Category: Discrete Math

Introduction to discrete math items.

  • 8-Complements of a set, Cumulative, associative of sets.

    8-Complements of a set, Cumulative, associative of sets.

    Complements Of A Set, Cumulative, Associative, Distributive Of Sets.

    The complement of a set.

    This is a new example-if we have a universal set of U={1,2,3,4,5,6,7,8,9,10}, and the set A, which is another set, where A={1,2,3,4,5}, What is the new set which is called A’?A’ represents all the elements that exist in U but not present in set A. If we draw U as a rectangle and draw the set A inside that rectangle.

    If we draw a circle around A, then A’ is called the complement of A. The complement of a set is the elements not present in set A, do exist in U.

    The complement of a set.

    The union and intersection of a set.

    The subject of union and intersection is a well-known subject. The intersection of two sets is the set of all elements that belong to both sets.

    For a set which is called P={1,2,3,4,5,6,7,8,9,10}.

    For another set which is called  Q={2,4,6,8,9,10,12,14,16,18,20}.

    The intersection of the two sets means the common elements of the two sets which are,2 and 4& 6& 8&10.

    P ∩ Q, objects that belong to set P and set Q, the symbols are close to the letter n. While the union is similar to the letter U.

    The definition of a union is the set of all elements that belong to either or both sets. We will write all the elements that are not repeated. P ∪ Q, will include1,2,3,4,5,6,7,8,9,10,12,14,16,18,20.

    Union and intersection

    how to use the Venn diagram?

    Using the Venn diagram, by drawing each set in a circle and examine the common elements for the intersection. This will enable us to do an investigation the relations in a form of a diagram.

    Another example, if we have P={3,6,9,12,15,18} and Q={2,4,6,8,10,12}. If we use the highlighter for elements that are common between the two sets, the values are (6,12). This can be expressed as P ∩ Q, or the intersection of sets P and Q. if we draw the set p in a circle, it contains the elements of 3,6,9,12,15,18.

    We draw the common elements (6,12) at the right edge. We draw the Q elements in another circle. What are the remaining elements of P, not included at the intersection?

    The elements are 3,9,15,18. While the remaining elements from Q are (2,4,8,10). For another expression 6 is a part of the area enclosed by the intersection between P and Q. The enclosed area is showing 6 and 12.

    Then it is a true statement that 6 belongs to the intersection of P and Q. But 8  does not belong to that intersection. It is written in this symbol 8 b∉P ∩ Q or the intersection of P and Q. While for the union relation for the universal set U, drawn as a rectangle.

    Inside the rectangle, we can draw two circles for P and Q. All of the elements of both P and Q are thus included. The U will include U ={3,6,9,12,15,18,2,4,8,10}.

    Using venn diagram for intersection and union

    Solved example#10.

    A new example. Let P= {3,9, 27}, Q={2,3,10,18,27,28}. R={2,10,28}. It is required the intersection between P and Q, or P∩ R & Q∩ R and P∩ R. What are the common elements between P and Q? These elements are (3,27). While between Q∩ R, these elements are (2, 28,10).

    The total of three elements. While between P and R, there is none. Then we can express that intersection between P and R. P∩ R=Ø. or Ø. It is called a disjoint set, which means that these sets are not joined together.

    Solved problem-10.

    Cumulative, associative, and distributive properties of a set.

    A new subject is the commutative, associative, and distributive properties of the set. The commutative relation can be expressed as A∩ B= B∩A.

    Also A ∪ B=B ∪ A. The order can be reversed between A and B Can be reversed. but keep in mind that both the left-hand side and the right-hand side are intersections and for the second case both are unions. While for the Associative relation if we have three sets of A&B&C, it is required to make a union between these sets. This can be written as A ∪ (B ∪ C) = (A ∪ B)∪ C. A ∩ (B ∩ C)= (A ∩ B) ∩C.

    The distributive property is the mix. The mix between union and intersection. It is called distributive A ∪ (B ∩ C)=(A ∪ B) ∩ C. Another relation is A ∩( B ∪ C)=(A ∩ B )∪C. We have discussed the cumulative, followed by the associative and the distributive.

    What is cumulative and associative?

    Set identity law.

    There is a new item which is the set Identity Law. The first relation is A ∪A=A. The second relation is A ∩A=A. The third relation is A∪Ø =A. This is a true statement. While A ∩Ø=Ø. the subject of Power set if we have set A = A ={1,2,3,4}. What is the subset?

    There is plenty of combinations of subsets as follows:
    The first four numbers are selected, from 1 to 4., And then select {1,2}& {1,3} {1,4} and also select {1},{2},{3},{4}, then & select the same set order {1,2,3,4}, {1,2}& {1,3}&, (1,4}, other collection {1,2,3} {1,3,4}, How many selections so far, we have a total of 10 selections.

    Starting with numbers 2, {2,3},{2,4}. Starting with the number 3, {3,4}, then select Ø. The total number of selections =5+5+6=16. In the end, if we have a set.
    The total number of subsets that can be created = a number of elements raised to the power of 2 or 4×4=16.Make as power 2, that is why the expression of power set is used, since 4^2=16. If one set has three elements for instance, then the number of subsets=3^2=9.

    Set identity law.

    Ordered pairs.

    The new item is the ordered pair, for the Cartesian coordinates x and y, we draw the x-axis first and then draw the y-axis. If we have a point A with a coordinate(5,2). The ordered pair is the sequence of the axes arrangement we go to the right 5 spaces, then come up 2 spaces. Here the sequence is important.

    This is called ordered pair. Pair means two. If we select (5,2), it will give another point not point A. Set A, with elements {a,b}, and set B, with elements {c, d} will be considered equal if a=c and b=d.
     

    Ordered pair

    The ordered pair can be used to solve equations, for the given example. For a given (x-3,y-2), given=set {4,5}. It is required to find the x,y values. Since both sides are equal, then x-3=4 and y-2=5.

    solved problem for ordered pair

    Cartesian product of two sets.

    The Cartesian product of two sets. If we multiply two sets by each other it will be written as A cross B.

    A new pair is constructed with x,y coordinates, such that x is a part of A and y is a part of B. As an example, if set A is ordered pair{7,8}, while set B is another the coordinates are {2,4,6}. It is required to find AxB. Following the order of operations, we can write,(7,2),(7,4),(7,6)& (8,2),(8,4),(8,6). It is the result of AXB. In the end, we have 6 ordered pairs.

    Set A is composed of two elements. Set B is composed of three elements. Then AXB=2×3=6.

    The ordered pair is the sequence of the axes arrangement we go to the right 5 spaces, then come up 2 spaces. Here the sequence is important.

    This is called ordered pair. Pair means two. If we select (5,2), it will give another point not point A. Set A, with elements {a,b}, and set B, with elements {c, d} will be considered equal if a=c and b=d.

    cartesian products of two sets

    We can use a table to do the cross product of AxB, by setting the element of A in the first column, while writing the elements of b as the second, third, and fourth columns, in our example, there are three elements for set B.

    Starting with 2, the first subset will be 2 with the second column value of b, then the same 2, will be with the third column of b and finally 2 will be with the last column of b element. Move to the second row, in which we have 3, then combine with 2, again with 4, and at last 2 will be combined with 6. The final answer is shown in the next slide image..

    Solved problem for cartesian product

    Another example for Cartesian Product Of Two Sets.

    In another example, part of the ordered part is given and it is required to get A x B in terms of the given ordered pair, three subsets. If A and B are two sets. The AxB consists of 6 elements, which means that one of the sets is of 3 elements, while the other is from two elements. since 2×3=6.

    Given only three elements of the product of AxB. it is required to get the full elements of AxB and AxA. From the given elements we call these elements as (a1,b1) ,(a2,b2) and (a3,b3).

    The first three terms are a’s terms, so the set A can be written as A={2,3,4}. Then the b should be of two elements only, since AxB=6, these elements can be written as B={5,7}, b can not be three elements, 7 is repeated. AxB ={(2,5)(2,7), (3,5)(3,7), (4,3),(4,7) }, each ordered pair is different from the other pairs.

    For the AxA={(2,2)(2,3), 2,4), (3,2)(3,3),( 3,4),(4,2)(4,3),( 4,4) }.

    What are the N-tubles

    Again the use of a table to facilitate the estimation of the cross product of AxB.

    Set of all orsered n-tubles

    Use of table to facilitate the estimation of the cross product of AxA.

    using table for AxA.

    N tuples.

    N-tuples, when we have three coordinates x,y,z. We want to show a point in space, so we need three coordinates.

    For the point in space, with the coordinate(a1,a2a3), will follow the order in which we start with a1 in the x-axis direction, then the distance used is a2 in the direction of y, and finally a3 in the z-direction.3 tuples. This can be represented by a vector from the origin pointing to the point.

    R3 is a linear algebra subject. Rn is in the n space. R3 is the set of all ordered triples of real numbers. R1 is a vector in the x-direction that can be also defined as the set of all real numbers in the 1-space.
    R2 is a vector in the 2 space (x&y), written as a square of R.
    The definition of it is the set of all ordered pairs, the definition is a bit different from the first definition of R1, which is a set of all real numbers.

    R2 could be positive or negative numbers, R2 can be represented by plenty of choices, and the vector will be in the (x,y) direction. R3 is shown in the next slide. A new term which is R4 is presented.

    n-tuples.

    The next image shows a definition of vectors in Rn quoted from Prof. Ron Larson’s handbook.

    Source from linear algebra textbook

    Definition of Vector space.

    The definition of R4 is a set of all ordered quadruples of real numbers. Numbers, the point in R4 can be written as (a1, a2, a3, a4).

    For Rn or vector in space, the point can be represented by (a1,a2,a3,….., an), and is called n-tuples. For the quadruples R4, the number of operations that can be performed in 10 operations, as we are going to see from the book of elementary linear algebra.

    The first operation is closure under addition, the second operation is the commutative property u+v=v+u, and the third operation is u(v+w)=u+(v+w), U+Ø=U, additive identity. U+(-U)=0, additive inverse. Cu is in V, where c is a constant. C(u+v)=Cu+Cv, as distributive property. (C+d)U=cu+du. Distributive property, C(du)=Cdu.1*(u)=u. These are the operations.

    Properties of vector space.

    Solved problem in R4.

    Our example for R4.The details of the points are given, and we will make the operation U+v+w, which will give (0,4,6,2).

    Find the set of all ordered triples of real number.

    For an external link, please find this link. The story of mathematics.

  • 7-Universal set Subsets of sets.

    7-Universal set Subsets of sets.

    Introduction to Universal set, and subsets of sets.

    What is the subset?

    I quote, if every element of A is also an element of B. if whenever x ∈A, as x is a part of A, then x ∈ B, then x is an element of B.

    The graph is showing A as a  small circle inside the big circle, which is  B. If all the elements in x, located at A also exist in B.

    It means that A is a subset of B, this is logic. The used shapes are called Venn diagram, a diagram style that shows the logical relation between sets, the symbol used is A⊆B, A is a subset of B. Set A is included in set B.

    Suppose A is intersecting with B, then A is partially part of B, but not fully. Then A is not a subset of B, to explain that, another symbol is used which is A⊄B, set A is not a subset of set B.

    What is the subset?

    A Solved problem-5.

    Solved problem # 5, if set A = {1,2,3,4,5,6}, while Set B={2,4,5} and set C = {1,2,3,4,5}, It is required to show the Venn diagram.

    We can draw each set as circles, Set A includes numbers from A to 6, and the second circle is for B, which includes 2,4,5. While C is another circle that includes 1,2,3,4,5.

    The biggest circle A is drawn first the numbers are from 1 to 6. We can select b part that includes 2,4,5 and make other shapes. We can select the C part that includes 1,2,3,4,5 as a new shape.

    The only remaining part for A is number 6. A is the biggest shape, B is the smallest figure, and C is the middle shape. if we need to give an expression, we start with the smallest to the biggest.

    Set B is a subset of A, and C is also a subset of A. Set B is a subset of C. If we start from the biggest 5 shapes toward the smallest shape A, the biggest shape is not a subset of B, also A is not a subset of C. C which is bigger than B is not a subset of B.

    Solved problem-5

    A Solved problem-6.

    A Solved problem number #6, if set A={1,2,3,4}, if set B={1,4}, while set C={1}.  Describe which is the subset. Check the biggest, which is A  followed by B then C is the smallest. and also check the smallest to biggest.

    For the smallest to biggest relation, B is a subset of A, since B is smaller than A. B contains 1,4 that are included in A. C is also a subset of A since element 1 is included in set A. C is a subset of B since element 1 is included in set B. But A is not a subset of B, A is >B. A is not a subset of C, A is >C.

    Solved example 6

    Solved Problems-7-8.

    Let us Check Solved problem #7, let set M ={a,b}, how many subsets?. What are the different alternatives? We have phi x, Ø. M has 4 substets,A={a}, B={b}, {a,b} ,Ø.

    Solved problem #8, how many subsets for M. ={a,b,c}?

    The answer is the following subsets are A={a}. B={b}, C={c}, later start to use the mix {a,b,c}, null, so far we have selected 5 choices. Add the alternatives {a,c},{a,b},{b,c}. We have a total of 8 subsets, all are {a},{b},{c},{a},Ø = {},{a,b,c},{a,b},{a,c}.

    Solved example -7

    Universal set.

    The new item is capital U is drawn as a rectangle, for any particular problem and is a set that contains all the possible elements of the problem.

    What is the universal set U

    A Solved Solved problem-9.

    Let us have a Solved problem-9, if U={1,2,3,4,5,6,7,8,9}, while A={1,2,5,6}. While B={5,6}.

     Draw a Venn diagram to represent these sets. If we draw each set separately. This is the circle that represents A={1,2,5,6}

    B={3,9}. This is the U={1,2,3,4,5,6,7,8,9}.for the Venn diagram draw the big box to represent U and draw shapes for A and B, which includes each element. For A, take {1,2,5,6} and draw a shape, then select B={3,9} and draw a shape. how many total elements that we have? we have 9 elements, already we have used 4 elements for A and 2 for B, so it is expected that the remaining are 3 elements. checking the remaining elements we will find that these elements are 4,7,8.

    Solved example-8

    For an external link, math is fun for Venn diagram.
    The next post, Complement of a set, Cumulative, associative, Distributive of sets.

  • 6- Easy approach what is Roster notation for sets?

    6- Easy approach what is Roster notation for sets?

    Introduction to Roster notation.

    Discrete math symbols.

    For the subject of discrete math, if we have a look at the NCEES handbook for the FE Exam ver. 5.5, we will find Discrete math subjects On page 21, the symbols are included. For the subject, Small x is part of X or a member of X. The second symbol is for the empty cell or phi. S is a subset of T. The next symbol for S is a proper subset of T. The next symbol is for the empty set, followed by a subset Of T.

    Discrtete math symbols from NCEES

    Roster notation.

    We start with the Roster notation. A roster means a list of people or things that belong to a particular group, from the Meriam webster. There are symbols with different shapes.

    List elements of a set inside braces separated by a comma. As discussed for the natural numbers according to notation, according to Roster notation, starts with braces at the left and right, enclosed is 1, included in the counting numbers 1,2,3,4,5,…  

    For the whole number, the previous list is enclosed but we add the 0. For both the natural and whole numbers, we have a continuation or Infinite.

    For the expression of Integers, we have already discussed, that integers are zero, positive and negative values.

    As for the Rational numbers, we have one brace { at the left and at the right, }, these braces include a/b,  where a and b belong to Integers. The expression deals with positive or negative.

    That is why it is written in the definition, that fractions are integers and it is possible to have a zero in the numerator and expressed as a/b, a, and b are Integers, but the denominators should not be = 0, it is written that b  does not=0  to avoid the expression of infinity.

    ∈ is an element that belongs to ∉ is not an element or does not belong to it is the same previous symbol ∈ but with an inclined line.

    What is Roster notation?

    Some samples for Roster notation.

    For example, we have 3 which belongs or an element of the group between two braces  1,2,3,4,5.

    It is a true statement since 3 is an element within that group. but1/3 does not belong to the previous group 1,2,3,4,5.

    Then it is written as ⊄. The number 50 is a part of X, such that, expressed by the symbol |. x is an Integer

    Provided that x, expressed by the symbol |, which starts by 0 and extends since 50 is a positive non-rational number, so the statement is true. It is said that 50 is an element of x, provided that x is an integer. 

    For -5. Does -5 belong to the family of rational numbers? It is possible, since a rational number may contain -5/1. Any number can be considered rational, if the denominator is=1, so -5 belongs to all x values such that x is a rational number.

    Roster notation examples.

    Solved Examples 1&2- for Roster notation.

    Let us check example #1. Let G is the set of whole numbers <10. We have explained that a whole number includes 0, and all positive numbers. To write that expression in Notation, we start with the left brace {, we start by writing 0, then followed by 1,2,3,4,5, and ends with 9, followed by the right brace }.

    For example #2. Let X is the set of all odd numbers that are <12. The odd numbers are 1,3,5,7,9,11,13. We consider these odd numbers 0 is not an odd number, it is not included accordingly, we start with the left brace {, we start by writing 1, then followed by 1,3,5, 7, 9,11 followed by the right brace }.

    Solved problems-1&2

    Example 3 for Roster notation.

    In example # 3, which of the following set of whole numbers < 10? There are three options, the first one is the Expression of capital c.

    It required to check the set of the whole number, as discussed, the whole numbers that include,0,1,2,3,4 without 0, without fraction. It is required only for the odd numbers, this is a true selection if it is required for the integers which are<10, but this is ok without writing only odd numbers.

    The first choice is not correct. We move to the second selection which is, {0,2,4,6,8}, this list includes all the even numbers<  10, this option is not {1,3,5,7,9}.

    This collection has all odd numbers which are <10 and positive Whole numbers.

    The fourth choice is not a true selection. The third option is the correct one.

    Solved problem-3

    Equality of sets.

    The next item is if we have two sets and we want to check whether they are equal or not.

    The first set is A= {1,3,5,7}. The first set is B= {3,7,1,5}.

    The two sets have the same number of elements, the same numbers are there. Since the arrangement is not important, then A=B. The selection is correct, why?

    since all elements of A are the same as the elements in b. Let us check example # 4, with God’s will. Let R all Set of whole numbers <5.

    Let S= {4,0,2,3,1}, same numbers of elements, and the next question will be what are the all set of whole numbers<5. The whole numbers start with 0 and continue with 1,2,3,4 and can be represented by R={0,1,2,3,4}.

    While for S= {4,0,2,3,1}, the same numbers of elements and same figures thus both R and S are equal sets.

    Equity of sets

    Examine the set whether finite or infinite.

    The next item is the examination of whether the set is finite or infinite. Finite is limited, while infinite is not defined. While writing the set of all Integers.

    We write the set as= {…,-2,-1,0,1,2,3,4,….}, the set of integers is infinite since it is continuous from both ends. While for item b) the set of all natural numbers between (0,5), {,1,2,3,4} is an example of a finite set.

    Finite and infinite sets

    For an external link, please find this link-Venn Diagram.
    For the next post, Subsets of sets, Venn diagram.

  • 5- 3 Solved Quizzes for rounding numbers-easy approach.

    5- 3 Solved Quizzes for rounding numbers-easy approach.

    3 Solved Quizzes for rounding numbers.

    The first quiz of the three Solved quizzes.

    This is Quiz-1, which is a multi-choice quiz. For Quiz -1, round 3.8439 to the nearest hundred.

    As was explained earlier, we will take two numbers. The different options are given as for option a) is 3.84 or option b) 3.844 or option c) 3.843 or option d) 3.85   to the nearest one hundred, make a line that separates the first two numbers at the right of the decimal from the other right.

    Since number 3 is less than 5, We will keep 84 unchanged, then the rounding of 3.8439 will be rounded to the hundred as equal to 3.84, which is option A is the correct answer. This is selected since two decimal points or the nearest hundred are required.

    Solved quiz number-1-solved Quizzes for rounding numbers

    The second quiz of the three Solved quizzes.

    For Quiz -2, round 12.6257 to the three decimal places.

    The different four options are given for option a) 12.6. For option b) 12.63. For option c) 12.626. For option d) 12.62.

    I will leave for a short time to check the answer. The answer is to consider three numbers at the right of the decimal point.

    let us count 1,2,3, and check at the right, we have 7 which is >5. So the 5 will be upgraded to 6,  so the final value is 12.626. it does not have the option a) nor option b) or option d).

    The correct answer is option d) with God’s will. Our last Quiz is quiz number 3, to return to come back to set and subset the main item. That was an opportunity to review the whole numbers, the natural number, and rational and irrational numbers since the data are included in the set subject.

    Solved quiz number-2

    The third quiz of the three Solved quizzes.

    Quiz 3, round 3.995 to two significant digits, there are 4 options, for option a) 3.9, for option b) 4.0, option c) 3.99, for option d) 4. For two significant digits.

    Starting from left to right, in case no zero entry exists.3.995 is 4 digits. Start to count from left to right, stopping between 9 and 9, since 9 is >5, this will make 39 as 40, but consider that, we have 4.0, not 40.

    The final answer that 3.995 to two significant figures becomes 4.0, which is option a). The question is why it is not 4 as of option d)? Since in the quiz, it is required to have two Significant digits. If it is required to have one significant digit, this will yield as in option d).

    Solved quiz number-3

    For an external link, math is fun for the Venn diagram.
    The next post, What is Roster notation for sets?

    You can use a calculator from calculator soup for any number you wish to find the corresponding rounding.

  • 4- How to round numbers to Decimal points? perfect guide

    4- How to round numbers to Decimal points? perfect guide

    Round numbers to Decimal points, Significant Digits.

    How do we round numbers?

    This is the fourth lecture about sets. This is the revision for the rounding of numbers. It is a part of discrete math. Make rounding, of the number of 28617 persons in a stadium. it is required to make rounding to the nearest 100.


    The easiest way is to leave two numbers on the right side and make a line. If we let only one number it will be to the nearest tenth only. We have a number 1, which is <5, so 6 stays as it is, with no change Then the number 28617, will be rounded to 28600 to the nearest one hundred.

    solved example , how to round a number?

    Rounding numbers to decimal points.

    Let us check a new subject, which is rounding decimals. What is the decimal point? this point differentiates between the whole number at the left and the decimals at the right of it.

    The number can be approximated to a decimal point. For counting numbers like 5  can be expressed as 5.0000.

    To round to the nearest decimal point. The decimal point. One decimal point means considering only one decimal point. 8 is the nearest 1/10, we are saying 8/10, while 6 is the nearest 1/100, we are saying 6/100.

    While for 4, we are saying 4/1000, 7, and 0.80, while the remaining is  86/100, 7.864, this 4/1000.

    Rounding to 1 decimal point, means the nearest 1/10 assume you make a line between 8 and 6, consider 6 is bigger than 5, and 8 can be upgraded to 9. The final rounding of 7.864 is 7.90 to the nearest 1/10. The dot is the decimal point and at the right side of the point is the fraction. The whole number is the number at the left of the decimal point.

    Rounding decimal.

    Solved examples for Rounding numbers to decimal points.

    Another example, is Round 5.574 to the nearest 2 decimal points, and consider the decimal point. For the first 1 decimal means consider only one number which is 5.

    For the two decimal points, consider the two numbers and check the third number which is 4. The rule is if the third number is < 5, then disregard it, if not increase the left number and upgrade. Sometimes, the expression of 2 (d.p) is used as an abbreviation for two decimal points.

    Rounding decimal to one decimal point.

    Another example:it is required to round 3.146. It is required to be rounded to hundreds, it will be rounded to 3.14 since it is required to have two decimal places. Why? because 1 is 1/10, while 4 is 4/100

    1 is < 5, then 4 will not be changed. The same 3.1416 is to be rounded to thousands or (3)(d.p), then 6 is >5, then 1 will be upgraded to 2, and the final value is 3.142. The next item is the Significant figures, which are useful in experiments for persons dealing with accurate measurements. That is why it is called significant figures. This can count from left to right.

    Rounding to two decimal points

    Rounding numbers to significant figures.

    As an example 1.239. It is required to round to three significant digits. For counting the three digits start from the left and consider that 1 is the first significant figure.

    Look at the right side of the third number and check that if the number is < 5, then the number will be unchanged. In our case, the 1.239 will be rounded to 1.24 since 9 is<5, which is three Significant digits.

    Let us count as 123, while the decimal point is not counted.

    For the same example round 134.9 to one scientific digit. Make a line separating the 1 and 3, since 3 is  <5, then 1 will remain, and the rest will be zeros. We will not 100.0.In the case that the significant figure for a number starts with a zero. In that case, do not count the zero, but start with the first non-zero number and consider it as the first significant figure.

    What are the significant digits?

    solved examples for Rounding significant figures.

    For example, a given number is 0.0043, for which it is required to round to one significant. To solve we will forget the zeros and start with the first figure which is 4, check whether 5, if less than 5 then, keep 4 without an upgrade, the final rounded is 0.004.

    No matter how are the numbers of zeros, start with the first non-zero element.

    Another example round 0.0165, is required to round to two significant figures. Forget the zeros, the first figure is 1 and the second number is 6, for the third number is 5, which is =5. The 6 will be upgraded to 7. The two significant figures are 0.017.

    Significant figures

    Rounding Numbers to the nearest whole number.

    To round to the nearest whole number, as we agree before the decimal point separates the whole number at the left from the fraction or decimals at the right.

    To round to the nearest whole number, check the number at the right side of the decimal point if<5 then disregard, but if=5>5 then upgrade. For our case, the 6 will be upgraded to 7. The nearest whole number will be 107. No fractions or decimals are to be included.

    Solved problem for the nearest whole number.

    There is another example, round 239.4556 to the nearest hundred, this is a mixed number. For the nearest 100, the answer is to neglect the fraction. For the Nearest hundredth.

    Make a line between 2 and 3, check 3 is<5, 2 will be left, and place 0 and 0 to the right, the round will be 200.  There will be no fractions. The idea is to consider the whole number at the left of the decimal. Another example: Round 78.546 to the two decimal points, the result is shown in the slide image.

    Round to nearest hundreds.

    For an external link, math is fun for the Venn diagram.
    The next post, Solved quizzes.

    You can use a calculator from calculator soup for any number you wish to find the corresponding rounding.

  • What is rounding of a number?
  • Definition of absolute value, rational numbers, fraction.

    What is the absolute value of a number?

    This is a review of the information about the absolute value, which is usually written for any number enclosed by two small vertical lines, which is the absolute value of ABS of -4=4. If we draw a number line, with equal distances, writing from 0 to 5 till infinity, and from the left side, we write (-1) to (-5) to infinity Absolute means How far from zero.
    For any number regardless of being positive or negative. When someone needs to know about the distance for any given number from zero. The answer will be  4 or 5 or 6 depending on that number. The absolute value of a number is the distance for that number from zero. We will not say(+4) or(+5), you will write a number without a sign, then the distance for the ABS of (-4), there are 4 spaces from the zero, which is the same distance for the positive value of 4.

    What is the Absolute value?

    The fraction of a number.

    Then the rational number is(1/5). But if the denominator is zero, we get infinity, when dividing any number by zero.

    What is a fraction of a number?

    The fraction of numbers like 1/2,3/4, and 7/10 are the family of fractions, the upper part is called the numerator and the lower part is the denominator. The fraction can be divided into parts, the first part is the proper fraction. If the denominator value is > the numerator value, this will give a value <1.

    Definition of fraction.

    If we have the reverse when the numerator is > denominator, for instance, this will give a value >1. Like (4/3) and 6/3=2 and 8/5 will give 1.60.  The second type is called an improper fraction. While the first type is called proper fraction. The next slide images explain the definition of fractions and the various types of fractions, quoted from Basic College Mathematics by Prof. Aufmann.

    Examples of fraction numbers

    What is factoring?


    The product and factoring When we have x^2 +4x +2, and it is required to make factoring, we can say (x^2+4x+2) its factor = (x+2)*(x+2), which means that it is returned into its major elements when multiplied we get the original element. For instance number 12, its factors are (112),(34),(2*6). These elements when multiplied will get 12. For elements, a and b when multiplied we will get c.

    The difference beween factoring and product.

    Then c is the product due by multiplying two factors a and b. Product is due to the multiplication of can be back to elements by factoring. What is an Irrational number? We have said that every number is a ratio.

    Integer 5 can be estimated as the ratio of 5/1. Also, 7 can be written as 7/1, which is an Improper fraction.

    Irrational numbers.

    What are the irrational numbers? these numbers are real numbers that cannot be written as simple fractions. We use the calculator to get the square root of 3, we get 1.732050808.

    irrational numbers.


    While the square root of 2 is 1.41412135. Pi is an example of an irrational number, its value=3.14159, and continues to infinity.


    Non-terminating -Non -recurring decimal. Non- Terminating item, to give an example of terminating number 1/2=0.50, there is a continuation of numbers.
    While for the square root of 3 or 2, the value is never terminating.

    Also in the Pi value, it is written as 3.14159 followed by dots. Recurring means repeated, for 1/3 it is=0.33333. While for the square root of 2, every decimal is not the same value or repeated numbers.

    So the definition of irrational numbers includes these two conditions and becomes not rational. These are the expansion for irrational numbers.

    Examples of irrational numbers.

    A general shape for all types of numbers.

    Here is a shape, he started with the Natural numbers, then a new square of Whole numbers after adding the zero.

    diagram for the relation of a real number and other types of a number

    Then add the negative and call that box Integers. As if moving up a higher degree. Then followed by the rational numbers afterward the irrational numbers, including for instance the negative value(-) of the square root of 8, sqrt(15), and PI. The big box is called the Real number and a symbol was given as capital R.

    The expanded form of a whole number.

    The next slide image shows the expanded form of a whole number and how we can express any number by a combination of ones, tens, hundreds, thousands, and more based on the value of that number.

    How to express a whole number?

    What is rounding of a number?

    The rounding subject is one of the important topics. Rounding numbers for instance, if the number is greater than 5, the rounding will increase the number to a higher number. On the contrary, if the number is less than 5, the number will be reduced. So numbers from 1,2,3,4 can be rounded down.

    While for number 5,6,7,8,9, the rounding will be rounding up, rounding will be to 10. For example for 27, it is required to round to the nearest 10.

    What is rounding?

    We put a line since the number 7, which is on the right side of the line is more than 5, then round up to 30. While for number=33, put a line at the 3. The 3 at the right of the line is<5 then the rounding of 33 is 30, and it is a round-down.

    Examples of rounding of a number?

    For Making round to the nearest one hundred and nearest one thousand and beyond. There are examples in the next slide, of God’s will.

    Round 4827 to the nearest ten. Since 7>5 then the number at the left of 7 will be upgraded to 3. and 7 will be 0. 7>5. The final number will be 4830. For the same number make a rounding to the nearest hundred. For 4827, put a line at the left of 2, since 2 is <5 then 4827 will be rounded down, then 4827 will be rounded to 4800.

    Examples of rounding of a number?

    While for 4827 to be rounded to the nearest thousands, take a line between 4 and 8, and check the right side of the line, we have 8 >5, then the number 4827 will be rounded up to 5000 that was for the nearest Thousands We cannot say 4900, since we upgrade 4 to 5.

    Examples of rounding of a number?

    For an external link, math is fun for the absolute value details.
    For the next post, how to round decimal numbers?