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Chapter 1

Chapter 1. Whole Numbers. 1.1 Introduction to Numbers, Notation and Rounding. Definitions Numbers : Amounts or quantities Set : A group of elements Natural Numbers : 1, 2, 3, 4, 5,…. Whole Numbers : 0, 1, 2, 3, …. Place Values.

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Chapter 1

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  1. Chapter 1 Whole Numbers

  2. 1.1 Introduction to Numbers, Notation and Rounding • Definitions Numbers : Amounts or quantities Set: A group of elements Natural Numbers : 1, 2, 3, 4, 5,…. Whole Numbers : 0, 1, 2, 3, ….

  3. Place Values Trillions period Billions period Millions period Thousands period Ones period • Procedure • Write a number in expanded form: • Write each digit multiplied by its place value • Express it all as a sum • Example • 68,456 = 6x 10,000 + 8 x 1000 + 4 x 100 + 5 x 10 + 6 x 1 • Standard form Expanded form • In words – 68 thousands, four hundred fifty six

  4. Use < , > , or = to make a true statement Definition Equation: A mathematical statement that contains an equal sign Example 14 = 14 is an equation, which is true 14 = 5 is not an equation , it is false, they are not equal Inequality: A mathematical statement that contains an inequality symbol. Greater than > Less than < 14 is greater than 12 12 is less than 14 14 > 12 12 < 14

  5. Round numbers to a specified place • To round a number to a given place value, consider the digit to the right of the desired place value. • If this digit is 5 or greater, round up If this digit is 4 or less, round down Example 45,685,923 45,685,923 45,685,923 45,685,923 45,685,923 Millions Hundred thousands Ten thousands Hundreds Tens

  6. 1.2 Adding, subtracting, and solving equations with whole numbers Definition Addition : Arithmetic operation that combines amounts Commutative property of addition a + b = b + a, where a and b are any numbers Associative property of Addition ( a + b) + c = a + (b + c), where a, b, c are any numbers To add whole numbers: • Stack with corresponding place values aligned • Add the digits Example 1 1 1 1 3456 62345 +906 + 9039 4362 71384

  7. Estimate sum Estimate the sum by rounding to the nearest Thousand, then find the actual sum. 52,407 +31,596 Estimate Actual 52,000 52,407 +32,000 +31,596 84,000 84,003

  8. Perimeter: The total distance around a shape. To find the perimeter of a shape, add the lengths of all the sides of the shape. 10 ft 8ft 8ft P = 8 + 10 + 8 + 10= 36 ft 10 ft Variable: A symbol that can vary or change in value Constant: Any symbol that does not vary in value.

  9. Subtraction To subtract whole number: • Stack the greater number on top of the smaller number, aligning the place values. • Subtract the digits in the bottom number from the digits directly above. If a digit in the top number is less than the digit beneath it, then rename the top digit. 14 9 9 3 4 10 10 13 Check by adding Check by estimating 45,00 3 36638 45,000 - 8,36 5 8365 - 8,000 37,000 36 6 3 8 45,00 3 Keywords for subtraction Subtract, minus, remove, decreased by, difference, take away, left, less than

  10. Procedure: To find a missing addend, write a related subtraction sentence. Subtract the known addend from the sum. Example 1 100 + x= 250 x = 250 – 100 = 150 Check 100+ 150 = 250 Example 2 65 + x= 93 x = 93 – 65 =28 Check 65 + 28 = 93 Definition Solution: A number that can replace the variable(s) in an equation and make the equation true

  11. 1.3 Multiplying whole numbers and exponents Commutative property of multiplication a.b= b.a Associative property of multiplication Grouping three or more factors differently will not affect the product (ab)c= a(bc), where a, b, c are any real numbers Multiplicative property of 0 The product of 0 and a number is always 0 0.n = 0 and n.0= 0, where n is any number Multiplicative property of 1 The product of 1 and a number is always the number 1.n = n and n.1 = n, where n is any number Distributive Property If a sum or difference is multiplied by a number, then each number inside parenthesis may be multiplied by the number outside the paranthesis a(b+c) = ab + ac a(b – c) = ab – ac, where a, b and c are any numbers

  12. Procedure To multiply two whole numbers, stack them and then apply the distributive property Multiply 503 x 62 1 503 X 62 1006 Multiply 2 times 503 + 3018 Multiply 6(tens) times 503 31,186 Key words for multiplication Multiply, times, product, each, of , by

  13. Exponent or power A symbol written to the upper right of a base number that indicates how many times to use the base as a factor. Base : The number that is repeatedly multiplied 28 Exponent = number of times the base is used as a factor Base 28 = 2.2.2.2.2.2.2.2 = 256 Exponential form Factored form Standard form

  14. Powers of 10 and Period Names Period names 103 =1,000 = One thousand 106 = 1,000,000 = One million 109= 1,000,000,000 = One billion 1012= 1,000,000,000,000 = One trillion 1015= (1 with 15 zeros) = One quadrillion The names continue using the preceding pattern. Some names are rather colorful. 10100 = (1 with 100 zeros) = googol 10googol = ( 1 with a googol of zeros)= googolplex

  15. Write in expanded form using powers of ten Example 1 3,029,408 = 3x 106 + 0 x 105 +2x 104 + 9 x 103 + 4 x 102 + 8 x 1 Example 2 24,902 2 x 104 +4 x 103 + 9 x 102 + 2 x 1

  16. 1.4 Dividing, square roots, and solving equations with whole numbers Division Property • When 1 is the divisor, the quotient is equal to the dividend n divided by 1 = n/1 =, where n is any number • When 0 is the divisor with any dividend other than 0, the quotient is undefined. n divided by 0, or n/0 , is undefined, when n = 0 • When 0 is the dividend, the quotient is 0 as long as the divisor is not also 0 0 divided n = 0/n , when n = 0 • If both dividend and divisor are 0, the quotient is indeterminate 0 divided 0 or 0/0 is indeterminate • When a number (other than 0) is divided by itself, the quotient is 1 n is divided by n = n/n = 1, when n = 0

  17. Divisibility Rules • 2 is an exact divisor for all even numbers. Even numbers have 0, 2, 4, 6, or 8 in the ones place Example- 2478 is divisible by 2 as even number 8 is in ones place b) To determine whether 3 is an exact divisor for a given number Add the digits in the dividend If the resulting sum is a number that is divisible by 3, then so is the dividend Example- 35,616 is divisible by 3 as sum of the digits = 3 + 5 + 6 + 1 + 6= 21 divisible by 3 c) 5 is an exact divisor for numbers that have 0 or 5 in the ones place Example – 71,315 is divisible by 5 because it has a 5 in the ones place

  18. Divide 387 42070 • 10836 93 3912517 84 372 243 192 224 186 196 651 196 651 0 07 Key words for division Divide, distribute, each, split, quotient, into, per, over

  19. Definition Square root : A base number that can be squared to equal a given number The symbol for square root is the radical sign. The number we wish to find the square root of is called the radicand Radical sign = 25 Perfect Square : A number that has a whole – number square root 25 2 = 625 Roots and their square Page 44

  20. Order of Operations • Grouping symbols. These include parentheses( ), brackets[ ], braces{}, absolute value , radicals , and fraction bars ___ 2. Exponents 3. Multiplication or division from left to right, in order as they occur • Addition or subtraction from left to right, in order as they occur G E M D A S or P E M D A S Grouping Exponents Multiplications Division Addition Subtraction Parenthesis Exponents Multiplications Division Addition Subtraction

  21. 1.6 Variables, Formulas, and Solving Equations Formula for Perimeter of a rectangle P = 2L + 2W To Use a formula • Replace the variables with the corresponding given values • Solve for the missing value L w w L

  22. Use the formula to find Parallel lines : Straight lines that never intersect Parallelogram : A four sided figure with two pair of parallel sides The area of a parallelogramA = bh Right angle: An angle that measures 900 Volume of a Cube which is a measure of the amount of space inside a three-dimensional object. V= lwh h b h w l

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