All Numbers in the Universe Real Numbers Imaginary Numbers 1, etc.. Rational Numbers (0, 3, -1, ½⅔,.524, etc..) Irrational Numbers, 2, 3, etc.. Integers (.-3,-2,-1,0,1,2,3..) Fractions (1/2, -4/3, %,.25, etc..) Whole Numbers {0,1,2,3, } Negative Integers {., -4, -3, -2, -1} Positive Integers (also called Natural Numbers) {1, 2, 3, 4, }
Section 1.2 WHOLE NUMBERS A set is a collection of objects. The set of natural numbers is {1,2,3,4,5,.} The set of whole numbers is {0,1,2,3,4,5, } Whole numbers are used for counting objects (such as money, but not cents!) However, they do not include fractions or decimals. The digits in a whole number have place value. Place Value 3 4 5 5 7 6 4 0 2 8 9 7 4 1 5 Three- Digit Groups (separated by commas) Billions group Thousands group Millions group Ones group Verbal Form: Trillions group 30,542 = Thirty thousand, five hundred forty-two. (Notice we don t use the word and.) Standard Notation: Uses only digits (0 through 9) and commas to state the number. 16 million = 16,000,000 Expanded Form: States the standard form of each place value. 31,567 3 is the place value for ten thousands so this represents 30,000 1 is the place value for thousands, so this represents 1,000 5 is the place value for hundreds, so this represents 500 6 is the place value for tens, so this represents 60 7 is the place value for ones, so this represents 7 ones or 7 31, 567 in Exanded Form is 30,000 + 1,0000 + 500 + 60 + 7
Section1.3 Adding and Subtracting Whole Numbers Addition Terms: The numbers being added are called addends. Subtraction Terms: The number you are taking away is the subtrahend, the number you are subtracting from is the minuend, and the answer is the difference. Mathematical properties are often used to simplify computation. Below are three addition properties stated in words, shown with a numeric example, and shown with an algebraic example. The Zero Property of Addition is also called the Identity Property of Addition. Associative Property of Addition When numbers or variables are added, for example (2 + 3) + 4 = 2 + (3 + 4) and (a + b) + c = a + (b + c) The addends can be grouped in different ways without changing the result. Commutative Property of Addition When numbers or variables are added, for example 2 + 3 = 3 + 2 and a + b = b + a, The order of the addends can be changed without changing the result. Zero Property of Addition When 0 is added to a number or variable, for example, 2 + 0 = 2 and a + 0 = a, the result is the same number or variable. These properties can be used when adding numbers in your head. Example: 337 + 18 = (300 + 30 + 7) + ( + 8) = 300 + (30 + ) + (7+ 8) = 300 + 40 + 15 = 355
Translating Verbal Expressions into Mathematical Expressions Verbal Expressions Examples Math Translation Addition added to 6 added to y 6+y more than 8 more than x 8+x the sum of the sum of x and z x+z increased by t increased by 9 t+9 the total of the total of 5 and y 5+y Subtraction minus x minus 2 x-2 less than 7 less than t t-7 subtracted from 5 subtracted from 8 8-5 decreased by m decreased by 3 m-3 the difference between the difference between y and 4 y-4
Perimeter Perimeter is the distance around a two-dimensional shape. Just remember to add up ALL the sides! Example 1: Find the perimeter of a rectangle whose length is 9 inches and whose width is 5 inches. The perimeter of this rectangle is 9+9+5+5 = 28 inches Example 2: What is the perimeter of the cross? All sides are of length inches. Example 3: Find the perimeter of a triangle whose sides are 21, 27, and 32. Perimeter = inches x 12 = 120 inches
Section 1.4 Rounding Whole Numbers: Step 1: Locate the rounding digit, which is the digit at the place value you are rounding to. Step 2: Look at the digit directly to the right of the rounding digit. This is the test digit. If the test digit is < 5 (less than 5), keep the rounding digit the same and change all digits to the right of it to 0. If the test digit is 5 (greater than or equal to 5), then increase the rounding digit by 1 and change all the digits to the right of it to 0. On a number line, numbers are written so that ascending numbers are to the right. is to the left of 20 on the number line, so 20 is greater than (20 > ). It can also be stated that descending numbers are to the left, so is less than 20 ( < 20) 0 20 30 40 50 Example: Round 48 to the nearest ten. Ask "is 48 closer to 40 or 50?" It is closer to 50, it is about 50. Round off the same way for larger numbers. Example: Round 1888 to the nearest thousand. Is 1888 closer to 00 or 2000? 1888 is closer to 2000. 0 Is 43,556 closer to 40,000, or 50,000? Compare the first two numbers. 43 is closer to 40 than 50. It is 40,000. 00 1888 2000 "Tricky" numbers are those with 5 as the number that decides. Is 650 closer to 600 or 700? On a number line 650 is half way. Math has a rule for this. When the digit is 5 or greater, round up. When the digit is less than 5, round down. Example: Round 24 to the nearest ten. Round 36 to the nearest ten 24 35 Rounds down 0 20 30 40 50 Rounds up 0 20 30 40 50 You may need to estimate to a certain place. Look at the number in the place to the right. Then round. Round 684 to the tens place. The number 4 in the ones place to the right of the tens. It rounds down to 680. Round 6423: to the nearest ten is 6420. to the nearest hundred is 6400. to the nearest thousand, it is 6000. Round 589,557: to the nearest ten, it is. to the nearest hundred, it is. to the nearest thousand, it is 590,000 to the nearest ten thousand is 590,000_. to the nearest hundred thousand is 600,000. You can round it to the nearest million, it is 1,000,000. 48
Estimating Sums and Differences When an exact answer is not necessary, an estimate can be used. The most common method of estimating sums and differences is called front-end rounding, which is to round each number to its largest place value, so that all but the first digit of the number is 0. Examples: Estimate 4,894 + 429 Round 4,894 to the nearest thousand. 4,894 5,000 Round 429 to the nearest hundred 429 400 Add the rounded numbers. 5,000 + 400 = 5,400 The actual answer is 4,894 + 429 = 5353, which is close to 5,400. If both addends are rounded up, the estimated sum will be greater than the actual sum, and if both addends are rounded down, the estimated sum will be less than the actual sum. Such generalizations are not possible with subtraction. Estimate 6,209 383. What is the largest place value of 6,209? Round to the nearest. 6,209 -> What is the largest place value of 383? Round to the nearest. 383 -> Subtract the rounded numbers. - = The actual answer is 6,209 383 = 5826 How close was your estimate?
1.5 Multiplying Numbers Multiplication: 5 x 4 = 5 4 = 5(4) = 20 factors Other terms for multiplication: times, multiplied by, of. product Multiplication Properties of 0 and 1: a 0 = 0 a 1 = a Commutative Property: a b = b a 5 4 = 4 5 Associative Property: (a b) c = a (b c) (2 4) 5 = 2 (4 5) 8 5 = 40 = 2 (20) Distributive Property: a (b+ c) = a b + a c 2 (4+5) = 2 4 + 2 5 2 (9) = 18 = 8 +
Example 2 Mileage Specifications for a Ford Explorer 4x4 are shown in the table below. How far can it travel on a tank of gas? mpg means Miles per Gal = miles/gal Engine Fuel Capacity Fuel economy (mpg) 4.0 L V6 21 gal 15 city/ 19 hwy Number of gallons in 1 tank We are being asked to find how far, which is a distance. Distance is measured in miles. We will assume this distance is traveled in the city so we ll use city mileage (15 mpg). Dimensional analysis is used to convert units. We are given miles per gal and gallons per tank. We want to know how many miles can be traveled on 1 tank. 15miles 21gal gal tank 15 21 315 miles/tank Note: if you don t know what 15 x 21 is doing mental math, you can do it using the distributive property: 15 x 21 = 15(20+1) = 15(20) + 15(1) = 300+15=315 Example 3: Calculating production The labor force of an electronics firm works two 8-hour shifts each day and manufactures 53 TV sets each hour. Find how many sets will be manufactured in 5 days. We are looking for the number of TV sets manufactured in 5 days. Use Dimensional Analysis. We are given TV sets manufactured per hour, and hours worked per shift, and shifts worked per day. If you included all the pertinent information, you should have cancelled out all the unnecessary units (like units on top cancel out like units on the bottom), and the units left should be TV sets, which is what we want. 2 shifts day 8 hours shift 53 TV sets hour 5 days 28535 4240 Using the commutative and distributive properties of multiplication we could we regroup these numbers for easier mental math. 2*5=, 8*53 = 8*(50+3) = 400+24=424 *424=4240
Rectangular Patterns If you have a rectangular pattern objects, such as rows and columns, you can determine the total number of objects by multiplying the number of rows times the number of columns (or objects per row). Example: Our classroom has 7 rows with 6 seats in each row. Therefore our classroom can hold 7x6 = 42 people. Area: The area of a rectangle is length X width= lw The is the amount of area covered within the rectangle. If length is in inches, and width is in inches, Area = length X width means the units of the Area are inch 2, or square inches Example 1) Find the area: Example 2) Find the area: Note that since a square is just a rectangle whose length and width equal each other, you just have to know 1 side, and then multiply it by itself. Example 3) Find the area: