Lesson Notes By Weeks and Term v4 - SHS 1
REAL NUMBER SYSTEM
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REAL NUMBER SYSTEM
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Subject: Mathematics
Class: SHS 1
Term: 1st Term
Week: 2
Grade code: 1.1.1.LI.3
Strand code: 1
Sub-strand code: 1
Content standard code: 1.1.1.CS.1
Indicator code: 1.1.1.LI.3
Theme: NUMBERS FOR EVERYDAY LIFE
Subtheme: REAL NUMBER SYSTEM
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In our daily lives, from buying items at the market to calculating distances, we use numbers. Mathematics is not just about getting answers; it has a set of fundamental rules or "laws" that govern how numbers behave. These rules, known as properties, make our calculations consistent, predictable, and often much easier. For example, when a market woman adds the cost of yam (GHS 20) and cassava (GHS 10), she knows that GHS 20 + GHS 10 is the same as GHS 10 + GHS 20. She is intuitively using a property of numbers. Today, we will formally learn the names of these properties and how to use them to our advantage in mathematics and real-life situations.
This section introduces the fundamental properties of real numbers under the operations of addition (+) and multiplication (×). A. Commutative Property This property tells us that the order in which we add or multiply two numbers does not change the result. The word "commute" means to travel or move around. So, the numbers can move or swap places. For Addition: `a + b = b + a` Example: Calculating the total cost of two items. 5 Cedis + 10 Cedis = 15 Cedis 10 Cedis + 5 Cedis = 15 Cedis Therefore, `5 + 10 = 10 + 5`. For Multiplication: `a × b = b × a` Example: Finding the area of a small farm plot measuring 7 metres by 6 metres. 7 m × 6 m = 42 m² 6 m × 7 m = 42 m² Therefore, `7 × 6 = 6 × 7`. Important Note: The commutative property does NOT work for subtraction or division. `10 – 5 = 5`, but `5 – 10 = -5`. So `10 – 5 ≠ 5 – 10`. `8 ÷ 4 = 2`, but `4 ÷ 8 = 0.5`. So `8 ÷ 4 ≠ 4 ÷ 8`. B. Associative Property This property tells us that when we add or multiply three or more numbers, the way we group them (using brackets) does not change the result. The word "associate" means to group or partner up. For Addition: `(a + b) + c = a + (b + c)` Example: Adding scores from three different assignments. Let the scores be 2, 8, and 5. Grouping the first two: `(2 + 8) + 5 = 10 + 5 = 15` Grouping the last two: `2 + (8 + 5) = 2 + 13 = 15` The result is the same. For Multiplication: `(a × b) × c = a × (b × c)` Example: Finding the volume of a box with sides 2m, 3m, and 4m. Grouping the first two: `(2 × 3) × 4 = 6 × 4 = 24 m³` Grouping the last two: `2 × (3 × 4) = 2 × 12 = 24 m³` The result is the same. Important Note: The associative property also does NOT work for subtraction or division. C. Identity Property This property is about finding a special number that doesn't change another number's identity when an operation is performed. Additive Identity: The additive identity is 0. When you add 0 to any number, the number remains the same. `a + 0 = 0 + a = a` Example: `18 + 0 = 18`. If you have 18 cedis and someone gives you 0 cedis, you still have 18 cedis. Multiplicative Identity: The multiplicative identity is 1. When you multiply any number by 1, the number remains the same. `a × 1 = 1 × a = a` Example: `25 × 1 = 25`. If you have 25 bags of maize and you multiply that by 1, you still have 25 bags. D. Inverse Property The inverse property is about "undoing" an operation to get back to the identity element. Additive Inverse: The additive inverse of a number `a` is the number that, when added to `a`, gives the additive identity, 0. It is simply the negative of the number. `a + (-a) = 0` Example 1: The additive inverse of `7` is `-7`, because `7 + (-7) = 0`. Example 2: The additive inverse of `-4` is `4`, because `-4 + 4 = 0`. Multiplicative Inverse (or Reciprocal): The multiplicative inverse of a non-zero number `a` is the number that, when multiplied by `a`, gives the multiplicative identity, 1. `a × (1/a) = 1` (for a ≠ 0) Example 1: The multiplicative inverse of `5` is `1/5`, because `5 × (1/5) = 5/5 = 1`. Example 2: The multiplicative inverse of `2/3` is `3/2`, because `(2/3) × (3/2) = 6/6 = 1`. Note: The number 0 has no multiplicative inverse, because division by zero is undefined. E. Distributive Property This is a very powerful property that connects multiplication with addition or subtraction. It tells us that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. General Form: `a × (b + c) = (a × b) + (a × c)` Think of it as the number `a` being "distributed" to `b` and `c`.