Number System

Whole Numbers

Whole numbers are a fundamental concept in mathematics. They represent a set of numbers that includes zero and all positive counting numbers. In simpler terms, they are the numbers you use for counting objects when you can have zero objects. The set of whole numbers is often denoted by the symbol 'W'.

The set of whole numbers can be written as: W = {0, 1, 2, 3, 4, 5, ...}.

Properties of Whole Numbers:

• Closure Property: The sum of any two whole numbers is a whole number. For example, 5 + 7 = 12, and both 5, 7, and 12 are whole numbers. Similarly, the product of any two whole numbers is a whole number. For example, 6 * 4 = 24.
• Commutative Property: The order of numbers does not affect the sum or product.
  • Addition: a + b = b + a (e.g., 3 + 8 = 8 + 3 = 11)
  • Multiplication: a * b = b * a (e.g., 4 * 9 = 9 * 4 = 36)
• Associative Property: The grouping of numbers does not affect the sum or product.
  • Addition: (a + b) + c = a + (b + c) (e.g., (2 + 5) + 6 = 2 + (5 + 6) = 13)
  • Multiplication: (a * b) * c = a * (b * c) (e.g., (3 * 2) * 7 = 3 * (2 * 7) = 42)
• Distributive Property: Multiplication distributes over addition.
  • a * (b + c) = (a * b) + (a * c) (e.g., 5 * (3 + 4) = (5 * 3) + (5 * 4) = 15 + 20 = 35)
• Identity Property:
  • Additive Identity: 0 is the additive identity for whole numbers because a + 0 = a for any whole number 'a'. (e.g., 9 + 0 = 9)
  • Multiplicative Identity: 1 is the multiplicative identity for whole numbers because a * 1 = a for any whole number 'a'. (e.g., 7 * 1 = 7)
• Existence of Zero: Zero is the smallest whole number.

Operations on Whole Numbers:

• Addition: Combining two or more quantities.
• Subtraction: Finding the difference between two numbers. It's important to note that subtraction is not always closed within whole numbers (e.g., 5 - 8 is not a whole number).
• Multiplication: Repeated addition.
• Division: Repeated subtraction. Division by zero is undefined. If a whole number 'a' is divided by a non-zero whole number 'b', and the remainder is 0, then 'b' is a factor of 'a'.

Integers

Integers are a broader set of numbers that include all whole numbers, their negative counterparts, and zero. They are used to represent quantities that can be positive, negative, or zero, such as temperature, altitude, or financial balances. The set of integers is often denoted by the symbol 'Z'.

The set of integers can be written as: Z = {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.

Integers can be categorized into three types:

• Positive Integers: These are the integers greater than zero (1, 2, 3, ...). They are the same as the natural numbers or counting numbers.
• Negative Integers: These are the integers less than zero (-1, -2, -3, ...). They are represented by a minus sign (-) preceding the number.
• Zero: Zero is neither positive nor negative. It is an integer.

Number Line Representation:

Integers are often visualized on a number line. Zero is placed at the center. Positive integers are to the right of zero, and negative integers are to the left of zero. The distance from zero to any integer is its absolute value.

Properties of Integers:

• Closure Property: Integers are closed under addition and multiplication. The sum or product of any two integers is an integer. (e.g., -5 + 3 = -2; -4 * 6 = -24). Integers are also closed under subtraction (a - b is always an integer).
• Commutative Property: Holds for addition and multiplication.
  • Addition: a + b = b + a (e.g., -2 + (-7) = -7 + (-2) = -9)
  • Multiplication: a * b = b * a (e.g., -3 * 5 = 5 * (-3) = -15)
• Associative Property: Holds for addition and multiplication.
  • Addition: (a + b) + c = a + (b + c) (e.g., (-1 + 4) + (-3) = -1 + (4 + (-3)) = 0)
  • Multiplication: (a * b) * c = a * (b * c) (e.g., (-2 * 3) * (-4) = -2 * (3 * (-4)) = 24)
• Distributive Property: Holds for multiplication over addition.
  • a * (b + c) = (a * b) + (a * c) (e.g., -4 * (2 + 5) = (-4 * 2) + (-4 * 5) = -8 + (-20) = -28)
• Identity Property:
  • Additive Identity: 0 is the additive identity (a + 0 = a).
  • Multiplicative Identity: 1 is the multiplicative identity (a * 1 = a).

Operations on Integers:

• Addition:
  • Same signs: Add the numbers and keep the common sign. (e.g., -5 + (-3) = -8; 6 + 4 = 10)
  • Different signs: Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. (e.g., -8 + 5 = -3; 10 + (-4) = 6)
• Subtraction: To subtract an integer, add its additive inverse. a - b = a + (-b). (e.g., 7 - 3 = 7 + (-3) = 4; -5 - 2 = -5 + (-2) = -7; 4 - (-3) = 4 + 3 = 7)
• Multiplication:
  • Positive * Positive = Positive (e.g., 3 * 5 = 15)
  • Negative * Negative = Positive (e.g., -4 * -6 = 24)
  • Positive * Negative = Negative (e.g., 7 * -2 = -14)
  • Negative * Positive = Negative (e.g., -8 * 3 = -24)
• Division: The rules are similar to multiplication regarding signs. Division by zero is undefined. (e.g., 12 / 3 = 4; -15 / -5 = 3; 20 / -4 = -5)

Fractions

Fractions represent a part of a whole. They are written in the form of a/b, where 'a' is the numerator and 'b' is the denominator. The denominator 'b' indicates the total number of equal parts the whole is divided into, and the numerator 'a' indicates how many of those parts are being considered. The denominator cannot be zero.

Types of Fractions:

• Proper Fraction: The numerator is smaller than the denominator (e.g., 1/2, 3/4, 7/10). These fractions represent a value less than 1.
• Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 5/3, 8/8, 11/4). These fractions represent a value greater than or equal to 1.
• Mixed Fraction: A combination of a whole number and a proper fraction (e.g., 1 2/3, 3 1/4). An improper fraction can be converted to a mixed fraction and vice versa.
• Unit Fraction: A fraction with a numerator of 1 (e.g., 1/5, 1/12).
• Like Fractions: Fractions with the same denominator (e.g., 2/7, 5/7, 1/7).
• Unlike Fractions: Fractions with different denominators (e.g., 1/3, 2/5).
• Equivalent Fractions: Fractions that represent the same value, even though they have different numerators and denominators. They can be obtained by multiplying or dividing the numerator and denominator by the same non-zero number. (e.g., 1/2 = 2/4 = 3/6).

Operations on Fractions:

• Addition and Subtraction:
  1. Like Fractions: Add or subtract the numerators and keep the common denominator. (e.g., 3/5 + 1/5 = 4/5; 7/9 - 2/9 = 5/9)
  2. Unlike Fractions: First, find a common denominator (usually the Least Common Multiple or LCM of the denominators). Convert each fraction to an equivalent fraction with the common denominator, then add or subtract the numerators.

     Example: 1/3 + 1/4

     LCM of 3 and 4 is 12.

     1/3 = (1*4)/(3*4) = 4/12

     1/4 = (1*3)/(4*3) = 3/12

     So, 4/12 + 3/12 = 7/12.

• Multiplication: Multiply the numerators together and multiply the denominators together. Simplify if possible.

  (a/b) * (c/d) = (a*c) / (b*d)

  Example: 2/3 * 4/5 = (2*4) / (3*5) = 8/15.

• Division: To divide by a fraction, multiply by its reciprocal (invert the second fraction).

  (a/b) ÷ (c/d) = (a/b) * (d/c) = (a*d) / (b*c)

  Example: 3/4 ÷ 1/2 = 3/4 * 2/1 = (3*2) / (4*1) = 6/4 = 3/2.

Simplifying Fractions:

A fraction is in its simplest form (or lowest terms) when its numerator and denominator have no common factors other than 1. To simplify, divide both the numerator and the denominator by their Greatest Common Divisor (GCD).

Example: Simplify 12/18.

The GCD of 12 and 18 is 6.

12 ÷ 6 = 2

18 ÷ 6 = 3

So, 12/18 simplifies to 2/3.

Decimals

Decimals are another way to represent fractions, specifically those with denominators that are powers of 10 (like 10, 100, 1000, etc.). They use a decimal point (.) to separate the whole number part from the fractional part. The digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on.

Place Value in Decimals:

Consider the number 123.456:

• 1 is in the hundreds place.
• 2 is in the tens place.
• 3 is in the ones place.
• . is the decimal point.
• 4 is in the tenths place (4/10).
• 5 is in the hundredths place (5/100).
• 6 is in the thousandths place (6/1000).

Types of Decimals:

• Terminating Decimals: Decimals that have a finite number of digits after the decimal point. For example, 0.5 (which is 1/2), 0.75 (which is 3/4), 0.125 (which is 1/8).
• Non-terminating Decimals: Decimals that have an infinite number of digits after the decimal point. These can be further divided into:
  • Repeating Decimals: The digits after the decimal point repeat in a predictable pattern. For example, 1/3 = 0.333... (written as 0.3̅), 1/7 = 0.142857142857... (written as 0.142857̅).
  • Non-repeating Decimals: The digits after the decimal point do not follow any predictable pattern (e.g., pi ≈ 3.14159265...). These represent irrational numbers.

Converting Fractions to Decimals:

Divide the numerator by the denominator.

Example: Convert 3/8 to a decimal.

3 ÷ 8 = 0.375

Example: Convert 2/3 to a decimal.

2 ÷ 3 = 0.666... (0.6̅)

Converting Decimals to Fractions:

1. Terminating Decimals: Write the decimal as a fraction with the digits of the decimal as the numerator and a power of 10 (10, 100, 1000, etc.) as the denominator, corresponding to the number of decimal places. Then, simplify the fraction.

   Example: Convert 0.65 to a fraction.

   0.65 has two decimal places, so the denominator is 100.

   0.65 = 65/100. Simplify by dividing by 5: 13/20.

2. Repeating Decimals: This involves algebraic manipulation.

   Example: Convert 0.3̅ to a fraction.

   Let x = 0.333...

   10x = 3.333...

   Subtracting the first equation from the second: 10x - x = 3.333... - 0.333...

   9x = 3

   x = 3/9 = 1/3.

   Example: Convert 0.12̅ to a fraction.

   Let x = 0.1222...

   10x = 1.222...

   100x = 12.222...

   Subtracting 10x from 100x: 100x - 10x = 12.222... - 1.222...

   90x = 11

   x = 11/90.

Operations on Decimals:

• Addition and Subtraction: Align the decimal points vertically and add or subtract as with whole numbers, carrying over or borrowing as needed.

  Example: 12.34 + 5.6 = 17.94

  Example: 20.5 - 8.75 = 11.75

• Multiplication: Multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and place the decimal point in the result so it has that total number of decimal places.

  Example: 2.5 * 1.2 = 3.00 = 3

  (2.5 has 1 decimal place, 1.2 has 1 decimal place. Total 2 decimal places. 25 * 12 = 300. Place decimal 2 places from the right: 3.00)

• Division:
  • Dividing by a whole number: Divide as usual, placing the decimal point in the quotient directly above the decimal point in the dividend.
  • Dividing by a decimal: Move the decimal point in the divisor to the right until it becomes a whole number. Move the decimal point in the dividend the same number of places to the right. Then, divide as usual.

    Example: 6.4 ÷ 0.8

    Move decimal 1 place right in both: 64 ÷ 8 = 8.

    Example: 1.25 ÷ 0.5

    Move decimal 1 place right in both: 12.5 ÷ 5 = 2.5.