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