Slide 1

Bahasa InggrisNUMBERPenyusunZulkifli Paldana AkbarTeknik Konstruksi Sipil1s2 sorePOLITEKNIK NEGERI JAKARTA2010

definition

Number is a mathematical object used incounting and measuring. A notational symbol whichrepresents a number is called a numeral, but in common usage the word is used for both the abstract and the symbol. As well as for the word. In addition to their use in counting and measuring. Numeral are often used for labels (telephone number), for ordering (serial number), and for codes (e.g ISBN). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.The Types of NumberNatural numbersb. Whole numbersc. Integersd. Rational numberse. Real numberf. Complex numbersa. Natural numbers is an ordinary counting numbers or is a number that occurs in commonly and obviously in nature. Natural numbers consist of positive integers or counting numbers (1,2,3, ) or, in some cases, non-negative numbers (0,1,2,3, ). Natural numbers have two main purposes : counting (there are 2 dogs), and ordering (that is the 4th largest planet).

Example : 1,2,3,4,5,

b. Whole numbers is an integer equal to or greater than zero. The whole numbers exactly does not have a consistent definition.Example : 0,1,2,3,4, The term whole number does not have a consistent definition. Various authors use it in one of the following senses:the nonnegative integers (0, 1, 2, 3, ...)the positive integers (1, 2, 3, ...)all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)

c. Integers is whole number (not a fraction) that can be positive, negative, or zero. Therefore, the numbers 10,0,-25, and 5,148 are all integers. unlike floating point numbers, integers can not have decimal place. When two integers are added, substracted, or multiplied, the result is also an integer. However, when one integer is divided into another, the result may be an integer or a fraction. For example : 6 divided by 3 equals 2, which is an integer, but 6 divided by 4 equals 1,5, which contains a fraction. Decimal number may either be rounded or truncated to produce an integer result.d. Rational numbers is a number that can be written as a simple fraction or A number capable of being expressed as an integer or a quotient of integers, excluding zero as a denominator.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.Arational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as a/b, where b is not zero.

e. irrational numbers is any real number that is not a rational number. Specifically, it is a number which cannot be expressed as a fraction m/n, where m and n non-zero. An example of an irrational number is 3.141592653589793... If written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without repetition. Pi and the square root of 2 (2) are irrational numbers.e. Real number is a number that can be written as a terminating or non-terminating decimal; a rational or irrational number. The numbers 2, -12.5, 3/7 , and pi () are all real numbers.is a value that represents a quantity along acontinum, such as 5 (an integer), 3/4 (a rational number that is not an integer), 8.6 (a rational number expressed in decimal representation), and pi (3.1415926535..., an irrational number). Real numbers are commonly opposed both to integers, such as 5 (whole numbers that express discrete rather than continuous quantities) and complex numbers (mathematical constructs that include real numbers as a special case). Real numbers can be divided into rational numbers, such as 42 and 23/129, and irrational numbers, such as pi and the square root of two. A real number can be given by an infinite decimal representation, such as 2.4871773339 ..., where the digits continue indefinitely. The real numbers are sometimes thought of as points on an infinitely long number line.The type of number we normally use, such as 1, 15.82, -0.1, 3/4, etcPositive or negative, large or small, whole numbers or decimal numbers are all Real Numbers. They are called "Real Numbers" because they are not Imaginary Numbers.

f. A complex number is a number consisting of a real and imaginary part. It can be written in the form a+bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = 1.[1] The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.[2] The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra.Complex numbers are plotted on the complex plane, on whichNumber SystemNatural(0), 1, 2, 3, 4, 5, 6, 7, ..., nIntegersn, ..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ..., nPositive integer1, 2, 3, 4, 5, ..., nrationalab where a and b are integers and b is not zeroRealThe limit of a convergent sequence of rational numberscomplexa + bi where a and b are real numbers and i is the square root of 1Here are some typical number-type questions (assuming that you haven't yet learned about imaginaries and complexes): Copyright Elizabeth Stapel 1999-2009 All Rights ReservedTrue or False: An integer is a rational number.Since any integer can be formatted as a fraction by putting it over 1, then this is true.True or False: A rational is an integer.Not necessarily; 4/1 is an integer, but 2/3 is not! So this is false.True or False: A number is either a rational or an irrational, but not both.True! In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types!

Classify according to number type; some numbers may be of more than one type.0.45This is a terminating decimal, so it can be written as a fraction: 45/100 = 9/20. Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real3.14159265358979323846264338327950288419716939937510...You probably recognize this as being pi, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so pi is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real3.14159Don't let this fool you! Yes, you often use something like this as an approximation of pi, but it isn't pi! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike pi which is irrational! The answer is: rational, real10Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real5/3This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real1 2/3This can also be written as 5/3, which is the same as the previous problem. The answer is: rational, realsqrt(81)Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: sqrt(81) = 9, which is just an integer. The answer is: integer, rational, real 9/3This is a fraction, but notice that it reduces to 3, so this may also count as an integer. The answer is: integer (possibly), rational, real

Except for the section where you have to classify numbers according to type, you really won't need to be terribly familiar with this hierarchy.It's more important to know what the terms mean when you hear them. For instance, if your teacher talks about "integers", you should know that the term refers to the counting numbers, their negatives, and zero.The exercisea. Natural numbers :5 + 5 = Fill in the missing number :___ten and 7ones = 17Between 9, _____, 11Ans : 10 (for all question)

b.Whole numbers :Write the number below in wordsEg : 4 159Ans : Four thousand one-hundred and fifty nine

Write each in expanded form Eg : 517 249 ans : 500 000 + 10 000 + 7 000 + 200 + 40 + 9

Give the place value of each digit in the following numbers.Eg : 2 719ans : 2(thousands), 7(hundreds), 1(tens), 9(ones)

c. IntegersIs 8 positive or negative?ans : yes, positiveWhat is the magnitude of 87?a.100b.97c.87ans :c. 87What is 5 - 8?a. 13b.3c.-3d.-13ans :c. -3

d. Rational numbersIs 3.2 a rational number?Is -5.321671 a rational number?ans : yes (for all question)

e. Real numbers|8| The number enclosed within the absolute value bars is a nonnegative number so the first part of the definition applies. This part says that the absolute value of 8 is 8 itself|-3| The number enclosed within absolute value bars is a negative number so the second part of the definition applies. This part says that the absolute value of3is the opposite of3, which is(3). By the double-negative property,(3)=3.|3|=3

f. Complex numbersSimplify 9 9 = 2 2 2 2 = (-1)(-1)(-1)(-1) = Find -49-49 = -1 49 = 7

endingnumber was classified according to the utility's own numbers. each number has an explanation in accordance with the part. turned out to not only learn the number adding and subtracting or dividing and multiplying numbers only. many explanations, and from which all numbers can be formed into integer, prime, fractional, integral, decimal and more. Thus the classification number that can be explained, may be understood and we all know about the various numbers and usefulness.

THANK YOU