In this checkpoint, we'll explore different number categories, including integers and the intriguing irrational numbers, each with its own special properties. We’ll also look at decimals and fractions – parts of numbers. After we've covered concepts, you'll get to put your new skills into action with a set of practice questions designed to challenge and solidify your understanding.

Let’s start with the easier concepts first!

Operations with Rational Numbers and Integers

Let’s look at the basic types of numbers there are:

• Rational Numbers: Numbers that can be expressed as a fraction a/b, where a and b are integers and b is not zero.
• Integers: Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...).

Do you know the four operators? Addition, subtraction, multiplication, and division. Let's go through some basic tips as a refresher (you may already know these):

Integers

• Addition: Same signs: Add the numbers and keep the sign. Example: -5 + (-2) = -7. Different signs: Subtract the smaller from the larger, use the sign of the larger. Example: -5 + 2 = -3
• Subtraction: Change to addition and change the sign of the second number. Example: -5 - 2 = -5 + (-2) = -7. Another example: 5 - (-2) = 5 + 2 = 7
• Multiplication: Same signs: Answer is positive. Example: (-5) * (-2) = 10. Different signs: Answer is negative. Example: (-5) * 2 = -10
• Division: Same signs: Answer is positive. Example: (-10) / (-2) = 5. Different signs: Answer is negative. Example: (-10) / 2 = -5

Fractions

• Addition/Subtraction: Must have a common denominator. Add or subtract the numerators, keep the denominator. Example: 1/5 + (-3/5) = -2/5. Example 1/2 - (1/3) = 3/6 - 2/6 = 1/6
• Multiplication: Multiple the numerators and then multiply the denominators. Example: (1/2) * (-2/3) = -2/6 = -1/3 (simplified).
• Division: Keep the first fraction. Change division to multiplication. Flip the second fraction (reciprocal). Then, multiply. *Example (-3/4) / (1/2) = (-3/4) * (2/1) = -6/4=-3/2

Practice Questions

Let’s do some basic questions just to reinforce our knowledge but look at ‘quick ways’ using mental math.

a) -5 + 8
b) 3/4 - 1/2
c) -6 * 4
d) 2.5 / 0.5
e) -12 + (-4)
f) A submarine is at a depth of -250 metres. It ascends 120 metres. What is its new depth?
g) Simplify: 2/3 * (1/4 + 5/6)
h) A recipe calls for 1 3/4 cups of flour. If you want to make half the recipe, how much flour do you need?

Solutions:

a) -5 + 8 = 3
b) 3/4 - 1/2 = 3/4 - 2/4 = 1/4
c) -6 * 4 = -24
d) 2.5 / 0.5 = 5
e) -12 + (-4) = -16
f) -250 + 120 = -130 metres
g) 2/3 * (1/4 + 5/6) = 2/3 * (3/12 + 10/12) = 2/3 * (13/12) = 26/36 = 13/18
h) 1 3/4 = 7/4 Half the recipe: (7/4) / 2 = 7/4 * 1/2 = 7/8 cups of flour

Irrational Numbers and π (Pi)

Irrational numbers are numbers that cannot be expressed as a fraction a/b, where a and b are integers and b is not zero. Their decimal representations neither terminate nor recur.

π (Pi): A famous irrational number representing the ratio of a circle's circumference to its diameter. It's approximately 3.14159, but its decimal goes on forever without repeating.

Examples: √2, √3, π are all irrational numbers.

Practice Questions

Let’s do some questions to reinforce our understanding of the concepts learned above.

Classify the following numbers as rational or irrational:

a) 0.25
b) √4
c) √7
d) π
e) 0.121212...

True or False: All square roots are irrational.
A circular pizza has a diameter of 14 inches. Using π ≈ 3.14, calculate the approximate circumference of the pizza.
Which is greater: 3.14 or π?
Can you write an irrational number between 1 and 2? If so, give an example.

Solutions:

a) Rational (Terminating decimal)
b) Rational (√4 = 2, which can be expressed as 2/1)
c) Irrational (Cannot be expressed as a fraction, non-terminating, non-recurring decimal)
d) Irrational (Non-terminating, non-recurring decimal)
e) Rational (Recurring decimal)
False. Some square roots are rational (e.g., √4 = 2, √9 = 3). Only square roots of non-perfect squares are irrational.
Circumference = π * diameter ≈ 3.14 * 14 inches ≈ 43.96 inches
π is greater than 3.14. π is approximately 3.14159...
Yes. Examples include √2, √3, 1 + π/10 (many other answers are possible)

Estimation

When working with numbers, sometimes, it’s time consuming to calculate an exact number and an estimation will do. Estimation is when you find an approximate answer, not an exact one.

Let’s say that you’re going grocery shopping with only $70 to spend. To make sure you don't go over budget, you round up the price of each item as you shop, keeping a running total in your head. This is estimation – a handy way to get a close-enough answer quickly. Professionals use estimation too. For example, an accountant might estimate a client's future tax burden to help them make informed financial choices.

Rounding is a common technique for estimation. Round numbers to the nearest whole number, ten, hundred, or a convenient decimal place.

When you estimate, always checking if your answer makes sense in the context of the problem. For example, 1,000 + 70 isn’t going to be a number less than 1,000. So you’ll be able to cut out such options if they’re provided when doing multiple choice.

Practice Questions

Let’s do some questions to reinforce our understanding of the concepts learned above.

a) Estimate the sum of 48 + 73 by rounding each number to the nearest ten.
b) Estimate the product of 3.8 and 12.2 by rounding each number to the nearest whole number.
c) A shirt costs $28.95. Approximately how much will 3 shirts cost?

Solutions:

a) 48 rounds to 50, 73 rounds to 70. 50 + 70 = 120
b) 3.8 rounds to 4, 12.2 rounds to 12. 4 * 12 = 48
c) $28.95 rounds to $30. 3 * $30 = $90

Decimals

A decimal is a way of representing a number that is not necessarily a whole number. It's a number that can include a part of a whole, in addition to a whole number. Think of it as a way to express numbers that fall in between the whole numbers you're familiar with (like 1, 2, 3, etc.). A decimal is a way of representing a number that is not necessarily a whole number. It's a number that can include a part of a whole, in addition to a whole number. Think of it as a way to express numbers that fall in between the whole numbers you're familiar with (like 1, 2, 3, etc.).

There are 3 key components:

• Whole Number Part: The digits to the left of the decimal point represent the whole number part.
• Decimal Point: The dot (.) that separates the whole number part from the fractional part.
• Fractional Part: The digits to the right of the decimal point represent the part of a whole.

Just like whole numbers, the digits in a decimal have place value. Here's how it works:

• The first digit to the right of the decimal point is in the tenths place (1/10).
• The second digit is in the hundredths place (1/100).
• The third digit is in the thousandths place (1/1000).
• And so on...

Let’s look at types of decimals:

• Terminating Decimals: These are decimals that have a finite number of digits. They end. Examples: 0.5, 0.75, 0.12345
• Recurring Decimals: These are decimals that have a pattern of digits that repeat forever. We show the repeating pattern by placing a dot or a bar over the repeating digits. Examples: 0.3333... (0.3 with a dot over the 3), 0.141414... (0.14 with a bar over 14)
• Identifying Terminating and Recurring: If the denominator of a fraction (in its simplest form) only has the prime factors 2 and/or 5, the decimal will terminate. Otherwise, it will recur. And lastly, we said that decimals are parts of numbers, much like fractions. You can convert fractions to decimals by dividing the numerator (top number) by the denominator (bottom number).

Let’s do some questions to reinforce our understanding of the concepts learned above.

Convert the following fractions to decimals and state whether they are terminating or recurring:

a) 3/8
b) 2/3
c) 7/20
d) 5/11
e) 9/25

Which of the following fractions will result in a terminating decimal?

f) 3/16
g) 4/25

Write the decimal 0.454545... as a fraction in its simplest form.

Express 0.7 (with a dot over the 7) as a fraction.

A tailor has a piece of fabric that is 5/6 of a metre long. He needs to divide it into pieces that are each 0.25 metres long. Will he have any fabric left over after cutting as many 0.25 metre pieces as possible? Why or why not?

Solutions:

a) 3/8 = 0.375 (Terminating)
b) 2/3 = 0.666... (0.6 with a dot over 6) (Recurring)
c) 7/20 = 0.35 (Terminating)
d) 5/11 = 0.4545... (0.45 with a bar over 45) (Recurring)
e) 9/25 = 0.36 (Terminating)

f) 3/16 (Denominator has only the prime factor 2)
g) 4/25 (Denominator has only the prime factor 5)

Let x = 0.454545... 100x = 45.454545... 100x - x = 45.454545... - 0.454545... 99x = 45 x = 45/99 = 5/11

Let x = 0.777... 10x = 7.777... 10x - x = 7.777... - 0.777... 9x = 7 x = 7/9

5/6 = 0.8333... (Recurring) 0.8333... / 0.25 = 3.333... The tailor can cut 3 whole pieces. 3 * 0.25 = 0.75 0.8333... - 0.75 = 0.08333... Yes, there will be fabric left over because 5/6 is a recurring decimal and cannot be divided evenly into 0.25 metre pieces.

Practice time!

Now, it's your turn to practice.

The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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