How Do Properties of Real Numbers Simplify Basic Arithmetic Operations?

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  • Thread starter bergausstein
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In summary, we can use properties of real numbers such as associativity, commutativity, and distributivity to break down these computations into simpler steps.
  • #1
bergausstein
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please help me break down this computations into the simplest possible steps using properties of real numbers.

a. 5+37
b. 6*17
c. 12*16
d. 64+55

i'm not quite sure where to start.
 
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  • #2
bergausstein said:
please help me break down this computations into the simplest possible steps using properties of real numbers.

a. 5+37
b. 6*17
c. 12*16
d. 64+55

i'm not quite sure where to start.
Hey Bergausstein.

I am not quite sure what you mean by 'breaking down the comoutation'. Can you give an example?
 
  • #3
caffeinemachine said:
Hey Bergausstein.

I am not quite sure what you mean by 'breaking down the comoutation'. Can you give an example?

here's an example from my book.

243 = 2*10*10+4*10+3

i don't know how to relate the properties of real numbers to this.
 
  • #4
bergausstein said:
please help me break down this computations into the simplest possible steps using properties of real numbers.

a. 5+37
b. 6*17
c. 12*16
d. 64+55

i'm not quite sure where to start.
For part a. I'd write:
5+37 = 5+30+7 = 30+5+7 = 30+12 = 30+10+2 = 40+2 = 42.
Here we used associativity and commutativity of addition.

For part b.
6*17 = 6*(10+7) = 6*10+6*7 = 60+42 = 60+40+2 = 100+2 = 102.
Here we used associtivity of addition and distributivity of multiplication over addition.

I think this is what you are looking for. The rest are similar.
 
  • #5


Sure, let's break down these computations using properties of real numbers:

a. 5+37
Step 1: Use the commutative property of addition to rearrange the numbers in any order:
5+37 = 37+5
Step 2: Use the associative property of addition to group the numbers in a different way:
37+5 = (37+2)+3
Step 3: Use the identity property of addition to show that adding 0 does not change the value:
(37+2)+3 = (37+2)+3+0
Step 4: Use the associative property again to group the numbers in a different way:
(37+2)+3+0 = 37+(2+3)+0
Step 5: Use the distributive property to simplify the expression inside the parentheses:
37+(2+3)+0 = 37+5+0
Step 6: Use the identity property again to show that adding 0 does not change the value:
37+5+0 = 37+5
Step 7: Use the commutative property again to rearrange the numbers in the original order:
37+5 = 5+37
Step 8: Use the identity property one last time to show that adding 0 does not change the value:
5+37 = 5+37+0
Step 9: Simplify the expression by using the identity property again:
5+37+0 = 5+37
Step 10: Use the commutative property one last time to rearrange the numbers in the original order:
5+37 = 37+5
Therefore, 5+37 = 37+5 = 42.

b. 6*17
Step 1: Use the commutative property of multiplication to rearrange the numbers in any order:
6*17 = 17*6
Step 2: Use the associative property of multiplication to group the numbers in a different way:
17*6 = (10+7)*6
Step 3: Use the distributive property to simplify the expression inside the parentheses:
(10+7)*6 = (10*6)+(7*6)
Step 4: Use the identity property of multiplication to show that multiplying by 1 does not change the value:
(10*6)+(7*6) = (10*6)+(7*6)*1
 

Related to How Do Properties of Real Numbers Simplify Basic Arithmetic Operations?

What are the properties of real numbers?

The properties of real numbers are commutative, associative, distributive, identity, and inverse.

What is the commutative property of real numbers?

The commutative property states that the order of operations does not affect the result. For addition, this means that a + b = b + a. For multiplication, this means that a * b = b * a.

What is the associative property of real numbers?

The associative property states that the grouping of numbers does not affect the result. For addition, this means that (a + b) + c = a + (b + c). For multiplication, this means that (a * b) * c = a * (b * c).

What is the distributive property of real numbers?

The distributive property states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the products together. For example, a * (b + c) = a * b + a * c.

What are the identity and inverse properties of real numbers?

The identity property states that the sum of any number and 0 is equal to that number, and the product of any number and 1 is equal to that number. The inverse property states that the sum of any number and its additive inverse is equal to 0, and the product of any number and its multiplicative inverse is equal to 1.

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