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:

1. Rational numbers2. Powers and exponents3. Zero and negative exponents4. Order of operations5. Perfect squares and square roots6. Pythagorean theorem7. Scientific notation

1.1: Real numbers

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

Rational numbers

Rational numbers are the following:1. Ratios (for example: 2/3);2. Decimal numbers that are

terminating (for example: 1,56)3. Decimal numbers that are non-

terminating and have a repeating pattern of digits (for example: 1,12121212…)

Irrational numbers

Irrational numbers are the following:1. Square roots (for example: the

square root of 2 but not the square root of 9)

2. Decimal numbers that are non-terminating and with digits that do not repeat in a fixed pattern (for example: pi = 3,141592…)

Real numbers continued

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…

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: 1. Do all operations in the brackets. 2. Do all your exponents. 3. Multiply and Divide terms from left to

right.4. Add and Subtract terms from left to

right.

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) )

The rules of integers: Addition and Subtraction

The rules of integers: Multiplication and Division

Rules for multiplying integers:1. The product of two positive integers is always a

positive integer. 2. The product of a positive integer and a negative

integer is always a negative integer. 3. The product of two negative integers is always a

positive integer.

Rules for dividing integers: 1. The quotient of two positive integers or two

negative integers is always a positive integer. 2. The quotient of a positive integer and a negative

integer is always a negative integer.

Substitution

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

How is substitution in Math done?

Substitution in an expression is done by following two rules:

1. Make your substitutions directly into your expression.

2. Evaluate your expression by following the order of operations

An example of substitution

Evaluate the expression, x2 + xy, if x=3 and y=6

1. Substitute: (3)2 + (3) x (6)2. Evaluate: (3)2 = 9 and (3) x (6) =

18 so 9 + 18 = 273. The final answer is 27

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.

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.

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…

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

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)

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.

The 7 Laws of Exponents

1. Multiplication rule: am x an = am+n

2. Division rule: am ÷ an = am-n

3. Power of a power: (am)n = amxn

4. Power of a product: (ab)m = am x bm

5. Power of a quotient: (a/b)m = am/bm

6. Zero exponents: a0 = 17. Negative exponents: a-n = a-n =1/an

=(1/a)n

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

Adding numbers in scientific notation

In order to add numbers in scientific notation:

1. The two numbers must have the same power 10.

2. Then, add the numbers together and keep the same power of 10.

3. Don’t forget that your final response must satisfy the criteria of scientific notation.

Subtracting numbers in scientific notation

In order to subtract numbers in scientific notation:

1. The two numbers must have the same power 10.

2. Then, subtract the numbers and keep the same power of 10.

3. Don’t forget that your final response must satisfy the criteria of scientific notation.

Multiplying numbers in scientific notation

In order to multiply numbers in scientific notation:

1. Multiply the two numbers.2. Then, multiply the two powers. Use

the laws of exponents to help evaluate this final exponent.

3. Don’t forget that your final response must satisfy the criteria of scientific notation.

Dividing numbers in scientific notation

In order to divide numbers in scientific notation:

1. Divide the two numbers.2. Then, divide the two powers. Use

the laws of exponents to help evaluate this final exponent.

3. Don’t forget that your final response must satisfy the criteria of scientific notation.

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.

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.

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.

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.

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.

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.

The summary of chapter 1

What did we learn about in Chapter 1? What concepts?