Math's Prep 1st 2024 First term, Unite 1 Rational Numbers Lesson 1 The rational numbers Lesson 2 Comparing the rational number Lesson 3 Adding and subtracting Lesson 4 Multiplying & dividing Lesson 5 Applications on rational numbers
1 Lesson:1 The rational numbers Notes: Set of counting numbers. = {1,2,3,……} Set of Natural numbers N = {0,1,2,3,….} Set of integers numbers Z = {….,-2,-1,0,1,2,….} Question..? How to represent them on the number line? كيف نمثلهم علي خط االعداد؟ Counting numbers = {1,2,3,……} {0,1,2,3,….} -4 -3 -2 0 1 2 3 4 Integer numbers Z = {….,-2,-1,0,1,2,….} Definition: The rational number is a number that can be expressed in the form of quatient of an integer divided by an integer other than 0. Any rational number can be expressed as ( a , b two terms ) X = and the set of rational number = Q Q = { X : X = , a € Z , b € Z , b ≠0 } Example of rational number form. , - , 0 , 0.7 , , 3 , 15% Positive Q Negative Q Decimal Integers Percentage
2 Based on the definition (3 notes) 1. All decimal are rational numbers ex (0.7,1.2) 2. All percentages are rational number ex (15%) 3. All integers are rational number ex (3,4) There for Z Q and N Z Then: N Z Q - Ex: show why each of the following is a rational number. 1. = = = form & b ≠0 2. -0.17 = - 3. 0.006 = 4. 27% = Ex: put each of the following numbers in the simplest form. = (Divided by 5 up and down) 15 ÷ 5 = 3 and 25 ÷ 5 = 5 - = - (Divided by 8 up and down) 24 ÷ 8 = 3 and 56 ÷ 8 = 7 = (Divided by 5 up and down) 45 ÷ 5 = 9 and 20 ÷ 5 = 4 - = - = - up and down ÷ 11 and ÷ 4 Ex: Write each of following as decimal and percentage. = = = = = 2.5 = 250% 7 = = = 7.1875 = 718.75%
3 Lesson2 Comparing the rational number Representing the rational number on the number line. Negative rational number Positive rational number 0 Ex: represent on the number line = 0 < < 1 Divided the distance between 0,1 to four equal parts 0 a 1 The point a represent = If X , Y two points X Y We notes (X < Y) and ( Y < X) Ex: represent the following numbers on the number line in ascending order. , zero , , 2 , -1 -1 0 1 2
4 To comparing two rational nambers Important Positive > negative ( 0.05 > ) denominator > numerator Q > 1 > . > 1 , 1 > The same positive denominator are the greatest numerator will be greater > The same positive numerator are the greatest denominator will be smaller > Different denominator & numerator, we convert the two terms. > = ( = = ) Ex: compare the two numbers in each of the following. , Ans < (The same positive denominator are the greatest numerator will be greater) , Ans + > (Positive > negative) , Ans > (The same positive numerator are the greatest denominator will be smaller) , Ans , = < (The same positive denominator are the greatest numerator will be greater) -3.2 , Ans = < = (Positive > negative)
5 Lesson:3 Adding and subtracting *** First: addition operation. Same denominator. = Ex , + = = Different denominator. + = + = = = Properties of the addition in Q. Clasury property االنغالق خاصيه Q + Q = Q + = ( Q= rational number ) Commutative االبدال a + b = b + a Associative الدمج ) a + b)+ c = a +(b + c) Neutral element الجمعي المحايد a +0 = 0+a = a Additive inverse الجمعي المعكوس a + ( -a ) = a - a = 0 *** Second: subtraction operation. If a , b are in Q Then: a - b = a + (-b) Ex: - = + (- ) = = Example: - Solution: = = = )G.C.F) Other solution: - = = (L.C.M)
6 Lesson:4 Multiplying & dividing The sign rulesIn multiplication :االشارات قاعده Different { + x - = - } , Same { + x + = + } or { - x - = + } Multiplication operation. If and two rational. Therefor : x = Example: x = = = x (-2) = x -2 = = -1 Properties of the multiplication. Closure property. االنغالق Q x Q = Q x = = Commutative property. االبدال a x b = b x a x = x =… Associative property. الدمج ) a x b ) x c = a x ( b x c) Multiplicative neutral. الضربي المحايـــد a x 1 = 1 x a = a Multiplicative inverse. الضربي المعكوس x = 1 x = 1 Distributing multiplication over addition & subtracting. a x ( b + c) = ( b + c) x a = a x b +a x c Ex: x + x - = x ( + –1) = x 0 = 0 عمليه القسمه.operation Division ÷ = x Ex: ÷ = x =
7 Lesson 5 Applications on rational numbers The distance between the two number X & Y | X – Y | = | Y – X | EX: The distance between 2 & 5 | 2 - 5| = | -3 | = 3 *3 length units* 2 3 4 5 6 Note: the number that lies at the mid-point of the way between two distances: = the smaller number + X the distance Or = the greater number + X the distance العدد الذي يقع في منتصف المسافه بين عددين = العدد االكبر - نصف المسافه او العدد االصغر + نصف المسافه Ex: find a rational number in half-way between & L.C.M = 35 = , = Since: <
8 Solution : First: by the smaller number + x | - | = + x = + = + = + = Second: the greater number - X | - | = - x = - = – = - = Note: The number that lies at one third the way between two numbers. العدد الذي يقع بين عددين في ثلث المسافه Ex: the number between 2 & 8 there are two numbers. First: the number at one third (side 2) 2 + | 8 – 2 | = 2 + x 6 = 4 Second: the number at one third (8) 8 - | 8 - 2 | = 8 - x 6 = 6 Ex: Find a rational number laying at one fifth of the way between & & = & The greater number = The smaller number = he distance between two number | - ( ) | = | | = Complate. end
Key words of lesson: االعداد النسبيه numbers Rational ارقام التعداد numbers Counting االرقام الطبيعيه numbers Natural رقم صحيح Integers Form of recurring متكرر شكل المسافه بين between Distance مقارنه Comparing يعبر Express نسبه مئويه Percent كسر عشري number Decimal طرفي الكسر terms Two توزيع Distributing أبسط صوره form simplest يقع laying مسافه distance نقطه المنتصف point-mid (G.F.C) greatest common factor أكبر مشترك مضاعف )L.C.M) least common multiple أدني مشترك مضاعف