What kind of number is 125

And one-half is the simplest fraction. Much more is involved in understanding and translating among representations of or rational numbers more generally. See Box 3—9 for an example. Perhaps the deepest translation problem in pre-K to grade 8 mathematics concerns the translation between fractional and decimal representations of rational numbers.

Successful translation requires an understanding of rational numbers as well as decimal and fractional notation—each of which is a significant and multifaceted idea in its own right. In school, children learn a standard way of converting a fraction such as to a decimal by long division. The first written step of the long division is dividing 30 tenths by 8.

After three divisions, the process stops because the remainder is zero. The quotient obtained, 0. The remainder at the seventh step is 2, which is where the first step began. Thus, a repeating decimal, where the horizontal bar is used to indicate which digits repeat. The process of using long division to obtain the decimal representation of a fraction will always be like one of the above cases: Either the process will stop or it will cycle through some sequence of remainders.

So the decimal representation of a rational number must be either a repeating or a terminating decimal. Thus a nonrepeating decimal cannot be a rational number and there are many such numbers, such as p and. In the process of converting a fraction to a decimal, all remainders must be less than the denominator of the fraction.

Because the list of possible remainders is finite, and because each subsequent step is always the same brings down a 0, etc. Understanding a mathematical idea thoroughly requires that several possible representations be available to allow a choice of those most useful for solving a particular problem.

And if children are to be able to use a multiplicity of representations, it is important that they be able to translate among them, such as between fractional and decimal notations or between symbolic representations and the number line or pictorial representations.

Addition is an idea—an abstraction from combining collections of objects or from joining lengths. Carrying out the addition of two numbers requires a strategy that will lead to the result.

For single-digit numbers it is reasonable to use or imagine blocks or cookies, but for multidigit numbers you need something more efficient.

You need algorithms. There are many such algorithms, as well as others that do not use pencil and paper. Years ago many people knew algorithms for computation on fingers, slide rules, and abacuses. Today, calculators and computer algorithms are widely used for arithmetic.

Indeed, a defining characteristic of a computational algorithm is that it be suitable for implementation on a computer. And in fact, most of algebra, calculus, and even more advanced mathematics may now be done with computer programs that perform calculations with symbols. When confronted with a need for calculation, one must choose an algorithm that will give the correct result and that can be accomplished with the tools available. Algorithms depend upon representations.

Note, for example, that algorithms for fractions are different from algorithms for decimals. And as was the case for representations, choosing an algorithm benefits from consideration of certain characteristics: transparency, efficiency, generality, and precision. The more transparent an algorithm, the easier it is to understand, and a child who understands an algorithm can reconstruct it after months or even years of not using it.

The need for efficiency depends, of course, on how often an algorithm is used. An additional desired characteristic is simplicity because simple algorithms are easier to remember and easier to perform accurately. Again, the key is finding an appropriate balance among these characteristics because, for example, algorithms that are sufficiently general and efficient are often not very transparent. It is worth noting that pushing buttons on a calculator is the epitome of a nontransparent algorithm, but it can be quite efficient.

In Box 3—10 , we show some examples of algorithms with various qualities. Algorithms are important in school mathematics because they can help students understand better the fundamental operations of arithmetic and important concepts such as place value and also because they pave the way for learning more advanced topics.

For example, algorithms for the operations on multidigit whole numbers can be generalized with appropriate modifications to algorithms for corresponding operations on polynomials in algebra, although the resulting algorithms do not look quite like any typical multiplication algorithms but rather are based upon the idea behind such algorithms: computing and recording partial products and then adding.

The polynomial multiplication illustrated below, for example, is somewhat like multiplication. The expanded method below shows the relationship a bit more clearly. The decimal place-value system allows many different algorithms for the four main operations.

The following six algorithms for multiplication of two-digit numbers were produced by a class of prospective elementary school teachers. They were asked to show how they were taught to multiply 23 by Note that all of these algorithms produce the correct answer. All except Method 4 are simply methods for organizing the four component multiplications and.

A more compelling justification uses the area model of multiplication. Note the correspondence between the areas of the four smaller rectangles and the partial products in Method 3. With more careful examination, it is possible to see the same four partial products residing in the four cells in Method 6. The 2 in the upper left cell, for example, actually represents Methods 1,. Any of the methods—and, in fact, any of the four justifications that followed— could serve as the standard algorithm for the multiplication of whole numbers because they are all general and exact.

Mathematically, these methods are essentially the same, differing only in the intermediate products that are calculated and how they are recorded. These methods, however, are quite different in transparency and efficiency. Methods 3 and 5 and the area model justification are the most transparent because the partial products are all displayed clearly and unambiguously.

The three justifications using the distributive law also show these partial products unambiguously, but some of the transparency is lost in the maze of symbols. Methods 1 and 2 are the most efficient, but they lack some transparency because the 23 and the 30 actually represent and , respectively. Method 4 takes advantage of the fact that doubling the factor 15 gives a factor that is easy to use. It is quite different from the others. For one thing, the intermediate result is larger than the final answer.

This method can also be shown to be correct using the properties of whole numbers, since multiplying one factor by 2 and then dividing the product by 2 has no net effect on the final answer. The usefulness of Method 4 depends on the numbers involved.

Clearly this method, although completely general, is not very practical. For most factors, it is neither simple nor efficient. The preceding sections have described concepts in the domain of number that serve as fundamental building blocks for the entire mathematics curriculum. Other fundamental ideas—such as those about shape, spatial relationships, and chance—are foundational as well. Students do not need to, and should not, master all the number concepts we have described before they study other topics.

Number is intimately connected with geometry, as illustrated in this chapter by our use of the number line and the area model of multiplication. Those same models of number can, of course, arise when measurement is introduced in geometry.

The connection between number and algebra is illustrated in the chapter by our use of algebra to express properties of number systems and other general relationships between numbers. Number is also essential in data analysis, the process of making sense of collections of numbers.

Using numbers to investigate processes of variation, such as accumulation and rates of change, can provide students with the numerical underpinnings of calculus. Some of the manifold connections and dependencies between number and other mathematical domains may be illustrated by the so-called handshake problem:. If eight people are at a party and each person shakes hands exactly once with every other person, how many handshakes are there? This problem appears often in the literature on problem solving in school mathematics, probably because it can be solved in so many ways.

This method of solution can be generalized to a situation with any number of people, which is what a mathematician would want to do. Because mathematicians are interested not only in generalizations of problems but also in simplifying solutions, it would be nice to find a simple way of adding the numbers.

Numbers that arise in this way are called triangular numbers because they may be arranged in triangular formations, as shown below. Therefore, 3, 6, 10, 15, 21, and 28 are all triangular numbers. This is a geometric interpretation, but can geometry be used to find a solution to the handshake problem that would simplify the computation?

One way to approach geometrically the problem of adding the numbers from 1 to m is to think about it as a problem of finding the area of the side of a staircase. The diagram on the right below includes a second copy of the staircase, turned upside down. So the area of the staircase is half that, or A closely related numerical approach to the problem of counting handshakes comes from a story told of young Carl Friedrich Gauss — , whose teacher is said to have asked the class to sum the numbers from 1 to , expecting that the task would keep the class busy for some time.

The story goes that almost before the teacher could turn around, Gauss handed in his slate with the correct answer. He had quickly noticed that if the numbers to be added are written out and then written again below but in the opposite. As can be seen below, each vertical sum is , and there are exactly of them. The handshake problem can be approached by bringing in ideas from other parts of mathematics.

If the people are thought of as standing at the vertices of an eight-sided figure octagon , then the question again becomes geometric but in a different way: How many segments sides and diagonals may be drawn between vertices of an octagon?

The answer again is 28, as can be verified in the picture below. But that multiplication counts each segment twice once for each endpoint , so there are really half as many, or 28, segments. In still another mathematical domain, combinatorics—the study of counting, grouping, and arranging a finite number of elements in a collection—the.

For example, in how many ways can a committee of two be chosen from a group of eight people? This is the same as the handshake problem because each committee of two corresponds to a handshake. It is also the same as the octagon problem because each committee corresponds to a segment which is identified by its two endpoints.

A critically important mathematical idea in the above discussion lies in noticing that these are all the same problem in different clothing. It also involves solving the problem and finding a representation that captures its key features. For students to develop the mathematical skill and ability they need to understand that seemingly different problems are just variations on the same theme, to solve the problem once and for all, and to develop and use representations that will allow them to move easily from one variation to another, the study of number provides an indispensable launching pad.

In this chapter, we have surveyed the domain of number with an eye toward the proficiency that students in grades pre-K to 8 need for their future study of mathematics. Several key ideas have been emphasized. First, numbers and operations are abstractions—ideas based on experience but independent of any particular experience. The numbers and operations of school mathematics are organized as number systems, and each system provides ways to consider numbers and operations simultaneously, allowing learners to focus on the regularities and the structure of the system.

Despite different notations and their separate treatment in school, these number systems are related through a process of embedding one system in the next one studied. All the number systems of pre-K to grade 8 mathematics lie inside a single system represented by the number line.

Second, all mathematical ideas require representations, and their usefulness is enhanced through multiple representations. Because each representation has its advantages and disadvantages, one must be able to choose and translate among representations. The number line and the decimal place-value system are important representational tools in school mathematics, but students should have experience with other useful interpretations and representations, which also are important parts of the content.

Third, calculation requires algorithms, and once again there are choices to make because each algorithm has advantages and disadvantages. And finally, the domain of number both supports and is supported by other. It is these connections that give mathematics much of its power. If students are to become proficient in mathematics by eighth grade, they need to be proficient with the numbers and operations discussed in this chapter, as well as with beginning algebra, measure, space, data, and chance—all of which are intricately related to number.

Some authors see, e. We are adopting the common usage of the U. The recognition that zero should be considered a legitimate number—rather than the absence of number—was an important intellectual achievement in the history of mathematics.

Zero as an idea is present in the earliest schooling, but zero as a number is a significant obstacle for some students and teachers. Abstraction is what makes mathematics work. Although negative numbers are quite familiar today, and part of the standard elementary curriculum, they are quite a recent development in historical terms, having become common only since the Renaissance.

Descartes, who invented analytic geometry and after whom the standard Cartesian coordinate system on the plane is named, rejected negative numbers as impossible. His coordinate axes had only a positive direction. His reason was that he thought of numbers as quantities and held that there could be no quantity less than nothing.

Now, however, people are not limited to thinking of numbers solely in terms of quantity. In dealing with negative numbers, they have learned that if they think of numbers as representing movement along a line, then positive numbers can correspond to movement to the right, and negative numbers can represent movement to the left.

This interpretation of numbers as oriented length is subtly different from the old interpretation in terms of quantity, which would here be un oriented length, and gives a sensible and quite concrete way to think about these numbers that Descartes thought impossible. Although rational numbers seem to present more difficulties for students than negative integers, historically they came well before.

The Greeks were comfortable. See also Behr, Harel, Post, and Lesh, In other notational systems, such as decimal representation, the rules will look somewhat different, although they will be equivalent. These numbers and many others are not rational because they cannot be expressed as fractions with integers in the numerator and denominator.

In the number-line illustrations throughout this chapter, the portion displayed and the scale vary to suit the intent of the illustration. Bruner, pp. Pimm, , suggests that people seek representational systems in which they can operate on the symbols as though the symbols were the mathematical objects.

Kaput, , argues that much of elementary school mathematics is not about numbers but about a particular representational system for numbers. See Cuoco, , for detailed discussions of various ways representations come into play in school mathematics.

What a tremendous labor-saving device! I went excitedly to my father to explain my discovery. Grouping is a common approach in measurement activities. For example, in measuring time, there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, approximately 30 days in a month, 12 months in a year, and so on.

For distance, the customary U. For example, IX means nine that is, one less than ten , whereas XI means eleven one more than ten. This generality was a significant accomplishment. In the third century B. The issue was serious enough that Archimedes, the greatest mathematician. Archimedes, however, did not go so far as to invent the decimal system, with its potential for extending indefinitely.

Behr, M. Rational number, ratio, and proportion. Grouws Ed. New York: Macmillan. Bruner, J. Toward a theory of instruction. Cambridge, MA: Belknap Press. Cuoco, A. The roles of representation in school mathematics Yearbook of the National Council of Teachers of Mathematics.

Duvall, R. Representation, vision, and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. Santos Eds. ED Freudenthal, H. Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel. Greeno, J. Practicing representation: Learning with and about representational forms.

Phi Delta Kappan , 78 , 1— Representation systems and mathematics. Janvier Ed. Hillsdale, NJ: Erlbaum. Knuth, D. Computer science and its relation to mathematics. American Mathematical Monthly , 81 , — Lakoff, G. The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics.

English Ed. Mahwah, NJ: Erlbaum. Morrow, L. The teaching and learning of algorithms in school mathematics Yearbook of the National Council of Teachers of Mathematics. Pimm, D. Symbols and meanings in school mathematics. London: Routledge.

Russell, B. Introduction to mathematical philosophy. Sfard, A. Commentary: On metaphorical roots of conceptual growth. Steen, L. Steen Ed. The perimeter of S is what fraction of the perimeter of R? The perimeter of R is and the perimeter of S is Therefore, the perimeter of S is of the perimeter of R.

To enter the answer you should enter the numerator 20 in the top box and the denominator 80 in the bottom box. Because the fraction does not need to be reduced to lowest terms, any fraction that is equivalent to is also considered correct, as long as it fits in the boxes. For example, both of the fractions and are considered correct. Thus the correct answer is or any equivalent fraction. For the large cars sold at an auction that is summarized in the table above, what was the average sale price per car?

What is the profit expressed as a percent of the merchant's cost? If you use the calculator and the Transfer Display button, the number that will be transferred to the answer box is You will need to adjust the number in the answer box by deleting all of the digits to the right of the decimal point. There are 8 octal characters, Obviously this can be represented by exactly 3 bits. Two octal digits can represent numbers up to 64, and three octal digits up to A byte requires 2. It is much less common today but is still used occasionally e.

Exercises: Convert from decimal to hexadecimal Answer 7D0 Convert 3C from hexadecimal to decimal Answer 60 Convert from binary to hexadecimal Answer A7B Convert 7D0 from hexadecimal to binary Answer If you shift a hexadecimal number to the left by one digit, how many times larger is the resulting number? Answer It was noted previously that we will not be using a minus sign - to represent negative numbers. We would like to represent our binary numbers with only two symbols, 0 and 1.

There are a few ways to represent negative binary numbers. The simplest of these methods is called ones complement, where the sign of a binary number is changed by simply toggling each bit 0's become 1's and vice-versa.

This has some difficulties, among them the fact that zero can be represented in two different ways for an eight bit number these would be and To represent an n bit signed binary number the leftmost bit, has a special significance.

The difference between a signed and an unsigned number is given in the table below for an 8 bit number. If Bit 7 is not set as in the first example the representation of signed and unsigned numbers is the same.

However, when Bit 7 is set, the number is always negative. For this reason Bit 7 is sometimes called the sign bit.

Signed numbers are added in the same way as unsigned numbers, the only difference is in the way they are interpreted. This is important for designers of arithmetic circuitry because it means that numbers can be added by the same circuitry regardless of whether or not they are signed.

To form a two's complement number that is negative you simply take the corresponding positive number, invert all the bits, and add 1. The example below illustrated this by forming the number negative 35 as a two's complement integer:. So is our two's complement representation of We can check this by adding up the contributions from the individual bits. The same procedure invert and add 1 is used to convert the negative number to its positive equivalent.

If we want to know what what number is represented by , we apply the procedure again. Since represents the number 3, we know that represents the number Exercises: Convert from binary to decimal Answer Convert from binary to decimal Answer 34 Convert from decimal to binary Answer In 'C', a signed integer is usually 16 bits.

What is the largest positive number that can be represented? Consider the following examples. Let's carefully consider the last case which uses the number As a 4 bit number this is represented as. This process is refered to as sign-extension, and can be applied whenever a number is to be represented by a larger number of bits.

In the C50 Digital Signal Processor, this typically occurs when moving a number from a 16 bit register to a 32 bit register. Whether or not sign-extension is applied during such a move is determined by the sign-extension mode bit. Note that to store a 32 bit number in 16 bits you can simply truncate the upper 16 bits as long as they are all the same as the left-most bit in the resulting 16 bit number - i. Most processors even have two separate instructions for shifting numbers to the right which, you will recall, is equivalent to dividing the number in half.

Firstly, work out the difference in the terms. This sequence is going up by four each time, so add 4 on to the last term to find the next term in the sequence. To work out the term to term rule, give the starting number of the sequence and then describe the pattern of the numbers. The first number is 3. The term to term rule is 'add 4'. Once the first term and term to term rule are known, all the terms in the sequence can be found. The first term is