Go Math Grade 8 Texas 1st Edition Solutions Chapter 1 Real Numbers
Page 7 Exercise 1 Problem 1
We divide the numerator by the denominator to convert a rational number to a decimal.
We simply convert a rational number to a decimal by converting it to the form of a fraction.
The numerator is then divided by the denominator, yielding the division’s exact value.
Because a/b is a non-terminating, non-repeating decimal, it cannot be used to represent irrational values.
In order to approximate the value of irrational numbers, students should know the perfect squares (1 to 15). , as well as square roots of numbers less than 225, are examples of irrational numbers.
In order to change a rational number to a decimal, we divide the numerator with the denominator or can be converted to a decimal by the division method.
Page 8 Exercise 2 Problem 2
Given: \(\frac{1}{8}\).
To convert fractions into decimals.
Method – We use the division method to convert fractions into decimals that means dividing the numerator by denominator.
It is given \(\frac{1}{8}\).
We have to convert fractions into a decimal.
Divide 1 by 8.
We will get
= \(\frac{1}{8}\)
\(\frac{1}{8}\) = 0.125.
The fraction \(\frac{1}{8}\) will be 0.125 in decimal.
Page 8 Exercise 3 Problem 3
Given: 2\(\frac{1}{3}\)
To convert fractions into decimals.
Method- Convert mixed fraction into an improper fraction.
It is given,2\(\frac{1}{3}\)
We have to convert fractions into a decimal.
First, we convert the mixed number to an improper fraction:
2\(\frac{1}{3}\) = 2 + \(\frac{1}{3}\)
= \(\frac{6}{3}\)+\(\frac{1}{3}\)
2\(\frac{1}{3}\) = \(\frac{7}{3}\)
To write 7/3 as a decimal, we divide the numerator by the denominator until the remainder is zero or until the digits in the quotient begin to repeat.
We add as many zeros after the decimal point in the dividend as needed.
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When a decimal has one or more digits that repeat indefinitely, we write the decimal with a bar over the repeating digit(s). In our case, 3 repeats indefinitely.
The decimal form of the given fraction is 2\(\frac{1}{3}\) = \(2 . \overline{3}\) or 2.3333
Page 8 Exercise 4 Problem 4
A positive number has two square roots because a positive number multiplied by itself is positive and a negative number multiplied by itself is also positive.
The principal square root is the non negative number that when multiplied by itself equals a.
The square root obtained using a calculator is the principal square root. The principal square root of a is written as √a.
The answer to the equation x2 = b is the square root of a number b.
It’s a number that equals b when multiplied by itself.
Every positive number b has two square roots, which are indicated by the letters √b and −√b.
The positive square root of b denoted b, is the major square root.
Page 8 Exercise 5 Problem 5
As, the number √2 be irrational because it is not an integer (2 is not a perfect square).
Any square root of any natural number that is not the square of a natural number is irrational.
Squares are integers obtained by multiplying one number by itself.
When you multiply a whole number by itself, the outcome is always another whole number, which is known as a perfect square.
As a result, perfect squares’ square roots are always whole numbers.
Page 9 Exercise 6 Problem 6
Given number: 64
To find out the two number roots of the 64
Method − For finding the square root prime factorization method.
It is given that,64
We have to find the two square roots of each number.
The positive square root and the negative square root are the two square roots of any positive number.
Therefore, 8 × 8 = 64
The positive square root of 64 is 8
While the negative square root is−8
⇒ (−8) × (−8) = 64
The two square roots of 64 is 8 and −8.
Page 9 Exercise 7 Problem 7
Given number: 100
To find out the two number roots of the 100
Method − For finding the square root prime factorization method.
It is given that, 100
We have to find the two square roots of each number.
The positive square root and the negative square root are the two square roots of any positive number.
Therefore, 10 × 10 = 100
The positive square root of 100 is 10
While the negative square root is −10
⇒ (−10) × (−10) = 100
The two square roots of 100 is 10 and −10.
Page 9 Exercise 8 Problem 8
Given: \(\frac{1}{9}\)
To find the two square root of \(\frac{1}{9}\).
Method – There must be two square roots of a positive real number, one is positive and another is the negative square root.
It is given,\(\frac{1}{9}\).
We have to find two square roots of a number.
For finding two square roots, we will take square root on both sides.
We will get
x2 = \(\frac{1}{9}\)
⇒ x = \(\sqrt{\frac{1}{9}}\)
⇒ x = ± \(\frac{1}{3}\)
= + \(\frac{1}{3}\), –\(\frac{1}{3}\).
The two square roots of \(\sqrt{\frac{1}{9}}\) will be + \(\frac{1}{3}\), –\(\frac{1}{3}\).
Page 9 Exercise 9 Problem 9
Given: Area of a square garden is 144 ft2.
To find the length of each side.
Method – By using the area formula of the square.
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It is given, area of a square garden is 144 ft2.
To find the length of each side of garden.
we will use the area of the square.
We will get
⇒ a2 = 144
⇒ a = \(\sqrt{144}\)
⇒ a = 12ft.
Each side of the square garden will be 12ft.
Page 10 Exercise 10 Problem 10
Given: \(\sqrt{2}\)
To find an estimation of \(\sqrt{2}\)
Method – Square root method
The square root of 2 or root 2 is written as√2 with a value of 1.414.
It is represented by the square root symbol.
The square root of 2 is the number which when multiplied with itself gives the result as 2. It is generally represented as √2 or 2\(\frac{1}{2}\).
The numerical value of square root 2 up to 50 decimal places is as follows:
\(\sqrt{2}\) = 1.41421356237309504880168872420969807856967187537694…
We can choose numbers with two decimal points instead of one and see in between which lies the number \(\sqrt{2}\)
Page 11 Exercise 11 Problem 11
An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal.
Instead, the numbers in the decimal would go on forever, without repeating.
The decimal numeral system is the standard system for denoting integer and non-integer numbers.
Since π is irrational, it means that its decimal representation goes on forever. It cannot be expressed as the ratio of two integers.
Page 11 Exercise 12 Problem 12
Given: The figure is img
Plot π on the number line.
Method – The number line method
The value of pi in decimal notation is about 3.14.
However, pi is an irrational number, which means that its decimal form does not terminate (such as \(\frac{1}{4}\) = 0.25) or become repetitious (such as \(\frac{1}{6}\) = 0.166666…).
As we have to plot the π on the given number line.
So, the value of π = 3.14
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Hence, the answer is
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