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Physics/Conceptual Physics Arithmetic Overview: Numbers and Operations Number Operations • Real numbers: all numbers on the number line. All numbers that appear on this test are real. • Integers: all numbers with no fractional or decimal parts, including negative whole numbers and zero; multiples of 1. See the number line below. Number Operations • Sum: the result of addition • Difference: the result of subtraction • Product: the result of multiplication • Quotient: the result of division Number Operations • Reciprocal: the result of switching the numerator and denominator of a fraction. • The reciprocal of 3/5 is 5/3. • The reciprocal of 2 is 1/2 because 2 can be considered to be the fraction 2/1 Numbers and the Number Line • A number line is a straight line, extending infinitely in either direction, on which real numbers are represented as points. • Decimals and fractions can also be depicted on a number line, as can numbers such as √2.. Numbers and the Number Line Numbers and the Number Line • The values of numbers get larger as you move to the right along the number line. • Numbers to the right of zero are positive; • Numbers to the left of zero are negative. • Zero is neither positive nor negative. Numbers and the Number Line • Any positive number is larger than any negative number. For example, -300 is less than 4. Assessment Question 1 All of the following statements are true EXCEPT: A. All numbers on the number line are real numbers. B. Integers are all numbers with no fractional or decimal parts. They are whole numbers C. The reciprocal of 3/7 is 7/3. D. Zero is neither positive nor negative. E. Division is used to determine the sum of two numbers. Number Operations • Operations: a process that is performed on one or more numbers. • The four basic arithmetic operations are addition, subtraction, multiplication, and division. Laws of Operation: Commutative Law • Commutative laws of addition and multiplication • a+b = b+a • axb = bxa Laws of Operation: Commutative Law • Addition and multiplication are both commutative. • Switching the order of any two numbers being added or multiplied together does not effect the result. • For example, 5 + 8 = 8 + 5; both sums equal 13. • Similarly 2 x 6 = 6 x 2; both products equal 12. Laws of Operation: Commutative Law • Subtraction and division are not commutative; switching the order of the numbers changes the result. • For instance, 3-2 ≠ 2-3; the left side yields a result of 1, whereas the right side yields a result of -1. • Similarly, 6÷2 ≠ 2÷6; the left side gives us 3, whereas the right side gives us ⅓ Laws of Operation: Associative Law • Associative laws of addition and multiplication: • (a + b) + c = a + (b + c) • (a x b) x c = a x (b x c) Laws of Operation: Associative Law • Addition and multiplication are both associative. • (3+5) + 8 = 3 + (5+8) • (8) +8 = 3 + (13) • 16 = 16 Laws of Operation: Associative Law • Addition and multiplication are both associative. • (4 x 5) x 6 = 4 x (5 x 6) • (20) x 6 = 4 x (30) • 120 = 120 Laws of Operation: Associative Law • Because addition and multiplication are both commutative and associative, numbers can be added or multiplied in any order. • Subtraction and division are not associative. Laws of Operation: Associative Law • Example: • 7 - (10 - 4) = 7 – (6) = 1 • (7 - 10) - 4 = (-3) - 4 = -7 • So 7 - (10 - 4) ≠ (7 - 10) - 4. Laws of Operation: Associative Law • Example: • 24 ÷ (12 ÷ 3) = 24 ÷ (4) = 6 • (24 ÷ 12) ÷ 3 = (2) ÷ 3 = ⅔ • So 24 ÷ (12 ÷ 3) ≠ (24 ÷ 12) ÷ 3 Assessment Question 2 All of the following statements are true EXCEPT: A. The four basic arithmetic operations are addition, subtraction, multiplication, and division. B. Numbers can be added, multiplied, subtracted or divided in any order to get the same result. C. 5 + 8 = 8 + 5 by Commutative law D. Subtraction and division are not commutative or associative. switching the order of the numbers changes the result. E. (3 x 5) x 7 = 3 x (5 x 7) by Associative law Laws of Operation: Distributive Law • In general, the distributive law is: • a(b + c) = ab + ac Laws of Operation: Distributive Law • The distributive law of multiplication allows you to "distribute" a factor over a group of numbers that is being added or subtracted, by multiplying that factor by each number in the group. Laws of Operation: Distributive Law Example: • 4(3 + 7) = • (4 x 3) + (4 x 7)= Laws of Operation: Distributive Law • Example: • 4(3 + 7) = (4 x 3) + (4 x 7) • 4 x (10) = (12) + (28) • 40 = 40 Laws of Operation: Distributive Law • Division can be distributed in a similar way, because dividing by a number is equivalent to multiplying by that number's reciprocal. Laws of Operation: Distributive Law • Example: • = (3 + 5) = • (8) == = = (3) + (5) Laws of Operation: Distributive Law • Don't get carried away, though. • When the sum or difference is in the denominator-that is, when you're dividing by a sum or difference-no distribution is possible. Laws of Operation: Distributive Law Laws of Operation: Distributive Law So their sum can not equal 1 Assessment Question 3 All of the following statements are true EXCEPT: A. A. B. C. D. In general, the distributive law is: a(b + c) = ab + ac Division can be distributed 5 x (8 x 7) = (5 x 8) + (5 x 7) 8/(3+2) = 8/3 + 8/2 Dividing by a number is equivalent to multiplying by that number's reciprocal. Operations with Signed Numbers • Numbers can be treated as though they had two parts: • A positive (+) or negative (-) sign • A number part Operations with Signed Numbers • For example: -3 • The sign of the number -3 is negative • The number part is 3. • Numbers without any sign are understood to be positive. Operations with Signed Numbers • To add two numbers that have the same sign, add the number parts and keep the sign. • Example: What is the sum of -6 and -3 ? Operations with Signed Numbers • • • • • Example: What is the sum of -6 and -3 ? To find (-6) + (-3) Add 6 and 3 Then attach the negative sign from the original numbers to the sum. • (-6) + (-3) = -9 Operations with Signed Numbers • • • • • Example: What is the sum of -6 and -3 ? To find (-6) + (-3) Add 6 and 3 Then attach the negative sign from the original numbers to the sum. • (-6) + (-3) = -9 Assessment Question 4 What is the sum of -7 and -2 ? A. B. C. D. E. -5 -9 14 27 -72 Operations with Signed Numbers • To add two numbers that have different signs, find the difference between the number parts, and keep the sign of the number whose number part is larger. • Example: What is the sum of -7 and +4? Operations with Signed Numbers • Example: What is the sum of -7 and +4? • To find (-7) + (+4), subtract 4 from 7 to get 3 • 7>4 • The number part of-7 is greater than the number part of +4 so the final sum will be negative. • (-7) + (+4) = -3 Operations with Signed Numbers • Subtraction is the opposite of addition. • You can rephrase any subtraction problem as an addition problem by changing the operation sign from a minus to a plus and switching the sign on the second number. • For instance: 8 - 5 = 8 + (-5). Operations with Signed Numbers • There's no real advantage to rephrasing if you are subtracting a smaller positive number from a larger positive number. • But the concept comes in very handy when you are subtracting a negative number from any other number, a positive number from a negative number, or a larger positive number from a smaller positive number. Operations with Signed Numbers • To subtract a negative number: • Rephrase as an addition problem and follow the rules for addition of signed numbers. • For instance: 9 - (-10) =9 + 10 =19 Operations with Signed Numbers • Here's another example: • (-5) - (-2) = (-5) + 2; • The difference between 5 and 2 is 3 • The number with the larger number part is -5 • So the answer is -3. Assessment Question 5 (-7) – (-2) = A. B. C. D. E. -5 -9 14 27 -72 Operations with Signed Numbers • To subtract a positive number from a negative number or from a smaller positive number: • Change the sign of the number that you are subtracting from positive to negative • Then follow the rules for addition of signed numbers. • For example, (-4) - 1 = (-4) + (-1) = -5. Operations with Signed Numbers • Example: Subtract 8 from 2. • 2 - 8 = 2 + (-8) • The difference between the number parts is 6, and the -8 has the larger number part. • So the answer is -6. Operations with Signed Numbers • Multiplication and division of signed numbers: • Multiplying or dividing two numbers with the same sign gives a positive result. • (-4) x (-7) = +28 • (-50) ÷ (-5) = +10 Operations with Signed Numbers • Multiplying or dividing two numbers with different signs gives a negative result: • (-2) x (+3) =-6 • 8 ÷ (-4) =-2 Assessment Question 6 All of the following statements are true EXCEPT: A. B. C. D. E. -8 - 9 = -17 8 x -8 = -64 (-8 x -7) = (8 x 7) -8/-2 = -4 2 - 7 = -5 Properties of Zero • Adding zero to or subtracting zero from a number does not change the number. • x+0=x • 0+x=x • x–0=x Properties of Zero • Example: • 5+0=5 • 0+(-3)=-3 • 4-0=4 Properties of Zero • Notice, however, that subtracting a number from zero switches the number's sign. • It's easy to see why if you rephrase the problem as an addition problem. Properties of Zero • Example: Subtract 5 from 0. • 0-5 = -5 • That's because 0 - 5 = 0 + (-5) • According to the properties of zero: • 0 + (-5) = -5. Properties of Zero • The product of zero and any number is zero: • 0xz=0 • z x 0 =0 • Example: • 0 x 12 = 0 Properties of Zero • Division by zero is undefined. • For practical purposes, that translates as "it can't be done." • Because fractions are essentially division (that is ¼ means 1 ÷ 4), any fraction with zero in the denominator is also undefined. Assessment Question 7 All of the following statements are true EXCEPT: A. B. C. D. E. 0 - 9 = -9 8x0=0 (-8 x 0) = (0 x 7) -8/0 = -8 2+0 =2 Properties of 1 and -1 • Multiplying or dividing a number by 1 does not change the number: • ax1=a • 1xa=a • a÷1=a Properties of 1 and -1 • Example: • 4x1=4 • 1 x (-5) =-5 • (-7) ÷ 1 =-7 Properties of 1 and -1 • Multiplying or dividing a nonzero number by -1 changes the sign of the number: • a x (-1) = -a • (-1) x a = -a • a + (-1) = -a Properties of 1 and -1 • Example: • 6 x (-1) = -6 • (-3) x (-1) = 3 • (-8) ÷ (-1) =8 Assessment Question 8 All of the following statements are true EXCEPT: A. B. C. D. E. 1x9 =9 -1(4) = -4 (-8 x -1) = (8 x 1) -8/-1 = 8 1+1 =1x1 Order of Operations • Whenever you have a string of operations, be careful to perform them in the proper order. • Otherwise, you will probably get the wrong answer. Order of Operations: PEMDAS • PEMDAS • The acronym PEMDAS stands for the correct order of operations: Order of Operations: PEMDAS Parentheses Exponents Multiplication Division Addition Subtraction }Multiplication & Division } in order from left to right }Addition & Subtraction } in order from left to right Order of Operations • If you have trouble remembering PEMDAS • Think of the mnemonic phrase: • Please Excuse My Dear Aunt Sally. Order of Operations • Example: • 66 x (3 -2) ÷ 11 Order of Operations Example: 66 x (3 -2) ÷ 11 • If you were to perform all the operations sequentially from left to right • Without regard to the rules for the order of operations • You would arrive at the answer 196/11 . Order of Operations Example: 66 x (3 -2) ÷ 11 • To do this correctly, do the operation inside the parentheses first: (3 – 2) = 1 • Now we have: • 66 x (1) ÷ 11 = • 66 ÷ 11 = 6 Assessment Question 9 90 - 4( 3+2) x 4 = A. B. C. D. E. -9 10 145 280 2064 Order of Operations • Example: • 30 - 5 x 4 + (7 -3)2 ÷ 8 = Order of Operations • Example: 30 - 5 x 4 + (7 -3)2 ÷ 8 = • First perform any operations within parentheses. (If the expression has parentheses within parentheses, work from the innermost out.) • 30 - 5 x 4 + (4)2 ÷ 8 = Order of Operations • Example: 30 - 5 x 4 + (7 -3)2 ÷ 8 = • Next, do the exponent: • 30 - 5 x 4 + (16) ÷ 8 = Order of Operations • Example: 30 - 5 x 4 + (7 -3)2 ÷ 8 = • Then, do all multiplication and division in order from left to right. • 30 - 20 + 2 = Order of Operations • Example: 30 - 5 x 4 + (7 -3)2 ÷ 8 = • Last, do all addition and subtraction in order from left to right. • 10 +2 = • The answer is 12. Assessment Question 10 25 ÷ ( 3+2)2 x 4 = A. B. C. D. E. 4 16 350 500 2500