Presented By: Ms. Poonam Anand  Know the different types of numbers  Describe positional notation  Convert numbers in other bases to base 10  Convert base 10 numbers into numbers of other bases  Describe the relationship between bases 2, 8, and 16  Fractions  Negative Numbers Representation  Floating Point Numbers Representation  Number categories ◦ Many categories: natural, negative, rational, irrational and many others important to mathematics but irrelevant to the understanding of computing  Number – unit belonging to an abstract mathematical system and subject to specified laws of succession, addition and multiplication ◦ Natural number is the number 0 or any other number obtained adding repeatedly 1 to this number. ◦ A negative number is less than 0 and it is opposite in sign to a positive number. ◦ An integer is any of positive or negative natural numbers ◦ A rational number is an integer or the quotient of any two integer numbers  is a value that can be expressed as a fraction  The base of number system represents the number of digits that are used in the system. The digits always begin with 0 and continue through one less than the base  Examples: ◦ There are two digits in base two (0 and 1) ◦ There are eight digits in base 8 (0 through 7) ◦ There are 10 digits in base 10 (0 through 9)  The base determines also what the position of the digits mean  It is a system of expressing numbers in which the digits are arranged in succession and, the position of each digit has a place value and the number is equal to the sum of the products of each digit by its place value  Example: ◦ Consider the number 954:  9 * 102 + 5 * 101 + 4 * 100 = 954 ◦ Polynomial representation - formal way of representing numbers, where X is the base of the number:  9 * X2 + 5 * X1 + 4 * X0  Formal representation – consider that the base of representation is B and the number has n digits, where d i represents the digit in the ith position. ◦ d * Bn-1 + d * Bn-2 + …+ d B +d n n-1 2 1 ◦ 642 is: 6 * 102 + 4 * 10 + 2 * 10 0 3 2 1 What if 642 has the base of 13? + 6 x 13² = 6 x 169 = 1014 + 4 x 13¹ = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 = 1068 in base 10 642 in base 13 is equivalent to 1068 in base 10  Decimal base has 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)  Binary is base 2 and has two digits (0 and 1)  Octal is base 8 and has 8 digits (0, 1, 2, 3, 4, 5, 6, 7)  Hexadecimal is base 16 and has 16 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)  What is the decimal equivalent of octal number 642? 6 x 8² = 6 x 64 = 384 + 4 x 8¹ = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2 = 418 in base 10 • Remember that octal base has only 8 digits (0, 1, 2, 3, 4, 5, 6, 7)  What is the decimal equivalent of the hexadecimal number DEF? D x 16² = 13 x 256 = 3328 + E x 16¹ = 14 x 16 = 224 + F x 16º = 15 x 1 = 15 = 3567 in base 10 • Remember that hexadecimal base has 16 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) • What is the equivalent decimal of the binary 10110 number? 1 x 2 = 1 x 16 = 16 4 + 0 x 23 = 0 x 8 = 0 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 0 x 20 = 0 x 1 = 0 = 22 in base 10  Remember that binary base has only 2 digits (0, 1)