbers.

 

Addition and subtraction of Radicals

You can only add or subtract like to like; we can add 2Ö3 and Ö3, but not 2Ö2 and Ö3.

 

Multiplication and Division of Radicals

The numbers inside radical symbols can be multiplied or divided, but must be multiplied or divided separately from the numbers outside radical symbols.

 

Simplifying Radicals

Find the factors of a number inside a radical symbol; if a factor or factors is a perfect square, you can pull out the square root, but any numbers which are not perfect squares must remain under the radical.