MODULE – 1 The Number System

Classification of numbers

RATIONAL NUMBERS A rational number is a number that can be expressed in the form where b 0 and where a and b are integers.

Example 1  

 

 

The halfway mark between 2, 5 and 3 is: This process can be continued indefinitely and we can find an infinite number of rational numbers lying between 2 and 3 and that get closer to 3 all the time. Therefore, between any two rational numbers, there are an infinite number of rational numbers.

EXERCISE 1  

Rounding off numbers to certain decimal places The rules for rounding off numbers to certain decimal places are as follows: Count to the number of decimal places after the comma that you want to round off to. Look at the digit to the right of this decimal place. If it is lower than 5, then truncate. If it is 5 or more than 5, then add one digit to the digit immediately to the left of it and drop it and all the digits to the right of it. If necessary, keep or add zeros as place- holders.

Example 2 Round off the following numbers to two decimal places: (a) 2, 31437 Answer: 2, 31 (b) 0, 77777 Answer: 0, 78 (c) 245, 13589 Answer: 245, 14 (d) 254, 2 Answer: 254, 10 (e) 11, 4963 Answer: 11, 50

EXERCISE 2 Round off the following numbers to the number of decimal places indicated: (a) 9, 23584 (3 decimal places) (b) 67, 2436 (2 decimal places) (c) 4, 3768534 (4 decimal places) (d) 17,247398 (5 decimal places) (e) 79, 9999 (3 decimal places) (f) 34, 2784682 (4 decimal places) (g) 5,555555 (5 decimal places)

The number : Definition:The number is the ratio of the circumference of a circle to its diameter forming a decimal, which neither terminates nor recurs.

It is an irrational number. The decimal version of is : 3,142857143 Properties of It is an irrational number. The decimal version of is : 3,142857143 The improper fraction version of is:

EXERCISE 4 1. State whether the following numbers are rational or irrational: (a) 8 (b) (c) 7 (d) (e) (f) (g) (h) (i) - 0,13 (j) 0,42 (k) (l) 0,2453756…

2. Classify the numbers by placing ticks in the appropriate columns: Real Rational Integer Whole Natural Irrational - 3 0, 3 8, 23647

Determining between which two integers an irrational number lies. Example 4 Without using a calculator, determine between which two integers lies.

Procedures to follow: Find an integer smaller than 11 that can be square rooted easily and an integer that is larger than 11 that can be square rooted easily. Then create an inequality and square root all three numbers to get the answer.

Therefore, lies between the integers 3 and 4. By calculating rounded off to four decimal places, determine between which two integers it lies: The decimal 3, 3166 lay between the integers 3 and 4. Let’s have fun with the next exercise!!!

EXERCISE 5 1. Without using a calculator, determine between which two integers the following irrational numbers lie: (a) (b) (c) (d) (e) (f)

2. By using a calculator, verify whether your answers in the previous question are correct.

REPRESENTING REAL NUMBERS ON A NUMBER LINE Set builder notation: Set builder notation is a useful way of representing sets of real numbers. Let us work through the following examples.

Example 5: Represent the following sets on a number line:

Example 6 Write the following in set builder notation:

Let’s look and work through the following examples: Interval notation: Definition: Interval notation is another way of representing real numbers on the number line. Let’s look and work through the following examples:

Example 7 Represent the following on a number line:

Example 8 Write the following in interval notation:

EXERCISE 6 1. Represent the following sets on a number line: (a) {x: x 6; Z} (b) {x: x > - 4; Z} (c) {x: x<4; N} (d) {x: x; Z} (e) {x: - 5 <; Z} (f) {x: l < x; N} (g) {x: x<10; N} (h) {x: x < - 7; R} (i) {x: - 4 <x; R} (j) {x: x < 4; R} (k) {x: x; R}

2. Write the following sets in set builder notation:

3. Represent the following on a number line: (c) (-2 ; 5) (d) (-8 ; 8) (e) ( -5 ; ) (f) (- ; 3) (g) ( ; ) (h) (- ; ) (i) ( )

INVESTIGATING LINEAR NUMBER PATTERNS AND CONJECTURES Example 9 Consider the linear number pattern 7; 9; 11; 13; 15 The first term ( ) is 7. The second term ( ) is 9. The third term ( ) is 11. And so on…

The pattern is formed by adding 2 to each new term. We say that the constant difference (d) between the terms is 2. We will now investigate a way of determining greater terms like without actually having to work out the pattern 50 times.

= 7 7 + 2 7 + 2 + 2 = 7 + (2) x 2 7 + 2 + 2 + 2 = 7 + (3) x 2 7 + 2 + 2 + 2 + 2 + 2 = 7 + (5) x 2 3 + 2 + 2 + 2 + 2 + 2 + 2 = 7 + (6) x 2 7 + (24) x 2

We can now make a conjecture regarding the above process of finding different terms in the number sequence. A conjecture is a mathematical statement which has evidence to support it. The conjecture in words can be stated as follows: Term you want = first term + (term number minus 1) x the number you add to get the next term. In mathematical language, the conjecture can be written as: 3 + (n-1)2

For example, the 100 term can be calculated as follows: This conjecture or formula (nth term) can be used to calculate any term. For example, the 100 term can be calculated as follows: Notice:The coefficient of n in the general term represents the constant difference of the number pattern.

In general, then, these linear number patterns are of the form where d represents the constant difference. We now focus on a really great method of determining the nth term or general term of a linear number pattern, instead of using the long method illustrated in example 9. Consider the linear number pattern 7; 9; 11; 13; 15 (Example 9) The constant difference is 2. in the general term now substitute d = 2 into the general term:

Then in order to get c, use the first term = 7

is the nth term. Note: the following symbols represent the following values: Number of terms common difference

Example 10 Determine the nth term of the following number pattern: 4 ; 9 ; l4 ; 19

Step 1: Determine the constant difference (d): The constant difference (d) = 5 Step 2: Now use the first term: to find the value of c.

Example 11 Chains of squares can be built with matchsticks as follows:

(a) Use matches and create a chain of 4 squares (a) Use matches and create a chain of 4 squares.How many matches were used? (b) Create a chain of 5 squares. How many matches were used? (c) Determine a conjecture for calculating the number of matches in a chain of n squares. (d) Now determine how many matches will be needed to build a chain of 100 squares.

Solutions: (a) 13 matches (b) 16 matches 1 (c) The number pattern is 4; 7; 10; l3 The constant difference is: (d) Matches.

Disproving conjectures: Not all conjectures are necessarily correct in all situations. Consider the following conjecture, which illustrates this nicely. “The cube of an integer is equal to the integer” In other words we want to prove that for all Z. For n = -1: For n = 0 : For n = l : We notice that the value substituted is the same as the answer!!! So it seems that this conjecture is always true. However, if n = 3, then .because, In other words, we have found a value of n for which the conjecture is clearly false. a = 3 is called a counter example to the conjecture. Therefore the conjecture “The cube of an integer is equal to the integer” is false. It is only true for a few values of n.

EXERCISE 7 Firstly determine a conjecture for the nth term of the following linear number patterns (sequences) Hence, determine the value of term in each case. (a) l; 3; 5; 7; 9;… (b) 3; 8; 13; 18;….. (c) -9; 0; 9; 18; … (d) 1; 2; ; …

Provide counter examples for the following conjectures: (a) The sum of two prime numbers is a prime number. (b) If a number is even then it can never be divisible by 9. (c) = for all real values of m and n. (d) for all real values of m and n. (e) for all real values of a.

3. The next question is an example of what is called an unseen, non-routine problem. Investigate the following pattern:

(a) Continue this pattern three more times. (b) Use the pattern to complete the following (c) Investigate this pattern for the squares of other consecutive numbers. (d) Formulate a conjecture. (e) Try to prove your conjecture. Hint: Let the one number be n and the other be n – 1 and use grade 9 algebra to prove that the left side equals the right side.

QUESTION 3 Consider this pattern: 3.1 Verify that the answers to the above calculations are correct by working out the powers. 3, 2 Now use the pattern to complete the following: (a) (b) (c) 3.3 Prove that the conjecture is true for all natural numbers.

4.1 Complete the following table: QUESTION 4 Fraction Recurring decimal 0, 045454545 4.1 Complete the following table: 4.2 Now investigate and report on any patterns you find in the table above.

ASSESSMENT TASKS

INVESTIGATION ON NUMBER PATTERNS QUESTION 1 1.1 Determine the 12th term of the following sequence: 1; 1; 2; 3; 5 ; 8; 13 (This sequence of numbers is called the Fibonacci sequence. Ask your teacher to tell you more about Fibonacci) 1.2 Determine a conjecture that will help you determine any term of the following sequences and hence use your conjecture to determine the 18 th term: (a) 5; 8; ll; l4; …. (b) 1; 2; 4; 8; 16; ….

QUESTION 2 2.1 Work out each of the following quotients and then show that the statements are true: (a) (b) (c) (d)

2.2 Write down the next quotient in the pattern. 2.3 Write the solution to: 2.4 Write the solution to: 2.5 Write down a conjecture from these examples. 2.6 Write down a formula to find the sum of consecutive numbers from 1 to n.