Mathematics Concepts: Rational Numbers & Angles

Rational Numbers

• A rational number is any number that can be written as a fraction, where both the numerator and denominator are integers
• Example: -3/7 is a negative rational number
• Example: 9/83 is a positive rational number
• Example: 5/-8 is a negative rational number

Complementary Angles

• Complementary angles are two angles that add up to 90 degrees
• Example: 51 degrees and 39 degrees are complementary angles because 51 + 39 = 90

Decimal Places

• The number of decimal places in a product is equal to the total number of decimal places in the numbers being multiplied
• Example: 99.02 has two decimal places, and 6.3 has one decimal place. The product will have three decimal places

Improper Fractions

• An improper fraction is a fraction where the numerator is greater than or equal to the denominator
• Example: 16/8 is an improper fraction because 16 is greater than 8

Associative Property

• The associative property states that the way numbers are grouped in addition or multiplication does not affect the result
• Example: -5 + (9 + -6) = (-5 + 9) + -6

Absolute Value

• The absolute value of a number is its distance from zero, regardless of direction
• Example: | -17| = 17 and |-4| = 4

Linear Pair

• A linear pair is a pair of adjacent angles that have a common side and whose non-common sides are opposite rays.
• Example: The angles BAG and BAT are a linear pair, and since the angles in a linear pair add up to 180 degrees, BAT = 180 - BAG = 180 - 55 = 125 degrees.

Reciprocal

• To find the reciprocal of a number, divide 1 by that number
• Example: The reciprocal of 5/6 is 6/5

Number Line

• A number line is a line that represents numbers as points
• The further to the right that a point is on the number line, the larger the number represented by that point
• The further to the left that a point is on the number line, the smaller the number represented by that point
• Example: -5/8 can be represented on the number line by placing a point located 5/8 of the distance between -1 and 0

Adjacent Angles

• Adjacent angles are two angles that share a common vertex and a common side.
• The two common sides of adjacent angles form a straight line
• Example: In a quadrilateral, two angles sharing a side are adjacent

Equivalent Fractions

• Equivalent fractions represent the same value, even though they have different numerators and denominators.
• Example: 1/2, 2/4, 3/6, 6/12, 9/18 are all equivalent fractions.

Ascending and Descending Order

• Ascending order means arranging from smallest to largest
• Descending order means arranging from largest to smallest

Adding Fractions

• To add fractions with the same denominator, add the numerators and keep the same denominator
• Example: 8/2 + 3/2 = (8 + 3)/2 = 11/2

Subtracting Decimals

• Line up the decimals and subtract vertically, borrowing if necessary

Dividing Decimals

• Convert the divisor to a whole number by moving the decimal point as many places to the right as necessary
• Move the decimal point in the dividend the same number of places to the right

Ratio

• A ratio is a comparison of two quantities by division
• Example: If the ratio of two supplementary angles is 5:7, then the angles are 5x and 7x. We know that the sum of supplementary angles is 180 degrees. So, 5x + 7x = 180, or x = 15. So, the angles are 515 = 75 degrees and 715 = 105 degrees

Product and Quotient

• The product of two numbers is the result of their multiplication.
• The quotient of two numbers is the result of their division

Case Study - Decimals

• The sum of 5.76 (decimal places: two) and 5.2 (decimal places: one) is 10.96 (decimal places: two).
• 5.1 (decimal places: one) times 2.4 (decimal places: one) is 12.24 (decimal places: two).
• 1.5 (decimal places: one) times 6.2 (decimal places: one) is 9.3 (decimal places: one).
• Since 12.24 is larger than 9.3, then 5.1 × 2.4 is greater than 1.5 × 6.2
• To find 46 × 0.093, we can write 0.093 as 93/1000. So, 46 x 0.093 = 46 x 93/1000. Since we know that 46 x 93 = 4278, then 46 x 0.093 = 4278/1000 = 4.278

This quiz covers essential mathematics concepts including rational numbers, complementary angles, decimal places, improper fractions, and the associative property. Test your understanding and enhance your skills in these foundational topics. Perfect for students looking to solidify their math knowledge.