Arithmetic Foundations: Integers, Number Line, and Basic Operations

Integers, Natural Numbers, and Whole Numbers

• Arithmetic foundation begins with integers, defined as whole numbers including negatives.

• layman definition: integers are numbers with no parts or pieces, i.e., whole numbers.

• Different categories to know:

  • Natural numbers (counting numbers): the numbers you can count on your fingers, starting at 1 and increasing without bound: 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots

  • Whole numbers: similar to natural numbers but include zero: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots

• Integers: include all natural numbers and all their negative counterparts: \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots

• Important caveat: infinity symbols ((\infty)) are not integers. Infinity is not a number; it is a concept.

Quick concept check: Which numbers are integers?

• 7: yes, an integer.

• 11: yes, an integer.

• -38: yes, an integer.

• 1/2: not an integer (not a whole number).

• -1.7: not an integer (has a fractional part).

• (\pi) (pi): not an integer.

• The symbol for infinity: not an integer.

• -\sqrt{2}: not an integer (irrational, non-integer).

The number line: a visual representation

• The number line is a visual tool to compare numbers and see which are bigger or smaller.

• Zero at the center; to the right: positive integers (1, 2, 3, 4, …).

• To the left: negative integers (−1, −2, −3, −4, …).

• Infinity to both ends: left goes to (-\infty), right goes to +\infty.

• Reading along the line: generally, as you move to the right, numbers get bigger.

• Example relationships:

  • 4 > 1 when read on the line.

  • 3 > 0.

  • 3 > -3.

• Interesting consequence: negative numbers follow the same order, but more left means smaller:

  • Negative four (−4) is less than negative two (−2) because it is further to the left.

  • Even though (4 > 2), we have (−4 < −2).

• The line is arbitrary in its orientation of “bigger” to the right. For example, reading direction is a convention (English left-to-right, Arabic right-to-left).

• To illustrate arbitrariness, an alternative vertical number line is possible:

  • Zero in the middle; numbers increase as you go up and decrease as you go down.

  • This vertical orientation connects with coordinate geometry later in the course.

Practical note on coordinate geometry

• The number line concept extends to coordinate geometry, where lines can be oriented differently (e.g., vertical orientation).

The four basic arithmetic operations (with quick connections)

• There are four major operations:

  • Addition

  • Subtraction

  • Multiplication

  • Division

• These form the core of arithmetic you need for the test, along with some related concepts (exponents, roots, factorials).

Addition

• Example: 10 + 4 = 14.

• Concept: you are taking one entity (the 10) and adding another entity (the 4) to it to form a total.

Subtraction

• Concept: subtraction is the opposite of addition; removing one element from another.

• Example: removing 7 from 15 gives 8: 15 - 7 = 8.

• This is the difference between two numbers.

Multiplication

• Example: 4 \times 5 = 20.

• Interpretation: this is essentially repeated addition:

  • You could write it as 4 + 4 + 4 + 4 + 4 = 20, i.e., the number 4 added 5 times.

  • Alternatively, you could write it as 5 + 5 + 5 + 5 = 20, i.e., the number 5 added 4 times.

• Key point: multiplication is a shorthand for repeated addition (and a more compact form overall).

• Note on notation: sometimes multiplication is written with a dot ((\cdot)) rather than an x; both denote multiplication.

• Another connection: multiplication is also tied to division (inverse operation): if 4 \times 5 = 20, then 20 \div 5 = 4.

Division

• Example: 10 \div 2 = 5.

• Concept: dividing 10 into groups of 2 yields 5 groups. Equivalently, how many times can you subtract 2 from 10 until you reach 0?

• Connection to multiplication: division is the inverse of multiplication; 2 \times 5 = 10.

• Additional representation: division can also be seen via fractions or repeated subtraction.

• Alternate forms: \dfrac{10}{2} = 5. Also, 5 + 5 = 10 or 2 + 2 + 2 + 2 + 2 = 10.

Exponents (shorthand for repeated multiplication)

• Example: 3^3 = 27.

• Meaning: multiply 3 by itself 3 times: 3 \times 3 \times 3 = 27.

• Concept: exponents are a shorthand for multiplication, just as multiplication is a shorthand for addition.

• Inline example: 4 = 2^2. If we square 2, we get 4.

• Teaser: exponent rules (not fully covered here) will be discussed later.

Factorials

• Notation: an integer followed by an exclamation point, e.g., n!.

• Definition: n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1.

• Example: 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720.

• Practical note: factorials grow very large very quickly.

Summary for the test: what you need to know

• The core concepts: integers, number line, and the four basic operations.

• In addition: exponents, roots, and factorials are also important for the test.

• Remember the relationship between operations:

  • Multiplication as repeated addition.

  • Division as inverse of multiplication.

  • Exponents as repeated multiplication.

• Be able to recognize integers and non-integers from examples:

  • Integers include: {…,-2,-1,0,1,2,3,…}

  • Non-integers include fractions, decimals, irrationals: \dfrac{1}{2}, -1.7, \pi, \sqrt{2} (not integers).

• Understand the arbitrariness of the number line and the possibility of vertical orientation:

  • Reading direction of bigger numbers is conventional.

  • Different representations can model similar relationships (e.g., vertical axis in coordinate geometry).

End of notes from transcript

Integers, Natural Numbers, and Whole Numbers

• Arithmetic foundation begins with understanding different categories of numbers, particularly integers.
• Integers are defined as whole numbers, including their negative counterparts, and zero. This means they are numbers with no fractional or decimal parts.
• Different categories to know:
  • Natural numbers (or counting numbers): These are the numbers used for counting items, starting at 1 and increasing without bound: 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots. They are a subset of whole numbers and integers.
  • Whole numbers: This set includes all natural numbers along with zero: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots. All natural numbers are whole numbers.
  • Integers: This comprehensive set includes all natural numbers, all their negative counterparts, and zero: \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots. Both natural numbers and whole numbers are subsets of integers.
• Important caveat: infinity symbols (\infty) are not integers. Infinity is not a number; it is a concept.

Quick concept check: Which numbers are integers?

• 7: yes, an integer.
• 11: yes, an integer.
• -38: yes, an integer.
• 1/2: not an integer (not a whole number).
• -1.7: not an integer (has a fractional part).
• \pi (pi): not an integer.
• The symbol for infinity: not an integer.
• - \sqrt{2}: not an integer (irrational, non-integer).

The number line: a visual representation

• The number line is a fundamental visual tool used to compare numbers and understand their relative magnitudes (which are bigger or smaller).
• Structure: Zero is at the center of the number line.
  • To the right of zero, positive integers are placed, increasing in value (1, 2, 3, 4, \ldots).
  • To the left of zero, negative integers are placed, decreasing in value (-1, -2, -3, -4, \ldots).
• Ends of the line: The number line extends infinitely in both directions.
  • To the left, it extends towards negative infinity (-\infty).
  • To the right, it extends towards positive infinity (+\infty).
• Reading along the line: A key principle is that as you move from left to right on the number line, the numbers generally get bigger.
• Example relationships:
  • 4 > 1 when read on the line, as 4 is to the right of 1.
  • 3 > 0, as 3 is to the right of 0.
  • 3 > -3, as 3 is to the right of -3.
• Interesting consequence for negative numbers: Negative numbers follow the same order principle; the number further to the left is smaller.
  • For example, -4 is less than -2 (-4 < -2) because -4 is positioned further to the left of -2 on the number line. Even though the absolute value of 4 is greater than 2, when negative, -4 is smaller than -2.

Practical note on coordinate geometry

• The number line concept extends directly to coordinate geometry, forming the axes (e.g., horizontal x-axis, vertical y-axis). Lines can be oriented differently, such as a vertical orientation for the y-axis, but the principle of increasing values (upwards for positive, downwards for negative) remains consistent.

The four basic arithmetic operations (with quick connections)

• There are four major arithmetic operations that form the core of mathematics:
  • Addition
  • Subtraction
  • Multiplication
  • Division
• These operations, along with related concepts like exponents, roots, and factorials, are crucial for mathematical proficiency.

Addition

• Example: 10 + 4 = 14.
• Concept: Addition involves combining two or more quantities (addends) to find their total sum. You are taking one entity (e.g., the 10) and joining another entity (e.g., the 4) to it to form a new, larger total.

Subtraction

• Concept: Subtraction is the inverse operation of addition; it involves removing one quantity from another to find the difference. It answers "how much is left?" or "what is the difference?"
• Example: Removing 7 from 15 results in 8: 15 - 7 = 8. This represents the difference between 15 and 7.

Multiplication

• Example: 4 \times 5 = 20.
• Interpretation: Multiplication is essentially a shorthand for repeated addition. Instead of adding the same number multiple times, you can multiply.
  • You could write it as 4 + 4 + 4 + 4 + 4 = 20 (the number 4 added 5 times).
  • Alternatively, you could write it as 5 + 5 + 5 + 5 = 20 (the number 5 added 4 times).
• Key point: Multiplication offers a more compact and efficient way to express repeated addition.
• Note on notation: Sometimes multiplication is written with a dot (\cdot) rather than an 'x'; both denote multiplication (e.g., 4 \cdot 5 = 20).
• Another connection: Multiplication is directly tied to division as its inverse operation. If 4 \times 5 = 20, then it logically follows that 20 \div 5 = 4 and 20 \div 4 = 5.

Division

• Example: 10 \div 2 = 5.
• Concept: Division involves splitting a quantity (dividend) into equal groups, determining the size of each group (divisor), or how many groups can be made. It can answer: "How many times does one number fit into another?"
  • For instance, dividing 10 into groups of 2 yields 5 groups. Equivalently, you can subtract 2 from 10 five times until you reach 0.
• Connection to multiplication: Division is the inverse of multiplication; for example, if 10 \div 2 = 5, then 2 \times 5 = 10.
• Additional representation: Division can also be represented using fractions, which is often its more common form (e.g., \dfrac{10}{2} = 5).

Exponents (shorthand for repeated multiplication)

• Definition: An exponent indicates how many times a base number is multiplied by itself.
• Example: 3^3 = 27.
• Meaning: This means you multiply the base number 3 by itself 3 times: 3 \times 3 \times 3 = 27.
• Concept: Exponents are a powerful shorthand for repeated multiplication, similar to how multiplication is a shorthand for repeated addition.
• Inline example: 4 = 2^2. If we square 2 (meaning 2 to the power of 2), we get 4.
• Teaser: Exponent rules (e.g., for multiplication or division of powers) will be discussed in more detail later.

Factorials

• Notation: A factorial is denoted by an integer followed by an exclamation point, e.g., n!.
• Definition: The factorial of a non-negative integer n (denoted by n!) is the product of all positive integers less than or equal to n. Mathematically, n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1. Note that 0! is defined as 1.
• Example: 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720.
• Practical note: Factorials represent rapid growth; even for relatively small n, n! becomes very large very quickly.

Summary for the test: what you need to know

• The core concepts for arithmetic on tests include a strong understanding of integers, how to use the number line, and proficiency in the four basic operations.
• In addition to these, concepts such as exponents, roots, and factorials are also important.
• Remember the fundamental relationships between operations:
  • Multiplication as repeated addition.
  • Division as the inverse of multiplication (and repeated subtraction).
  • Exponents as repeated multiplication.
• Be able to recognize and differentiate between integers and non-integers from examples:
  • Integers include: \ldots, -2, -1, 0, 1, 2, 3, \ldots (whole numbers and their negatives).
  • Non-integers include fractions, decimals, and irrational numbers: \dfrac{1}{2}, -1.7, \pi, \sqrt{2}$$ (these are not integers).
• Understand the versatility of the number line, including the possibility of vertical orientation in coordinate geometry, where it still models numerical relationships.
  • The convention of reading larger numbers as being to the right (or higher) is crucial for comparing values.