UC MATH 1A - Chapter 3 Math Fundamentals

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Chapter 3 Math Fundamentals DEALING WITH NUMBERS Whenever you decide to learn a new language what do they start with on the very first day Vocabulary Well math has as much of its own lexicon as any country s mother tongue so now is as good a time as any to familiarize yourself with the terminology These vocabulary words are rather simple to learn or relearn but they re also very important Any of the terms you ll read about in this chapter could show up in a GRE math question so you should know what the test is talking about For a more lengthy list you can consult the glossary in Chapter 14 We ll start our review with the backbone of all Arabic numerals the digit Digits You might think there are an infinite number of digits in the world but in fact there are only ten 0 1 2 3 4 5 6 7 8 and 9 This is the mathematical alphabet that serves as the building block from which all numbers are constructed Modern math uses digits in a decile system meaning that every digit in a number represents a multiple of ten For example 1 423 795 1 1 000 4 100 2 10 3 1 7 0 1 9 0 01 5 0 001 You can refer to each place as follows 1 occupies the thousands place 4 occupies the hundreds place 2 occupies the tens place 3 occupies the ones or units place 7 occupies the tenths place so it s equivalent to seven tenths or 9 occupies the hundredths place so it s equivalent to nine hundredths or 5 occupies the thousandths place so it s equivalent to five thousandths or When all the digits are situated to the left of the decimal place you ve got yourself an integer Integers When we first learn about addition and subtraction we start with integers which are the numbers you see on a number line Integers and digits are not the same thing for example 39 is an integer that contains two digits 3 and 9 Also integers are not the same as whole numbers because whole numbers are non negative which include zero Conversely integers include negative numbers Any integer is considered greater than all of the integers to its left on the number line So just as 5 is greater than 3 which can be written as 5 3 0 is greater than 4 and 4 is greater than 10 For more about greater than less than and solving for inequalities see Chapter 4 Consecutive Integers and Sequences Integers can be listed consecutively such as 3 4 5 6 or in patterned sequences such as odds 1 3 5 7 evens 2 4 6 8 and multiples of 6 6 12 18 24 The numbers in these progressions always get larger except when explicitly noted otherwise Note also that because zero is an integer a list of consecutive integers that progresses from negative to positive numbers must include it 2 1 0 1 Zero Zero is a special little number that deserves your attention It isn t positive or negative but it is even So a list of consecutive even integers might look like 4 2 0 2 4 Zero might also seem insignificant because it s what s called the additive identity which basically means that adding zero to any other number doesn t change anything This will be an important consideration when you start plugging numbers into problems in Chapter 5 Positives and Negatives On either side of zero you ll find positive and negative numbers For the GRE the best thing to know about positives and negatives is what happens when you multiply them together A positive times a positive yields a positive 3 5 15 A positive times a negative yields a negative 3 5 15 A negative times a negative yields a positive 3 5 15 Even and Odd As you might have guessed from our talk of integers above even numbers which include zero are multiples of 2 and odd numbers are not multiples of 2 If you were to experiment with the properties of these numbers you would find that any number times an even number yields an even number the product of two or more odd numbers is always odd the sum of two or more even numbers is always even the sum of two odd numbers is always even the sum of an even number and an odd number is always odd Obviously there s no need to memorize stuff like this If you re ever in a bind try working with real numbers After all if you want to know what you get when you multiply two odd numbers you can just pick two odd numbers like 3 and 7 for example and multiply them You ll see that the product is 21 which is also odd Digits Quick Quiz Explanations for Digits Quick Quiz 1 2 3 If x y and z are consecutive even integers and x 0 and z 0 then x must be 2 y must be 0 and z must be 2 Therefore their product is 0 and you would enter this number into the box Take the answer choices and switch the hundreds digit and ones digit When the result is 396 less than the old number you have a winner Choices A B and C are out because their ones digits are greater than their hundreds digits therefore the result will be greater for example 293 becomes 392 If you rearrange 713 the result is 317 which is 396 less than 713 The answer is D Pick three consecutive digits for a b and c such as 2 3 and 4 Quantity A becomes 2 3 4 or 24 and Quantity B becomes 2 3 4 or 9 Quantity A is greater so eliminate B and C But if a b and c are 1 0 and 1 respectively then both quantities become 0 so eliminate A Therefore the answer is D MORE ABOUT NUMBERS Prime Numbers Prime numbers are special numbers that are divisible by only two distinct factors themselves and 1 Since neither 0 nor 1 is prime the least prime number is 2 The rest as you might guess are odd because all even numbers are divisible by two The first ten prime numbers are 2 3 5 7 11 13 17 19 23 and 29 Note that not all odd numbers are prime 15 for example is not prime because it is divisible by 3 and 5 Said another way 3 and 5 are factors of 15 because 3 and 5 divide evenly into 15 Let s talk more about factors Factors As we said a prime number has only two distinct factors itself and 1 But a number that isn t prime like 120 for example has several factors If you re ever asked to list all the factors of a number the best idea is to pair them up and work through the factors systematically starting with 1 and itself So for 120 the factors are 1 and 120 …


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