Introduction to Rational Numbers Practice answers
1. True. A rational number is a number that can be written as the quotient of two integers, where the denominator is not zero. For example, 2/3 and -5/2 are rational numbers.
2. False. A rational number can be both positive and negative. For example, -1 is a rational number.
3. True. The decimal form of a rational number either terminates after a certain number of digits or repeats a pattern of digits. For example, 1/4 can be written as 0.25, which terminates, while 1/3 can be written as 0.333..., which repeats the digit 3.
4. False. Irrational numbers cannot be expressed as the quotient of two integers, whereas rational numbers can. For example, the square root of 2 (√2) is an irrational number.
5. True. The sum of two rational numbers is always a rational number. For example, 1/2 + 3/4 = 5/4, which is a rational number.
6. False. The product of two rational numbers is always a rational number. For example, 1/2 * 2/3 = 1/3, which is a rational number.
7. True. The reciprocal of a rational number is also a rational number. For example, the reciprocal of 1/2 is 2/1, which is a rational number.
8. False. The additive inverse of a rational number is also rational. For example, the additive inverse of 3/4 is -3/4, which is a rational number.
9. True. A rational number can be expressed in different forms, such as fractions and decimals. For example, 3/2 can also be written as 1.5 in decimal form.
10. False. All integers are rational numbers. For example, the number 5 is an integer and also a rational number.
Sure! Here are the step-by-step answers to a practice exercise on the introduction to rational numbers.
Question 1: Identify whether the following numbers are rational or irrational.
a) 0.25
b) √3
c) -5
d) 1.333...
Step-by-step answers:
a) To determine if 0.25 is rational or irrational, we need to check if it can be expressed as a fraction. 0.25 can be written as 25/100, which reduces to 1/4. Since it can be expressed as a fraction, 0.25 is rational.
b) To determine if √3 is rational or irrational, we need to check if it can be expressed as a fraction. Since √3 cannot be expressed as a fraction, it is irrational.
c) To determine if -5 is rational or irrational, we need to check if it can be expressed as a fraction. -5 can be written as -5/1, which is a fraction. Therefore, -5 is rational.
d) To determine if 1.333... (repeating decimal) is rational or irrational, we need to convert it into a fraction. Let x = 1.333...
Multiplying x by 10 gives 10x = 13.333...
Subtracting both equations, we get 10x - x = 13.333... - 1.333...
This simplifies to 9x = 12
Dividing both sides by 9, we get x = 4/3
Since 1.333... can be expressed as a fraction 4/3, it is rational.
Therefore, the answers are:
a) Rational
b) Irrational
c) Rational
d) Rational
Question 2: Express the following rational numbers in decimal form.
a) 3/4
b) -2/5
c) 5/8
Step-by-step answers:
a) To express 3/4 as a decimal, divide 3 by 4.
3 ÷ 4 = 0.75
Therefore, 3/4 in decimal form is 0.75.
b) To express -2/5 as a decimal, divide -2 by 5.
-2 ÷ 5 = -0.4
Therefore, -2/5 in decimal form is -0.4.
c) To express 5/8 as a decimal, divide 5 by 8.
5 ÷ 8 = 0.625
Therefore, 5/8 in decimal form is 0.625.
Therefore, the answers are:
a) 0.75
b) -0.4
c) 0.625
I hope this helps! Let me know if you have any further questions.