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Chapter 1

Chapter 1. Number Sense See page 8 for the vocabulary and key concepts of this chapter. Chapter 1: Get Ready!. These are the necessary concepts to be reviewed before beginning Chapter 1: Rational numbers Powers and exponents Zero and negative exponents Order of operations

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Chapter 1

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  1. Chapter 1 Number Sense See page 8 for the vocabulary and key concepts of this chapter.

  2. Chapter 1: Get Ready! • These are the necessary concepts to be reviewed before beginning Chapter 1: • Rational numbers • Powers and exponents • Zero and negative exponents • Order of operations • Perfect squares and square roots • Pythagorean theorem • Scientific notation

  3. 1.1: Real numbers • Real numbers can be classified in two big categories: rational numbers and irrational numbers.

  4. Rational numbers • Rational numbers are the following: • Ratios (for example: 2/3); • Decimal numbers that are terminating (for example: 1,56) • Decimal numbers that are non-terminating and have a repeating pattern of digits (for example: 1,12121212…)

  5. Irrational numbers • Irrational numbers are the following: • Square roots (for example: the square root of 2 but not the square root of 9) • Decimal numbers that are non-terminating and with digits that do not repeat in a fixed pattern (for example: pi = 3,141592…)

  6. Rational numbers have 3 sub-groups: Integers are numbers of the group …-3, -2, -1, 0, 1, 2, 3… Whole numbers are numbers of the group 0, 1, 2, 3, 4… Natural numbers are numbers of the group 1, 2, 3, 4… Real numbers continued

  7. 1.2: Order of Operations with rational numbers • Rational numbers follow the same rules of order of operations as integers and whole numbers. • Here is the order of operations: • Do all operations in the brackets. • Do all your exponents. • Multiply and Divide terms from left to right. • Add and Subtract terms from left to right.

  8. The rules of integers: Addition and Subtraction • Rule of adding integers #1: If the terms have the same sign, add the digits together and keep the same sign. • Rule of adding integers #2: If the terms have different signs, subtract the little number from the big number and keep the sign of the bigger number. • Rule of subtracting integers #1: If we subtract two integers, two negative signs are going to change to a positive sign. • Rule of subtracting integers #2: In order to subtract integers, subtracting a number is equal to adding the opposite. (for example, 5-6 is the same as 5 + (-6) )

  9. The rules of integers: Multiplication and Division • Rules for multiplying integers: • The product of two positive integers is always a positive integer. • The product of a positive integer and a negative integer is always a negative integer. • The product of two negative integers is always a positive integer. • Rules for dividing integers: • The quotient of two positive integers or two negative integers is always a positive integer. • The quotient of a positive integer and a negative integer is always a negative integer.

  10. Substitution • Substitution in mathematics means to replace a letter (an unknown value) with a exact number value.

  11. Substitution in an expression is done by following two rules: Make your substitutions directly into your expression. Evaluate your expression by following the order of operations How is substitution in Math done?

  12. An example of substitution • Evaluate the expression, x2 + xy, if x=3 and y=6 • Substitute: (3)2 + (3) x (6) • Evaluate: (3)2 = 9 and (3) x (6) = 18 so 9 + 18 = 27 • The final answer is 27

  13. 1.3: Square roots and their applications • A perfect square is a number which is the product of two identical factors. • For example, 16 is a perfect square because 16 = 4 x 4 • A square root of a number is the factor which multiplies together with itself to give that specific number. The symbol is √. • For example, since 9 x 9 = 81, the square root of 81 is 9.

  14. Principal Square Root • A square root can be a positive or negative number. • For example, 81 = 9 x 9 and 81 = (-9) x (-9). So, la square root of 81 is ±9 (+9 or -9) • The principal square root is the positive square root of a number.

  15. Estimating the value of square roots • The square root of 72 is not easy to evaluate without a calculator but you can estimate its value by using your knowledge of the square roots of perfect squares. • For example, you know that the square root of 64 = 8 and the square root of 81 = 9. • 72 lies between 64 and 81 so the square root of 72 is between 8 and 9. • Verify your estimate with a calculator and you will see that the square root of 72 is equal to 8.485 281 374…

  16. Pythagorean Theorem • A significant application of square roots is the Pythagorean theorem, a subject discussed in Grade 8 Math. • The equation of the Pythagorean theorem is a² + b² = c² • The following web sites will help explain this concept: • http://argyll.epsb.ca/jreed/math8/strand3/3202.htm • http://www.arcytech.org/java/pythagoras/history.html

  17. 1.4: Exponents • A power is a short form for writing repeated multiplication of the same number. • 53, 107, x2 are all powers. • The base (of a power) is the number being repeatedly multiplied. For example, in 63, 6 is the base. • The exponent is the raised number in the power that indicates how many times to multiply the base. For example, in 63, 3 is the exponent. (63 = 6x6x6)

  18. The Laws of Exponents • The « laws of exponents » are the rules used to evaluate expressions that have exponents in them. • Attention: For the powers am or an, a is the base; m et n are the exponents.

  19. The 7 Laws of Exponents • Multiplication rule: am x an = am+n • Division rule: am ÷ an = am-n • Power of a power: (am)n = amxn • Power of a product: (ab)m = am x bm • Power of a quotient: (a/b)m = am/bm • Zero exponents: a0 = 1 • Negative exponents: a-n = a-n =1/an =(1/a)n

  20. 1.5: Scientific notation • Scientific notation is a way to represent really large or really small numbers. • Scientific notation always has 2 parts: a number from 1 to 10 and a power of 10. • For example, 123000 = 1.23 x 105 • For example, 0.000085 = 8.5 x 10-5

  21. Adding numbers in scientific notation • In order to add numbers in scientific notation: • The two numbers must have the same power 10. • Then, add the numbers together and keep the same power of 10. • Don’t forget that your final response must satisfy the criteria of scientific notation.

  22. Subtracting numbers in scientific notation • In order to subtract numbers in scientific notation: • The two numbers must have the same power 10. • Then, subtract the numbers and keep the same power of 10. • Don’t forget that your final response must satisfy the criteria of scientific notation.

  23. Multiplying numbers in scientific notation • In order to multiply numbers in scientific notation: • Multiply the two numbers. • Then, multiply the two powers. Use the laws of exponents to help evaluate this final exponent. • Don’t forget that your final response must satisfy the criteria of scientific notation.

  24. Dividing numbers in scientific notation • In order to divide numbers in scientific notation: • Divide the two numbers. • Then, divide the two powers. Use the laws of exponents to help evaluate this final exponent. • Don’t forget that your final response must satisfy the criteria of scientific notation.

  25. 1.6: Matrices • We can represent data in a diagram, a table or a matrix. • A matrix is a rectangular group of numbers which are organized in rows and columns. These rows and columns are surrounded by square brackets.

  26. The dimensions of a matrix • If a matrix is composed of two rows of numbers and 3 columns of numbers; it is a 2x3 matrix. • The dimensions of a matrix is also called the order of a matrix.

  27. The elements of a matrix • Each number in a matrix is called an element. • We can determine the number of elements in a matrix by multiplying the matrix’s dimensions together. • For example, a matrix with dimensions 5x4 is going to have 20 elements.

  28. Adding matrices • We can determine the sum of two matrices by adding each element of the first matrix with the corresponding elements of the second matrix.

  29. Subtracting matrices • We can determine the difference between two matrices by subtracting each element of the first matrix from the corresponding elements of the second matrix.

  30. Multiplying matrices by a scalar • A scalar is a numerical quantity. • To multiply a matrix by a scalar, multiply each element of the matrix by the scalar.

  31. The summary of chapter 1 • What did we learn about in Chapter 1? What concepts?

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