Arithmetic in the Hindu-Arabic System.
From our previous lesson you know that the Hindu-Arabic system is a positional numeration system with base ten, which means that each numeral is a string of symbols that have a face and a place value and each place value is ten times larger than the one to the right of it.
Example: In the numeral 851 the symbol “5” has a face value equal to 5 but because of the place it occupies in this particular numeral it means 50. In the number 5367 it means 5000.
Once humans had this system of denoting numbers they needed to be able to perform arithmetic operations with them in an efficient manner. You already know the algorithms that allow you to add or subtract numbers in columns. In this lesson you will become acquainted with the expanded form of a Hindu-Arabic numeral. Working with the expanded form will allow you to understand why the well-known algorithms of addition and subtraction in columns actually work. You will also become acquainted with calculation devices used in the past. It is expected that you master the following techniques:
- adding and subtracting numerals in expanded form;
- reading an abacus;
- the lattice method for products;
- the nines complement method for subtracting.
In order to accomplish that you must
- read section 4.3 “Arithmetic in the Hindu-Arabic System” on page 153 of the textbook (you can skip the Napier’s rods);
- watch the videos in Lesson 3 on MyMathLab (this lesson is a prequisite to your HW #3);
- do HW #3 on MyMathLab.
Converting between bases different from 10. Computer mathematics.
In our last lesson homework you converted numbers from base 10 to nondecimal bases and viceversa. If you need to convert from one nondecimal base to another one (for example, a number in base three to base five), then as a general rule you would have to convert the number to base ten first.
Example: Suppose we have 2120three and we want to write this number in base five. Let us convert it first to base ten using for instance the shortcut method.
In the previous calculation we used the order of operations that is often referred as PEMDAS. The number 69 (which is written in base 10) has to be converted to base five now. We will use the other shortcut method you learned in our last lesson:
69÷5=13 R 4
13÷5=2 R 3
2÷5=0 R 2
That is why 69=234five. We obtain that 2130three=69=234five. That is 2130three=234five.
Following this example please convert 412six to base four and deposit your solution in the Assignments like you did in the case of Egyptian Multiplication. Make sure you include all the steps as in the previous example.
Computer mathematics: This area of mathematics is concerned, among other things, with converting numbers to and between bases that are powers of two. Read pages 164-169 paying close attention to examples. You will see that not always you have to resort to base ten while converting between nondecimal bases. To practice what you learned solve some of the exercises 47-56 on page 170 of the textbook.
Prime and composite numbers. Divisibility and Prime Factorization.
In this lesson you will be reminded of several definitions related to the divisibility of numbers. The defined concepts will be in bold. They will be of help when you read the textbook and solve the exercises.
The integer a is said to be divisible by the integer b if a=b×k, where k is also an integer.
A positive integer is said to be prime if it is divisible only by 1 and by itself. 1 is not considered to be a prime number since prime numbers are used to factor numbers and factoring a as a×1 is considered trivial. The smallest prime number is 2. A number is said to be composite if it is not prime or, in other words, if it can be written as a product of two integers different from 1.
Examples: 2, 3, 5, 7 and 11 are the first five prime numbers. 9 is not a prime because 9=3×3.
Factoring means writing as a product. The prime factorization of a positive integer is its representation as a product of powers of prime numbers. The prime factorization of a number is unique.
Examples of prime factorizations of numbers:
At the end of this lesson, besides knowing the concepts listed above, you will have to be able to:
- determine if a number is prime or composite;
- list the factors of a number;
- apply divisibility criteria;
- find the prime factorization of a number.
Read section 5.1 on page 178 paying close attention to all examples.
Do HW#6 on MayMathLab.
Greatest Common Factor and Least Common Multiple.
In this lesson you will review two concepts you are already familiar with: the greatest common factor and the least common multiple.
Read Section 5.4. You will be acquainted with three different methods of finding the greatest common factor. It is important that you master the following
- Prime factors method
- Euclidean Algorithm
When you read about the Least Common Multiple pay special attention to the prime factors methods.
It is worth noticing that the so-called formula to find the least common multiple on p. 246 can be written in the following way
LCM(m,n) x GCF(m,n) = m x n.
So every time you find the least common multiple and the greatest common factor of two numbers multiply them and verify that the product equals the product of the numbers themselves.
Example: The greatest common factor of the numbers 6 and 15 is 3. Their least common multiple is 30. 3 x 30=90 and 6 x 15 = 90. The formula verifies.
Do not forget to do HW#8 before its due date.
Fibonacci numbers and the Golden Ratio.
Read Section 5.5 of the textbook. You will explore the Fibonacci numbers. If you want to know the origin of the Fibonacci numbers you can read section 1.3 of the textbook, pages 20 and 21. The Fibonacci numbers are the solution of the problem listed in Example 1 , page 20. In this lesson you will learn that the Fibonacci numbers are involved in many number patterns. You will practice learning about mathematics and using your knowledge to solve exercises.
Complete Lesson #8 and HW #8 on MyMathLab.