COMPETENCY 1.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY

Size: px

Start display at page:

Download "COMPETENCY 1.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY"

Marshall Morton
5 years ago
Views:

1 SUBAREA I. NUMBERS AND OPERATIONS COMPETENCY.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY Skill. Analyze the structure of the base ten number system (e.g., decimal and whole number place value) Rational numbers can be expressed as the ratio of two integers, a b where b 0, for example 2, - 4 5, 5 = 5. The rational numbers include integers, fractions and mixed numbers, terminating and repeating decimals. Every rational number can be expressed as a repeating or terminating decimal and can be shown on a number line. Integers are positive and negative whole numbers and zero....-6, -5, -4, -, -2, -, 0,, 2,, 4, 5, 6,... Whole numbers are natural numbers and zero. 0,, 2,,,4,5,6... Natural numbers are the counting numbers., 2,, 4, 5, 6,... Irrational numbers are real numbers that cannot be written as the ratio of two integers. These are infinite non-repeating decimals. Example: 5 = pi = = A fraction is an expression of numbers in the form of x y, where x is the numerator and y is the denominator, which cannot be zero. Example: 7 is the numerator; 7 is the denominator If the fraction has common factors for the numerator and denominator, divide both by the common factor to reduce the fraction to its lowest form. Example: = = 9 Divide by the common factor A mixed number has an integer part and a fractional part.

2 Example: 2, 5, Percent = per 00 (written with the symbol %). Thus 0% 0 = = Decimals are portions of ten (deci = part of ten). To find the decimal equivalent of a fraction, use the denominator to divide the numerator as shown in the following example. Example: Find the decimal equivalent of 7 0. Since 0 cannot divide into 7 evenly = Whole Number Place Values are where the digits fall to the left of the decimal point. Consider the number 792. We can assign a place value to each digit. Reading from left to right, the first digit (7) represents the hundreds place. The hundreds place tells us how many sets of one hundred the number contains. Thus, there are 7 sets of one hundred in the number 792. The second digit (9) represents the tens place. The tens place tells us how many sets of ten the number contains. Thus, there are 9 sets of ten in the number 792. The last digit (2) represents the ones place. The ones place tells us how many ones the number contains. Thus, there are 2 sets of one in the number 792. Therefore, there are 7 sets of 00, plus 9 sets of 0, plus 2 ones in the number 792. Decimal Place Value is where the digits fall to the right of the decimal point. More complex numbers have additional place values to both the left and right of the decimal point. Consider the number 4.87.

3 Reading from left to right, the first digit, 4, is in the ones place and tells us the number contains 4 ones. After the decimal, (8) is in the tenths place and tells us the number contains 8 tenths. (7) is in the tenths place and tells us the number contains 7 sets of ten. The fourth digit () is in the hundredths place and tells us the number contains sets of one hundredth. Each digit to the left of the decimal point increases progressively in powers of ten. Each digit to the right of the decimal point decreases progressively in powers of ten. Example: occupies the following powers of ten positions: The Expanded Form of a number is an alternative method of writing a number. To write a number in expanded form each digit is multiplied by the power of ten that represents its place value. Example: Write in expanded form. We start by listing all the powers of ten positions Multiply each digit by its power of ten. Add all the results. 0 Thus = 4 2 (7 0 ) + ( 0 ) + ( 0 ) + (6 0 ) (9 0 ) + (0 0 ) + (0 0 ) + (5 0 ) ( 0 ) + (7 0 ) Example: Determine the place value associated with the underlined digit in

4 The place value for the digit 9 is 0 or 000. The Standard Form is a convenient method for writing very large and very small numbers. It employs two factors. The first factor is a number between -0 and 0. The second factor is a power of 0. This notation is a shorthand way to express large numbers (like the weight of 00 freight cars in kilograms) or small numbers (like the weight of an atom in grams). Example: Write 2,000 in standard form. 2. x 0,000 = 2. x 0 4 Example: Write in standard form. 79 = 79 = = Skill.2 Demonstrate knowledge of the characteristics of whole numbers (e.g., prime/composite, divisibility) In number theory, the fundamental theorem of arithmetic states every natural number either is itself a prime number, or can be written as a unique product of prime numbers. Factors are whole numbers that can be multiplied together to get another whole number. Prime numbers are whole numbers greater than that have only 2 factors, and the number itself. Examples of prime numbers are 2,, 5, 7,,, 7, and 9. Note that 2 is the only even prime number. Composite numbers are whole numbers that have factors other than and the number itself. For example, 9 is composite because is a factor in addition to and is also composite because, besides the factors of and 70, the numbers 2, 5, 7, 0, 4, and 5 are also all factors. Remember that the number is neither prime nor composite. The following are some rules for divisibility: a. A number is divisible by 2 if that number is even (which means it ends in 0,2,4,6 or 8).

5 ,54 ends in 4, so it is divisible by ,685 ends in a 5, so it is not divisible by 2. b. A number is divisible by if the sum of its digits is evenly divisible by. The sum of the digits of 964 is = 9. Since 9 is not divisible by, neither is 964. The digits of 86,54 is = 24. Since 24 is divisible by, 86,54 is also divisible by. c. A number is divisible by 4 if the number in its last 2 digits is evenly divisible by 4. The number,6 ends with the number 6 in the last 2 columns. Since 6 is divisible by 4, then,6 is also divisible by 4. The number 5,627 ends with the number 27 in the last 2 columns. Since 27 is not evenly divisible by 4, then 5,627 is also not divisible by 4. d. A number is divisible by 5 if the number ends in either a 5 or a ends with a 5 so it is divisible by 5. The number 470 is also divisible by 5 because its last digit is a 0. 2,58 is not divisible by 5 because its last digit is an 8, not a 5 or a 0. e. A number is divisible by 6 if the number is even and the sum of its digits is evenly divisible by. 4,950 is an even number and its digits add to 8. ( = 8) Since the number is even and the sum of its digits is 8 (which is divisible by ), then 4950 is divisible by is an even number, but its digits add up to. Since is not divisible by, then 26 is not divisible by ,5 is not an even number, so it cannot possibly be divided evenly by 6. f. A number is divisible by 8 if the number in its last digits is evenly divisible by 8. The number,6 ends with the -digit number 6 in the last places. Since 6 is divisible by 8, then,6 is also divisible by 8. The number 465,627 ends with the number 627 in the last places. Since 627 is not evenly divisible by 8, then 465,627 is also not divisible by 8.

6 g. A number is divisible by 9 if the sum of its digits is evenly divisible by 9. The sum of the digits of 874 is = 9. Since 9 is not divisible by 9, neither is 874. The digits of 6,54 are = 8. Since 8 is divisible by 9, 6,54 is also divisible by 9. h. A number is divisible by 0 if the number ends in the digit ends with a 5 so it is not divisible by 0. The number 2,00,270 is divisible by 0 because its last digit is a 0. 42,978 is not divisible by 0 because its last digit is an 8, not a 0. Skill. Apply the Fundamental Theorem of Arithmetic to determine the prime factorization of numbers The Fundamental Theorem of Arithmetic states that every composite (non-prime) number can be written as a product of primes in one, and only one way. Prime factorization of number is when a number is written as the product of prime numbers or prime factors. To get the prime factors of a number, the number is factored into any 2 factors. The resulting factors are checked to see if either can be factored again. Continue factoring until all remaining factors are prime. This is the list of prime factors. Regardless of what way the original number was factored, the final list of prime factors will always be the same. Example: Factor 0 into prime factors. Factor 0 into any 2 factors, e.g. 5 and Now factor the These are all prime factors. Or factor 0 into any 2 factors, and 0. 0 Now factor the These are the same prime factors even though the original factors were different. Example: Factor 240 into prime factors. Factor 240 into any 2 factors Now factor both 24 and Now factor both 4 and These are prime factors. 4 This can also be written as 2 5.

7 Skill.4 Use principles of number theory (e.g., greatest common factor, least common multiple) to solve problems in applied contexts The greatest common factor (GCF) is the greatest number that is a factor of all the numbers given in a series. The GCF can be no greater than the smallest number given in the series. To find the GCF, list all possible factors of the least number given (including the number itself). Starting with the greatest factor (which is the number itself), determine if it is also a factor of all the other given numbers. If so, that is the GCF. If that factor does not work, try the same method on the next factor. Continue until a common factor is found. If no other number is a common factor, then the GCF will be the number. Note: There can be other common factors besides the GCF. Example: Find the GCF of 2, 20, and 6. The smallest number in the problem is 2. The factors of 2 are, 2,, 4, 6, and 2. The greatest factor is 2, but it does not divide evenly into 20. Neither does 6, but 4 will divide into both 20 and 6 evenly. Therefore, 4 is the GCF. Example: Find the GCF of 4 and 5. Factors of 4 are, 2, 7, and 4. The greatest factor is 4, but it does not divide evenly into 5. Neither does 7 or 2. The only factor that is common to both 4 and 5, is, therefore is GCF. The least common multiple (LCM) of a group of numbers is the smallest number into which all of the given numbers will divide evenly. Example: Find the LCM of 20, 0, and 40. The greatest number given is 40, but 0 will not divide evenly into 40. The next multiple of 40 is 80 (2 x 40), but 0 will not divide evenly into 80 either. The next multiple of 40 is is divisible by both 20 and 0, so 20 is the LCM. Example: Find the LCM of 96, 6, and 24. The largest number is 96. The number 96 is divisible by both 6 and 24, so 96 is the LCM.

FUNDAMENTAL ARITHMETIC

FUNDAMENTAL ARITHMETIC Prime Numbers Prime numbers are any whole numbers greater than that can only be divided by and itself. Below is the list of all prime numbers between and 00: Prime Factorization