Understanding 0.9 Repeating Equals 1

The video tutorial explains why 0.9 repeating is equal to 1 using three methods: fractions, equations, and infinite geometric series. It begins by showing that 1/3 as a decimal is 0.3 repeating, and thus 3/3 equals 0.9 repeating, which is 1. The second method uses algebraic equations to demonstrate that if x equals 0.9 repeating, then 10x equals 9.9 repeating, leading to x equaling 1. The final method involves expressing 0.9 repeating as an infinite geometric series and calculating its sum, which also equals 1. Each method reinforces the concept that 0.9 repeating is mathematically equivalent to 1.

What is the main idea introduced about 0.9 repeating in the video?

• It is less than 1.

• It is greater than 1.

• It is a negative number.

• It is equal to 1.

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CCSS.7.NS.A.2D

CCSS.8.NS.A.1

How is the fraction 1/3 related to the concept of 0.9 repeating?

• 1/3 is greater than 0.9 repeating.

• 1/3 is unrelated to 0.9 repeating.

• 1/3 is used to show that 0.9 repeating equals 1.

• 1/3 is equal to 0.9 repeating.

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CCSS.7.NS.A.2D

CCSS.8.NS.A.1

What is the decimal representation of 1/3?

• 0.333...

• 0.3333

• 0.33

• 0.3

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CCSS.7.NS.A.2D

CCSS.8.NS.A.1

What equation is set up to prove that 0.9 repeating equals 1?

• x = 0.99

• x = 1.9 repeating

• x = 0.9 repeating

• x = 1.0

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CCSS.7.NS.A.2D

CCSS.8.NS.A.1

In the algebraic proof, what operation is performed after setting x equal to 0.9 repeating?

• Division

• Addition

• Multiplication

• Subtraction

What mathematical concept is used in the third proof to show 0.9 repeating equals 1?

• Arithmetic series

• Infinite geometric series

• Harmonic series

• Finite series

What is the common ratio (r) in the infinite geometric series used in the proof?

• 1/3

• 1/10

• 1/100

• 1/2

What is the formula used to find the sum of an infinite geometric series?

• a sub 1 minus 1 minus r

• a sub 1 times 1 minus r

• a sub 1 divided by 1 minus r

• a sub 1 plus 1 minus r

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CCSS.6.NS.C.6C

What is the result of the infinite sum of the series representing 0.9 repeating?

• 1

• 0.99

• 0.9

• 1.1

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CCSS.6.NS.C.7A

CCSS.6.NS.C.7B

What is the final conclusion about 0.9 repeating?

• It is greater than 1.

• It is equal to 1.

• It is not a number.

• It is less than 1.

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CCSS.6.NS.C.7A

CCSS.6.NS.C.7B