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- 1.
- 2. Classifying numbers<br />Natural Numbers are counting numbers<br />It includes 1,2, 3, and so forth 1,2,3… the trois dots at the end indicate that the set continues on indefenitely, repeating the pattern.<br />Whole numbers include natural number plus zero: 0,1,2,3,.. <br />Integers: are the whole numbers and their opposites, where 0 is its own opposite. …-2,-1, 0, 1,2,3,..<br />
- 3. Classifying numbers<br />Rational number is a number that can be written as a simple fraction (i.e. as a ratio). <br />Example 1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction) <br />7 is rational, because it can be written as the ratio 7/1<br />Example 0.317 is rational, because it can be written as the ratio 317/1000<br />Here are some more examples:<br />The square root of 2 cannot<br /> be written as a simple fraction! <br />A rational number is a number that can be in the form p/qwhere p and q are integers and q is not equal to zero.<br /> 7/0 No! "q" can't be zero!<br />
- 4. Irrational number: is a real number that cannot be written as a simple fraction. Irrational <br /> means not Rational<br /> Example: π(Pi) is an irrational number. <br />π = 3.1415926535897932384626433832795 (and more...)<br />You cannot write down a simple fraction that equals Pi. <br />The popular approximation of 22/7 = 3.1428571428571... is close but not accurate.<br />Many square roots, cube roots, etc are also irrational numbers. <br />Examples:<br />√3 = 1.7320508075688772935274463415059 (etc) <br />√99 = 9.9498743710661995473447982100121 (etc) <br />But √4 = 2 (rational), and √9 = 3 (rational) ...<br />... so not all roots are irrational.<br />
- 5. Complex number system is the entire number system and it is made up of <br /> real numbers and imaginary numbers.<br />Real Number (-4, -1/2, 0, 3) <br />Imaginary –complex (-2i, -i, 5+ 4i)<br />Rational (-4, -1/2, o, -5/2<br />Irrational (V---2, V---11pi<br />Integers (-3, -2, 0, 1,2)<br />Whole number (0, 1, 2, 7, 389)<br />Natural number(2,6,10)<br />
- 6. Consider the numbers: 4, 0, 7/8, 11/5, -6. Find:<br />a. The natural numbers<br />b. The whole numbers<br />c. The integers:<br />d. The rational numbers<br />
- 7. Practice<br />Problem 1. Which of the following are rational numbers and which are irrational? <br />Irrational<br />rational<br />c)<br />a)<br />rational<br />Irrational<br />d)<br />b)<br />3.5<br />Irrational<br />3√2 <br />b)<br />0<br />24/8<br />-5<br />61 02/3<br />1/2<br />√9 <br />
- 8. Place Value<br />Each digit is said to have a place value<br />ones<br />Billions<br />
- 9. Practice<br />What is the place value of the number 4 in the following problems?<br /> <br />1.) 567.431 6.) 654<br />2.) 981.042 7.) 754,90<br />3.) 453.2 8.) 390,410,8<br />4.) 784.32 <br />5.) 7849<br />
- 10. Place Value<br />Numbers, such as 84, have two digits. Each digit is a different place value. <br />The left digit is the tens' place. It tells you that there are 8 tens. <br />The last or right digit is the ones' place which is 4 in this example. Therefore, there are 8 sets of 10, plus 4 ones in the number 84. <br />8 4 <br /> | |__ones' place <br /> |_________tens' place<br /> 7 8 4 <br />| | |__ones' place <br /> | |_________tens' place <br /> |________________hundreds' place<br />4 9 5,7 8 6 <br /> | | | | | |__ digits' place value one’s place<br /> | | | | |____ tens' place value<br /> | | | |______ hundreds' place value <br /> | | |________ thousands' place value<br /> | |__________ ten-thousands' place value<br /> |____________ hundred-thousands' place value<br />
- 11. Place Values of Decimals<br />Decimal numbers, such as O.6495, have four digits after the decimal point. <br /> Each digit is a different place value. <br /> <br />The first digit after the decimal point is called the tenths place value. <br />There are six tenths in the number O.6495. <br />The second digit tells you how many hundredths there are in the number. <br />The number O.6495 has four hundredths. <br />The third digit is the thousandths place. <br />The fourth digit is the ten-thousandths place which is five in this example. <br />Therefore, there are six tenths, four hundredths, nine thousandths, and five ten-thousandths in the number 0.6495. <br />0. 6 49 5 <br /> | | | | |<br /> | | | | |____ ten-thousandth' place value<br /> | | | |______ thousandths' place value <br /> | | |________ hundredths' place value<br /> | |__________ tenths' place value<br /> |____________ place value<br />
- 12. How to round a Number<br />Rounding off is a kind of estimating. <br />Decide to what place you will round the number.<br />Underline the number that is in the place<br />Look at eh number to the right of the underlined number<br />If the number to the right is 5 or greater, add 1 to the underlined number and substitute zeros of all the numbers to the right of the underline number.<br />If the number to the right of the underlined number is less than 5, leave the underlined number as it is and substitute zeros for all the numbers to the right of the underlined number.<br />
- 13. Absolute value of integers<br />The absolute value of a number is its distance <br />from zero on the number line.<br />Absolute value is denoted by I I<br />Both 3 and −3 are a distance of 3 units from 0. <br /> |3| = |−3| = 3. Distance, in mathematics, is never negative. <br />
- 14. Absolute value equations<br />Absolute value equations<br />|a| = 5<br />What values could a have?<br />a could be either 5 or −5. For, if we replace a with either of those values, the statement -- the equation -- will be true.<br />And so, any equation that looks like this --<br />|a| = b<br />-- has the two solutions<br />|x − 2| = 8 <br />x − 2 = 8, or x − 2 = −8. <br />We must solve these two equations. The first implies<br /> x = 8 + 2 = 10.<br />The second implies<br /> x = −8 + 2 = −6. <br />These are the two solutions: x = 10 or −6.<br />
- 15. Problem 3.<br />a) An absolute value equation has how many solutions? <br />Two.<br />b) Write them for this equation: <br />|x| = 4.<br />|x + 5| = 4<br />|1 − x| = 7<br />
- 16. Understand basic operation on number<br />The four basic operation<br /><ul><li>Addition (+)
- 17. Subtraction (-)
- 18. Multiplication (x)
- 19. Division (/)</li></li></ul><li>Addition of signed Numbers<br />Sum is the result of addition<br />Addends are the number being added<br /> Rules of addition:<br />Numbers with like signs: <br />add the absolute values and give the answer the common <br />Sign<br />Example: 9+3 = (9) + (3) = +12 = 12<br /> (-6) + (-2) = -8 Add -6 and -2 and give the answer the common sign.<br />Numbers with unlike signs: <br />subtract the smaller absolute value from the larger . Give the answer <br />the sign of the number with the greater absolute value.<br />Example: (-3) + (+8) = + (8-3) + +5=5 +8 > -3 , so the answer is positive<br /> (+2) + (-9) = -(9-2) = - 7 -9 > +2 , so the answer is negative<br />
- 20. The commun<br />
- 21. Rules for addition<br />Adding zero to any number results in the sum being the <br />number itself. <br />Zero is called additive identity since the result of adding <br />zero is the identical number<br />Example 3+ 0 =3 -5 +0=-5 x2 + 0 =x2 + 0 =<br />Additive inverse is the sum of a number and its opposite<br />Example: 3 and -3 are additive inverses since <br /> 3 + (-3) =0 <br />Zero is neither positive nor negative. The opposite of zero is zero.<br />
- 22. Rules for subtraction<br />The difference is the result of a subtraction problem.<br />
- 23. The Commutative and associative laws of addition<br />For any real numbers a and b,<br />a+b=b+aCommutative law of addition<br />The commutative law of addition says that we may add in any order. Notice that the order of the letter is different on each side of the equal sign. Parentheses tell us to do the operation inside them first.<br /> 6+(5+3)=6+8=14<br />The associative law of addition says that when we are only doing additions , we may move the grouping symbols. The grouping does not affect the sum. Notice that the order of the letter does not change. <br />For any real numbers a , b, and c <br /> a+(b + c)= (a + b) + c Association law of addition<br />Example: Is the commutative or associative law of addition illustrated by each equation?<br />a.7+5=5+7 Commutative law of addition<br />The order of the numbers changes<br />7+5=12 Notice that the result is the same. <br />5+7=12<br />3 + (8+9) = (3+8) + 9 Association law of addition<br />The order of the number does not change. We do the calculation inside the parentheses first but <br />we get the same result<br />
- 24. Multiplication and Division<br />like signs<br />Example:<br /> a. (- 3) . (- 4) = +12 =12 b. (+4) . (+2) = +8<br />Different signs <br />Example: a) +4(-2) = -8 b) -15/3 = -5<br />Note: When the factors have alike signs, the product is positive. When the factors have unlike signs, the product is negative.<br />To multiply or divide two real numbers with the same signs multiply or divide their absolute values. Give the answer a positive sign.<br />Rule<br />(+ )(+) = +<br />(- ) (-) = +<br />(+ )( -) = -<br />(- )(+) = -<br />To multiply or divide two real numbers with different signs multiply or divide their absolute values. Give the answer a negative sign.<br />
- 25. Multiplication<br /><ul><li>Product is the result of multiplication
- 26. Multiplicand is the number being multiplied
- 27. Multiplier is the number doing the multiplying.
- 28. Symbol used to indicate Multiplication
- 29. Time sign 8 x 2
- 30. Raise dot 8 . 2
- 31. Parenthesis 8(2) or (8)2 or (8)(2)
- 32. When we use variables: 3p = 3 times p
- 33. Any numbers multiply by zero is equal to zero
- 34. 4 x 0 = 0 (Z) (0) = 0
- 35. Any number multiply by 1 results in the number itself
- 36. 1 is called multiplicative identity
- 37. Examples: 10 x 1 = 10 -6 x 1= -6 (Q)( 1) = Q</li></li></ul><li>Division<br />Quotient is the result of Division<br />Dividend is the number being divided<br />Divisor is the number doing the division<br />Symbol for division<br />Division sign<br />Fraction bar<br />Long division symbol<br />Dividing zero to any number (except zero 0) is zero<br />Example: 0/7 = 0 0/-3 = 0 0/K = 0, When K is not equal 0<br />Example 12/0 = undefined<br />Division by zero is undefined<br />
- 38. Division<br />Reciprocal is when any number except zero is inverted. It is called multiplicative inverse<br />To invert a number it must be expressed in a fraction first then you turn the fraction upside down. The numerator become the denominator and the denominator becomes the numerator. <br />To find the reciprocal of a number, invert the number.<br />zero does not have a reciprocal, since 1/0 is undefined<br />
- 39. Every nonzero real number A has a reciprocal 1/A<br />Examples: Write the reciprocal of each number <br />Any time a number is multiply by its reciprocal the product is 1<br />Example: 1/3 x 3/1 = 1 -4/5 x 5/-4 = 1<br />
- 40. Operation on Fractions<br />When adding or subtraction fractions each must have a common denominator<br />Multiplying and dividing fractions do not required a common denominator. <br />To multiply fractions you just need to multiply the numerators and multiply the denominators. If possible , reduce the product.<br />Examples: 7/9 x 2/5 = 14/45 =2/3 x 6/11 = -12/33 (reduce to -4/11) <br />To divide fractions we need to invert the second fraction and change <br />the sign to multiplication<br />
- 41. Operations on mixed numbers<br />To change an improper fraction to a mixed or whole number, divide the numerator by the denominator<br />Example 15/5 = 3 23/7 = 3 2/7<br />To change a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction and add the numerator of the fraction to this product.<br />Example: 7 ¼ = 29/4<br /><ul><li> Multiply 7 by 4, then add 1 to the product
- 42. Write this sum, 29 as the numerator
- 43. Keep the denominator, 4, of the given fraction</li></li></ul><li>Addition of mixed numbers<br />When adding mixed numbers, add the whole numbers first. Then add the fractions . Next, add the sum of the whole numbers to the sum of the fractions. When possible, simplify the results. <br />Example:<br />4 1/5 + 2 1/5 = 6 2/5<br />
- 44. Addition of mixed numbers<br />Changed the mixed numbers to improper fractions. Then add using the procedure for adding proper fraction<br />Example:4 1/5 + 2 1/5 =<br />Changed to an improper fraction is 12/5 and 11/5<br />Simplify: 12/5 + 11/5 = 32/5 = 6 2/5<br />To subtract, multiply and divide mixed numbers, first change them to improper fractions. Then simply follow the rules for subtracting, multiplying or dividing proper fractions.<br />
- 45. Decimal patterns<br />You can use patterns to multiply decimals mentally <br />by 10, 100, 1,000<br />
- 46. Place value patterns<br />You can name the numbers30,000 in several different ways<br />3 ten thousand (or 3x10,000)<br />30 thousand (or 30x1,000)<br />300 hundreds (or300x100)<br />3,000tens (or 3,000x10)<br />30,000 ones (or 30,000x1)<br />
- 47. Comparing Real Number-- Inequalities<br />Number line help us to see the relationship between two numbers.<br /> If a number is to the right of another number on the number line, it is the greater of the two numbers. <br /> If a number is to the left of another number, it is the smaller of the numbers. <br />The symbol > stands for “greater than” <br /> < means “less than”<br />Example: <br />On the number line below, <br />the integer -6 is to the left of 3. <br />Therefore -6 is less than 3<br />Positive real numbers<br />Negative real number<br />***All positive real numbers are greater than zero and all negative real numbers are less than zero<br />
- 48. Rational<br />Rational numbers, more commonly know as fractions, have a numerator and an denominator.<br />The numerator is the number on top<br />The denominator is the number on the bottom<br />Example 2 /3 2 is the numerator and the 3 is the <br /> denominator<br />Zero can never be the denominator. <br /> Division by zero is impossible or is said to be <br />undefined<br />
- 49. Decimals<br />Decimal are like fractions except that the denominator<br />must be power of ten. <br />This includes denominator of 10,100,1,000 and so<br />forth.<br />Because of this restriction, decimal are written with a<br />decimal point instead of a denominator.<br />Example: <br />0.8 = 8/10 0.003=3/100<br />0.12 =12/1,000 1.3 mans 13/10<br />
- 50. Decimals<br />
- 51. Decimal vs. fraction<br />If you want to change a fraction to a decimal, divide the denominator into the numerator<br />Example: 2/5 =0.4 11/4 = 2.75<br />When you want to change a decimal to a fraction, you must use the correct power of 10 in the denominator. Reduce the fraction if possible.<br />Example: 0.56 = 56/100 = 14/25 <br /> Since there is two decimal places to the right of the decimal point, the numerator will be 100<br />
- 52. Percent<br />Percent comes from the Latin meaning “per on <br />hundred.” The symbol of percent is %<br />To change decimal point to a percent move the <br />decimal point to two places to the right <br />Example: 0.47 = 47%<br />If we want to change a percent back to a<br />decimal move the decimal point two places to <br />the left and remove the percent sign.<br />
- 53. Fill in the blank in the table<br />
- 54. Operation on decimals<br />When adding or subtract decimals, the decimal points must be lined up<br />It not necessary to line up the decimal points when multiplying decimals. Follow these steps:<br />Set up the number as if the numbers being multiplied were intergers and multiply.<br />Count the number of places to the right of the decimal point in each number in t eproblem and add them together<br />In the answer, start from the right and count over the total number of places arrived at in the step above. This is where you place the decimal point for the answer.<br />Example: 56.13 x 8.1<br />56.13<br /> 8.1<br /> 5613<br />44904<br />454.653<br /> 2 places<br />+<br /> 1 place<br />3 places<br />
- 55. Ordering Numbers<br />A number line help us to see the relationship between tow numbers<br />
- 56. Variables In algebra <br />Variable: is a letter that is used to represent one or more numbers value in an <br /> expression or an equation. <br /> We call these letters "variables" because the numbers they represent can <br /> vary—that is, we can substitute one or more numbers for the letters in the <br /> expression.<br />Coefficients are the number part of the terms with variables<br />Variable expression is a collection of numbers, variable, and operations.<br />Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes <br />Variabe Expression: 8x + 4<br />constant<br />Variable<br />coefficient<br />
- 57. Evaluating variable expression<br />Simplify the <br />numeral expression.<br />Write the variable expression.<br />Substitute values <br />for variables.<br />Example: Evaluate the expression when x = 2<br />8x <br />Solution: <br />8x = 8(2) <br />=16 <br />Substitute 2 for y<br />simplify<br />
- 58. Exponents and Powers<br /> An expression like 46 is called a power. The exponent6 represent the number of times the base 4 is used as a factor. <br />base 46=4.4.4.4.4.4<br /> 6 factors of 4 <br />exponent<br />power<br />http://www.purplemath.com/modules/exponent2.htm<br />
- 59. Reading and Writing Powers<br />Express the meaning of the power in the word and then with number or variables.<br />Example: Evaluate the expression x3 when x =5<br />Solution: <br />X3 = 53<br /> = 5.5.5.<br /> = 125<br />Substitute 5 for x<br />Write factors.<br />Multiple.<br />
- 60. Evaluating Powers<br />
- 61. Scientific Noation<br />Whe we want to convey way to represent very large (2,340,000) or very small (0.000056) numbers given in a standard form, scientific notation is often used<br />Scientific notation is written as a number less than 10 but greater than or equal to 1 times 10 power (positive or negative). In order to change a number to this form, it is necessary to move the decimal point either to the right or left. Each place moved represents a power of 10<br />Example: 2.340000 = 3.34 X 106<br />In this case the decimal point is located after the last zero. Move the decimal six place to the left so that it is located between the 2 and 3.<br />
- 62. Scientific Notation<br />Example: In the number 0.000056, move the decimal point five places to the right so that it is located between the 5 and 6<br />0.000056 5.6 x 10-6<br />Note: <br />When the decimal point is moved to the left, the power of 10 is positive. When the decimal point is moved to the right, the yypower of 10 is negative<br />

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