PROJECT REPORT

ON

GOVT. GIRLS SEC. SCHOOL

JANDIALA GURU, DISTT. AMRITSAR

NUMBER SYSTEM

Submitted by: -Mrs. KamayniMath Mistress

NUMBERS

Number (mathematics), word or symbol used to designate quantities or entities that behave like quantities.

NATURAL NUMBERS

The simplest numbers are the natural numbers, 1, 2, 3, .... The natural numbers are also called the whole numbers, positive integers, or positive rational integers. The natural numbers are closed with respect to addition and multiplication—that is, the sum and product of two natural numbers are always natural numbers.

RATIONAL NUMBERS

The sum and product of two natural numbers are always natural numbers. Because the quotient (the result of dividing) of two natural numbers, however, is not always a natural number, it is convenient to introduce the positive fractions to represent the quotient of any two natural numbers. The natural number n is identified with the fraction n/1. Furthermore, because the difference of two positive fractions is not always a positive fraction, it is expeditious to introduce the negative fractions (including the negative integers) and the number zero (0). The positive and negative integers and fractions, and the number 0, comprise the rational number system.

IRRATIONAL NUMBERS

The development of geometry indicated the need for more numbers; the length of the diagonal of a square with sides one unit long cannot be expressed as a rational number. Similarly, the ratio of the circumference to the diameter of a circle is not a rational number. These and other needs led to the introduction of the irrational numbers. A decimal expansion that is neither of the two types described above represents an irrational number. For example, Ã = 1.4142135623 ... and p = 3.1415926535 ... are irrational

COMPLEX NUMBERS

The product of a real number multiplied by itself is 0 or positive, so the equation x2 = -1 has no solutions in the real number system. If such a solution is desired, new numbers must be invented. Let i = Á be a new number representing a solution of the preceding equation. All numbers of the form a + bi, in which a and b are real numbers, belong to the complex number system. If b is not 0, the complex number is called an imaginary number; if b is not 0 but a is 0, the complex number is called a pure imaginary number; if b is 0, the complex number is a real number. Imaginary numbers (the term must not be used in a literal sense but in the technical sense just described) are extremely useful in the theory of alternating currents and many other branches of physics and natural science.

Number Systems

A number system is defined by the base it uses, the

base being the number of different symbols required by

the system to represent any of the infinite series of

numbers. Thus, the decimal system in universal use

today (except for computer application) requires ten

different symbols, or digits, to represent numbers and is

therefore a base-10 system

OPERATIONS WITH POSITIVE INTEGERS

1. Adding Positive Integers

The arithmetic operation of addition is basically a means of counting quickly and is indicated by the plus sign (+). We could place 4 apples and 5 more apples in a row, then count them individually from 1 to 9. Addition, however, makes it possible to count all of the apples in a single step (4 + 5 = 9)

We can easily add long lists of numbers with more than one digit by repeatedly adding one digit at a time. For example, if the numbers 27, 32, and 49 are listed in a column so that all the units are in a line, all the tens are in a line, and so on, finding their sum is relatively simple:

27 [TENS ] + 32 [ UNITS] 49 18 90 108 First add the units (7 + 2 + 9); they total 18.

Then add the digits in the tens place (2 + 3 + 4); they

total 9, but this means 9 tens, or 90. In the last step, add

the total of the units to the total of the tens:

We can skip the second step, adding the sum of the

units to the sum of the tens, by using a shortcut called

carrying. Carry the 1 in 18, which stands for 1 ten, over

to the tens column and add it directly to the digits there

2.Subtracting Positive Integers

•The arithmetic operation of subtraction is the opposite of addition and is indicated by the minus sign (-). If we take 5 apples away from 9 apples, subtraction tells how many apples remain without our actually counting them. The simple sums memorized for addition are used in reverse for subtraction. We can subtract large numbers by repeatedly subtracting one digit at a time. First align the numbers under one another, units under units, tens under tens, as in addition

example, the result of 9 minus 5 is 4 because 4 is the number we would have to add to 5 for a sum of 9. The end result of subtraction is called the difference.

It is possible to subtract 23 from 66 by counting backward 23 integers from 66, one number at a time, or by taking away 23 items from a collection of 66 and counting the remainder Either way we would reach 43. The rules of arithmetic for subtraction, however, provide a much quicker method for obtaining the answer. We can subtract large numbers by repeatedly subtracting one digit at a time. First align the numbers under one another, units under units, tens under tens, as in addition:

66

-23

43

Subtract the units: 6 - 3 = 3. Then subtract the tens column: 6 – 2 = 4. The results of these two single-digit subtractions, written side by side, provide the answer:

Subtraction is a bit more complicated if we need to subtract a larger digit from a smaller one. For example, when subtracting 47 from 92, the units value (7) of 47 is greater than the units value (2) of 92. We can handle this situation using a procedure called borrowing, which is like carrying in reverse. Ten units can be borrowed from the tens column—that is, from the 9 of 92—leaving 8 in the tens column. Bring the 10 over to the units column and add it to the 2 already there, giving 12 in that column from which 7 can then be subtracted:

8 12

92

47

45

Complete the subtraction by taking 4 away from 8 in the tens column, which gives 4. The answer, or difference, is 45

3. Multiplying Positive Integers

• Multiplication is simply repeated addition and is often indicated by the times sign (×). The expression 3 × 4 means that 3 is to be added to itself 4 times or, similarly, that 4 is to be added to itself 3 times. In either case, the answer is the same: 12. For example, 3 sets of 4 apples together contain a total of 12 apples. When large numbers are involved, however, such repeated addition is tedious. Multiplication provides a procedure for simplifying repeated addition. Sometimes a dot or an asterisk is used instead of a times sign to indicate the multiplication of two or more numbers, and sometimes parentheses are used. For example, 3 × 4, 3 · 4, 3 * 4, and (3)(4) all indicate 3 times 4. The end result of multiplication is called the product.

.

To multiply two numbers if the number with the most digits is placed on the top:

386 x 4 24(4x6 in unit place) 320(4x8 in the tens place,or 4x80) 1200(4x3 in hundreds place or 4x300) 1544

We then multiply each digit of the top number by the bottom number, in this case, 4. Adding the results of all these multiplications together gives the product 1,544

.

Carrying tens and hundreds, as in addition, shortens this operation:

2

386 4 4 Multiply the 6 by 4, giving 24. Write the 4 in the units place

of the product and carry the 2, which stands for 2 tens, or 20. Multiply the 8 in the tens place by 4, giving 32, then add the carried 2, giving 34. (We actually multiplied 80 by 4, giving 320, and carried 20, which we then added to 320.) Write the 4 in the tens place just to the left of the 4 in the units place, and carry the 3 (which stands for 3 hundreds), placing it over the hundreds column.

To finish, multiply the 3 in the hundreds place by 4, giving 12 (actually 1,200), and add the 3 that we carried, giving 15 (actually 1,500)

3 2

386 4 1544

We can follow a similar procedure when both numbers to be multiplied have more than one digit.

To multiply 36 by 52, for example, begin by multiplying the top number, 36, by the unit 2 of the bottom number:

1

36 x 52 72 (partial product) Next multiply the 6 by 5, giving 30, and put the 0 under

the number 7 in the tens place of the partial product. This placement is chosen because the 5 in the bottom number is in the tens place and actually represents 50. Carry the number 3 as usual.

Multiply the 3 in the tens place of the top number by 5, giving 15, and add the carried 3, giving 18 (really 5 times 30, plus 30, for a total of 180). Now write the 8 in the hundreds place (directly to the left of the 0 in the tens place), and carry the 1 into the thousands place. We obtain the total product by adding the two partial products:

3

36

x 52

72 (partial product)

+180 (partial product)

1872 (total product)

4. Dividing Positive Integers

The arithmetic operation of division is the opposite, or inverse, of multiplication. Using the example of 12 divided by 4, we may indicate division by the division sign (12 ÷ 4), a bar (Ž), a slash (12/4), or the notation p. Division determines how many times one number is contained in another number. For example, 4 is contained 3 times in 12; thus, 12 apples could be divided into 3 sets of 4 apples, so 12 divided by 4 is 3. The number to be divided is called the dividend, the number the dividend is divided by is called the divisor, and the end result of division is called the quotient:

4)12 (3

divisor) divident (quotient• Simple divisions such as 12 ÷ 4 may be carried out

mentally, but more complicated cases require a procedure known as long division. Long division involves the repetition of simple operations

For example, to divide 4,518 by 6, consider the divisor (6) and the first digit (4) of the dividend to see whether the divisor is contained in that first digit one or more times:

6 ) 4518 ( If the first digit is too small (6 is not contained in 4 even

once), try to divide the first two digits of the dividend (45) by the divisor (6) . To determine how many 6s are contained in 45, make a guess. If we guess 8, we can check our guess by multiplying 6 by 8, which yields 48. Since 48 is more than 45, the guess was too big. Guessing 6 and multiplying 6 × 6 yields 36—too small

We know it is too small because when we subtract 36 from 45 (45 – 36) we get 9, which indicates that 45 contains another 6. Therefore 6 will go into 45 no more than 7 times (6 × 7 = 42). Write the number 7 in the quotient over the 5 in the dividend, 4,518, and write the 42 (the product of the divisor, 6, and the first number of the quotient, 7) under the 45 and subtract from it, yielding 3. What we have actually determined so far is that 4,518 contains at least 700 6s, and that 318 is left when these are taken away. In the next step of the division process, bring the 1 in the dividend down and write it to the right of the 3 to give 31:

6 ) 4518 ( -42 31

The 6 in the divisor will go into 31 no more than 5 times (6 × 5 = 30). Write the 5 in the quotient to the right of the 6, above the 1 in the dividend. Place the product of 5 × 6, or 30, under the 31 and subtract, yielding 1. Bring the 8 from the dividend down and write it to the right of the 1 to give 18. The 6 in the divisor will go into 18 exactly 3 times, so write the number 3 in the quotient above the 8 in the dividend

6 ) 4518 ( 753 -42 31 - 30 18 -18 0 The answer to how many times 6 will divide 4,518 is therefore

753. We can verify this solution by multiplying 6 × 753, which yields the dividend 4,518.