About Properties of Real Numbers:
When we work with real numbers, there are a few basic properties that we must know. First and foremost, the commutative property tells us that we can add or multiply in any order without changing our answer. Additionally, we have the associative property, which tells us we can group our addition or multiplication of three or more numbers in any way and not change the answer. The identity properties give us special properties for zero and one. We know that zero is the additive identity and one is the multiplicative identity. If we add zero to a number or if we multiply a number by one, the number will be unchanged. The inverse properties give us special properties that allow us to achieve a product of 1 or a sum of zero. If we add a number and its opposite (additive inverse) the result is always zero. If we multiply a non-zero number by its reciprocal, the result is always 1. Lastly, we will think about the distributive property which tells us that multiplication is distributive over addition or subtraction.
Test Objectives
- Demonstrate an understanding of the commutative property
- Demonstrate an understanding of the associative property
- Demonstrate an understanding of the inverse properties
- Demonstrate the ability to rewrite a product as a sum using the distributive property
- Demonstrate the ability to rewrite a sum as a product using the distributive property
#1:
Instructions: Rewrite each using the commutative property.
$$a)\hspace{.2em}6 + 5$$
$$b)\hspace{.2em}(-2) \cdot (-8)$$
$$c)\hspace{.2em}x + y$$
$$d)\hspace{.2em}a \cdot b$$
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#2:
Instructions: Rewrite each using the associative property.
$$a)\hspace{.2em}(4 + 9) + 3$$
$$b)\hspace{.2em}2(7 \cdot 5)$$
$$c)\hspace{.2em}(x + y) + z$$
$$d)\hspace{.2em}a(b \cdot c)$$
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#3:
Instructions: Find the number required to obtain a 0 through addition and a 1 through multiplication.
$$a)\hspace{.2em}{-12}$$
$$b)\hspace{.2em}\frac{1}{5}$$
$$c)\hspace{.2em}x$$
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#4:
Instructions: Rewrite each product as a sum.
$$a)\hspace{.2em}{-3}(x - y)$$
$$b)\hspace{.2em}6(x^2 - 2y + z)$$
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#5:
Instructions: Rewrite each sum as a product.
$$a)\hspace{.2em}8x - 4xy$$
$$b)\hspace{.2em}2x^2 + 10xz$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}6 + 5=5 + 6$$
$$b)\hspace{.2em}(-2) \cdot (-8)=(-8) \cdot (-2)$$
$$c)\hspace{.2em}x + y=y + x$$
$$d)\hspace{.2em}a \cdot b=b \cdot a$$
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#2:
Solutions:
$$a)\hspace{.2em}(4 + 9) + 3=4 + (9 + 3)$$
$$b)\hspace{.2em}2(7 \cdot 5)=(2 \cdot 7) \cdot 5$$
$$c)\hspace{.2em}(x + y) + z=x + (y + z)$$
$$d)\hspace{.2em}a(b \cdot c)=(a \cdot b) \cdot c$$
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#3:
Solutions:
$$a)\hspace{.2em}12, -\frac{1}{12}$$
$$b)\hspace{.2em}-\frac{1}{5}, 5$$
$$c)\hspace{.2em}{-x}, \frac{1}{x}, x \ne 0$$
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#4:
Solutions:
$$a)\hspace{.2em}{-3}x + 3y$$
$$b)\hspace{.2em}6x^2 - 12y + 6z$$
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#5:
Solutions:
$$a)\hspace{.2em}4x(2-y)$$
$$b)\hspace{.2em}2x(x + 5z)$$
