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Ch. 1: Number Relationships

Ch. 1: Number Relationships. Factoring. Powers. Multiples. Square Roots. Order of Operations. And More…. Multiples. For example: 7 x 1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 So, the multiples of 7 are 7, 14, 21, 28, and so on.

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Ch. 1: Number Relationships

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  1. Ch. 1: Number Relationships Factoring Powers Multiples Square Roots Order of Operations And More…

  2. Multiples • For example: • 7 x 1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 • So, the multiples of 7 are 7, 14, 21, 28, and so on. • The Product of a number with the natural numbers 1, 2, 3, 4, 5, ...are called the multiples of the number. • Note:  • The multiples of a number are obtained by multiplying the number by each of the natural numbers. • For example: • multiples of 2 are 2, 4, 6, 8, … • multiples of 3 are 3, 6, 9, 12, … • multiples of 4 are 4, 8, 12, 16, … Example 1: Write down the first ten multiples of 5. Solution: The first ten multiples of 5 are 5, 10 15, 20, 25, 30, 35, 40, 45, 50.

  3. Common Multiples Multiples that are common to two or more numbers are said to be common multiples E.g. Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …   Multiples of 3 are 3, 6, 9, 12, 15, 18, … So, common multiples of 2 and 3 are 6, 12, 18, … Example 2: Find the common multiples of 4 and 6. Solution: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, … Multiples of 6 are 6, 12, 18, 24, 30, 36, … So, the common multiples of 4 and 6 are 12, 24, 36, …

  4. Lowest Common Multiple (LCM) E.g. Multiples of 8 are 8, 16, 24, 32, …   Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, … The smallest common multiple of two or more numbers is called the lowestcommon multiple (LCM). Therefore, LCM of 3 & 8 = 24 In general: To find the lowest common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number.  This is the LCM. Example: Find the lowest common multiple of 6 and 9. Solution: List the multiples of 9 and stop when you find a multiple of 6. Multiples of 9 are 9, 18, … Multiples of 6 are 6, 12, 18, … Therefore, LCM of 6 & 9 = 18 Example: Find the lowest common multiple of 5, 6 and 8. Solution: List the multiples of 8 and stop when you find a multiple of both 5 and 6. Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, … Stop at 120 as it is a multiple of both 5 and 6. So, the LCM of 5, 6 and 8 = 120.

  5. Factors For example, 20 ÷ 4 = 5 So, 4 is a factor of 20 as it divides exactly into 20. A whole number that divides exactly into another whole number is called a factor of that number. Likewise, 20 ÷ 5 = 4 So, 5 is a factor of 20 as it divides exactly into 20. Note: If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number. For example, 20 = 1 x 20 = 2 x 10 = 4 x 5 So, the factors of 20 are 1, 2, 4, 5, 10 and 20. Example: List all the factors of 42. Solution: 42 = 1 x 42 = 2 x 21 = 3 x 14 = 6 x 7 So, the factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. Example: Is 7 a factor 15? Solution: 15 ÷ 7 = 2 1/7 Clearly, 7 does not divide exactly into 15. SO, 7 is not a factor of 15.

  6. Common Factors For example: 4 = 1 x 4 = 2 x 2  Factors of 4 are: 1, 2, 4 6 = 1 x 6 = 2 x 3  Factors of 6 are: 1, 2, 3, 6 So, the common factors of 4 & 6 are: 1 & 2 Factors that are common to two or more numbers are said to be common factors. Example: Find the common factors of 10 and 30. Solution: 10 = 1 x 10 = 2 x 5  Factors of 10 are: 1, 2, 5, 10 30 = 1 x 30 = 2 x 15 = 3 x 10 = 5 x 6  Factors of 30 are: 1, 2, 5, 10, 3, 6, 15, 30 So, the common factors of 10 and 30 are 1, 2, 5 and 10. Example: Find the common factors of 26 and 39. Solution: 26 = 1 x 26 = 2 x 13  Factors of 26 are: 1, 2, 13, 26 39 = 1 x 39 = 3 x 13  Factors of 39 are: 1, 3, 13, 39 So, the common factors of 26 and 39 are 1, 13

  7. Greatest Common Factor (GCF) The largest common factor of two or more numbers is called the greatest common factor (GCF). For Example: 8 = 1 x 8 = 2 x 4 Factors are: 1, 2, 4, 8 12= 1 x 12 = 2 x 6 = 3 x 4 Factors are: 1, 2, 3, 4, 6, 12 So, the common factors of 8 & 12 are: 1, 2, 4 and clearly 4 is the largest common factor. Therefore, GCF = 4 Example: Find the highest common factor of 14 and 28. Solution: 14 = 1 x 14 = 2 x 7 28 = 1 x 28 = 2 x 14 = 4 x 7 GCF = 14

  8. Prime & Composite Numbers Prime Number: A number that has only two different factors, 1 and itself. For example, 7 = 1 x 7; 7 is a prime number since it has only two different factors. Clearly 2= 1 x 2; 3 = 1 x 3; 5 = 1 x 5; 11 = 1 x 11 …SO 2, 3, 5, 11, are prime numbers Composite Number: A number that has more than two factors. Ex. 14 = 1 x 14 = 2 x 7; So, 14 is a composite number as it has more than two factors. Example:State which of the following numbers are a prime: a) 46 b) 19 Solution: a)  46 is not a prime because 46 = 2 × 23. b)  19 is a prime since it has only two different factors, 1 and 19. Example:Express 90 as a product of prime numbers Solution:90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 OR 90 = 2 x 3² x 5 Alternatively, we can use a factor tree to express 90 as a product of prime numbers as illustrated below. 90 10 9 2 x 5 3 x 3 So, 90 = 2 x 3 x 3 x 5 OR 90 = 2 x 3² x 5

  9. Powers and Roots Square Roots: To find the the square root of a given number is to think of what number when multiplied by itself is equal to the given number. For example, to find the square root of 4, we have to think of a number that when multiplied by itself gives 4.  The number is 2 since 2 × 2 = 4. So the square root of a given number is the positive number whose square produces the given number. 

  10. Powers and Roots (cont.) Example: What number do you have to square to get each of the following numbers? a) 144 b)  400 • Square Roots: Solution: Ex. Solution:

  11. Powers • We know that: That is: So, 10 squared has 2 of the same factors.

  12. Powers (cont.) 73 is read as '7 to the power of 3' or '7 to the 3' or 7 cubed'. This suggests that the power (or index) tells us how many times the base has to be multiplied by itself. For example: Ex: Write 5 × 5 × 5 × 5 in index form. Solution: Expanded Form: If a number is written as a product of factors, then it is said to be in expanded form. Ex: Write 36 in expanded form. Solution: Ex: Evaluate 23. Solution: Ex: Solution: Ex: Solution:

  13. Powers of 10 • Look at this pattern: From this we can infer that: EX: Solution: EX: Solution: Ex: Express 5000 as a number between 1 and 10 multiplied by a power of 10. Solution:

  14. Standard Form (Scientific Notation) Astronomers, biologists, engineers, physicists and many others encounter quantities whose measures involve very small or very large numbers.  For example, the distance of the earth from the sun is approximately 144,000,000,000 metres and the distance that light will travel in 1 year is 5,870,000,000,000 metres. It is sometimes tedious to write or work with such numbers.  This difficulty is overcome by writing such numbers in standard form. E.g.  144,000,000,000 = 1.44 × 10¹¹ 5,870,000,000,000 = 5.87 × 10¹² If a quantity is written as the product of a power of 10 and a number that is greater than or equal to 1 and less than 10, then the quantity is said to be expressed in standard form (or scientific notation).  It is also known as exponential form. For example, 65 = 6.5 × 10¹  Note that we have expressed 65 as a product of 6.5 and a power of 10.  Clearly, 6.5 is between 1 and 10.  So the standard form of 65 is 6.5 × 10¹. Ex: Write 643 in standard form. Solution: Note: The decimal point is shifted to the left by 2 places, and 2 appears as the positive index in the power of 10. In general: In converting a number to standard form, if the decimal point is shifted to the left p places, then p appears as a positive index in the power of 10.

  15. Problem Solving Unit Problem 2.1  Squares • A square of a number is obtained by multiplying the number by itself. 1.  Find the value of: 2.  Find the digit(s) in which the square numbers can never end in.

  16. Rules For Exponents • First, any number raised to the power of "one" equals itself. This makes sense, because the power shows how many times the base is multiplied by itself. If it's only multiplied one time, then it's logical that it equals itself. • Secondly, one raised to any power is one. This, too, is logical, because one times one times one, as many times as you multiply it, is always equal to one. • Rules of 1:There are two simple "rules of 1" to remember. x1 = x 31 = 3 1m = 1 14 = 1 x 1 x 1 x 1 = 1 Rule # 2: Product Rule:The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut! Rule # 3: Power Rule:The "power rule" tells us that to raise a power to a power, just multiply the exponents. Here you see that 52 raised to the 3rd power is equal to 56.

  17. Rules of Exponents (cont.) Rule # 4: Quotient Rule:The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown. Rule # 5: Zero Rule:According to the "zero rule," any nonzero number raised to the power of zero equals 1. Rule # 6: Negative Exponents:The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.

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