Number Systems Chapter 1
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Questions and Answers

State whether every irrational number is a real number.

true

Is every point on the number line of the form m, where m is a natural number?

false

Is every real number an irrational number?

false

Are the square roots of all positive integers irrational?

<p>No, square root of 4 is a rational number.</p> Signup and view all the answers

How can the number 5 be represented on the number line?

<p>Locate the point on the number line corresponding to 5.</p> Signup and view all the answers

According to the content, what happens when the remainder when dividing rationals by a q becomes zero?

<p>The decimal expansion terminates.</p> Signup and view all the answers

When the remainder never becomes zero when dividing rationals by q, what kind of decimal expansion do we get?

<p>Non-terminating recurring decimal expansion.</p> Signup and view all the answers

Can every non-terminating recurring decimal be expressed as a rational number? True or false?

<p>True</p> Signup and view all the answers

Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

<p>Yes, 0 is a rational number and it can be written in the form of 0/1</p> Signup and view all the answers

Find six rational numbers between 3 and 4.

<p>3 1/2, 3 2/3, 3 3/4, 3 4/5, 3 5/6, 3 6/7</p> Signup and view all the answers

Find five rational numbers between 5/5 and 6/6.

<p>5 1/6, 5 2/6, 5 3/6, 5 4/6, 5 5/6</p> Signup and view all the answers

Every natural number is a whole number.

<p>True</p> Signup and view all the answers

Every integer is a whole number.

<p>False</p> Signup and view all the answers

Every rational number is a whole number.

<p>False</p> Signup and view all the answers

Is 3 a rational number?

<p>Yes</p> Signup and view all the answers

What is the Pythagoras theorem known for?

<p>The Pythagorean theorem is known for relating the lengths of the sides of a right triangle.</p> Signup and view all the answers

Define irrational numbers.

<p>A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.</p> Signup and view all the answers

Express $3.3333...$ in the form of a fraction.

<p>1/3</p> Signup and view all the answers

Show that $1.272727...$ can be expressed as $1.27$ in fractional form.

<p>14/11</p> Signup and view all the answers

Express $0.2353535...$ in the form of a fraction.

<p>233/990</p> Signup and view all the answers

What is the property of the decimal expansion of an irrational number?

<p>Non-terminating non-recurring</p> Signup and view all the answers

Find an irrational number between $1/7$ and $2/7$.

<p>0.150150015000150000...</p> Signup and view all the answers

The decimal expansion of a rational number is either terminating or ___________ recurring.

<p>non-terminating</p> Signup and view all the answers

What is (a)0 equal to?

<p>1</p> Signup and view all the answers

Let a > 0 be a real number and n a positive integer. Then a^n = b, if bn = a and b > 0. In the language of exponents, we define a^n as ____.

<p>a</p> Signup and view all the answers

Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then a^m/n = n^m a^n. This is known as the law of exponents for _____.

<p>rational numbers</p> Signup and view all the answers

Which of the following identities hold for positive real numbers a and b? Select all that apply.

<p>ab = a^b</p> Signup and view all the answers

What operation is used to rationalize the denominator of (1 / (a + b)), where a and b are integers?

<p>multiplication</p> Signup and view all the answers

What is the definition of taking square roots of real numbers?

<p>If a is a natural number, then a = b means b^2 = a and b &gt; 0. The same definition can be extended for positive real numbers.</p> Signup and view all the answers

How can x for any positive real number x be found geometrically?

<p>By marking point B on the number line such that AB = x units, marking point C such that BC = 1 unit, finding the midpoint of AC, marking it as O, drawing a semicircle with center O and radius OC, and drawing a line perpendicular to AC passing through B and intersecting the semicircle at D.</p> Signup and view all the answers

What is the radical sign used in mathematics?

<p>The symbol '√' used in expressions such as √2, 3√8, n√a, etc., is called the radical sign.</p> Signup and view all the answers

Rationalize the denominator of 1/(2+3).

<p>1/(2+3) = 2-3/13</p> Signup and view all the answers

Which identity is used to rationalize the denominator of 1/(3-5)?

<p>(a + b)(a - b)</p> Signup and view all the answers

What is the process of converting an expression to have a rational denominator called?

<p>Rationalizing the denominator.</p> Signup and view all the answers

Classify 2-√5 as ______ or ______.

<p>2-√5 is irrational.</p> Signup and view all the answers

What are the laws of exponents used for?

<p>The laws of exponents are used to simplify expressions involving powers and exponents.</p> Signup and view all the answers

Study Notes

Number Systems

Introduction to Number Systems

  • Students have learned about the number line and representation of various types of numbers on it.
  • The number line has infinitely many numbers, including natural numbers, whole numbers, integers, and rational numbers.

Natural Numbers, Whole Numbers, and Integers

  • Natural numbers are denoted by the symbol N and include numbers like 1, 2, 3, and so on.
  • Whole numbers are denoted by the symbol W and include natural numbers and 0.
  • Integers are denoted by the symbol Z and include whole numbers and negative integers.

Rational Numbers

  • Rational numbers are denoted by the symbol Q and include numbers that can be written in the form p/q, where p and q are integers and q ≠ 0.
  • Rational numbers include natural numbers, whole numbers, and integers.
  • Rational numbers do not have a unique representation in the form p/q, where p and q are integers and q ≠ 0.
  • Examples of rational numbers include 1/2, 3/4, and 22/7.

Irrational Numbers

  • Irrational numbers are numbers that cannot be written in the form p/q, where p and q are integers and q ≠ 0.
  • Examples of irrational numbers include √2, √3, and π.
  • Irrational numbers were first discovered by the Pythagoreans in ancient Greece.
  • Irrational numbers cannot be expressed as a finite decimal or fraction.

Real Numbers

  • Real numbers are denoted by the symbol R and include all rational and irrational numbers.
  • Real numbers can be represented on the number line.
  • Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number.

Decimal Expansions of Real Numbers

  • Rational numbers have a finite or recurring decimal expansion.
  • Irrational numbers have a non-recurring and non-terminating decimal expansion.
  • Examples of decimal expansions of rational numbers include 0.5, 0.333..., and 0.142857...Here are the study notes for the text:
  • Rational and Irrational Numbers*
  • The decimal expansion of a rational number is either terminating or non-terminating recurring.
  • A number with a non-terminating non-recurring decimal expansion is irrational.
  • Examples of rational numbers: 3.142678, 0.3333...
  • Examples of irrational numbers: π, √2
  • Operations on Rational Numbers*
  • Rational numbers satisfy commutative, associative, and distributive laws for addition and multiplication.
  • The sum, difference, product, and quotient of two rational numbers are always rational.
  • Operations on Irrational Numbers*
  • Irrational numbers also satisfy commutative, associative, and distributive laws for addition and multiplication.
  • The sum, difference, product, and quotient of a rational and an irrational number are always irrational.
  • Example: 2 + √3, 2√3, π - 2 are all irrational numbers.
  • Square Roots and nth Roots*
  • For any positive real number x, √x is a real number that satisfies (√x)^2 = x.
  • Similarly, for any positive real number x and positive integer n, x is a real number that satisfies (^n x)^n = x.
  • Geometric construction of square roots and nth roots is possible using circles and triangles.
  • Identities for Square Roots*
  • Some useful identities relating to square roots:
    • √ab = √a × √b
    • √(a/b) = √a / √b
    • (√a)^2 = a
    • √(a^2) = a
    • (√a + √b)(√a - √b) = a - b### Algebraic Identities
  • $(a+b)(a-b) = a^2-b^2$
  • $(a+b)^2 = a^2 + 2ab + b^2$
  • $(a+b)(c+d) = ac + ad + bc + bd$
  • $(a-b)^2 = a^2 - 2ab + b^2$

Rationalising the Denominator

  • To rationalise the denominator of $\frac{1}{a+b}$, multiply by $\frac{a-b}{a-b}$
  • To rationalise the denominator of $\frac{1}{a-b}$, multiply by $\frac{a+b}{a+b}$
  • Example: Rationalise the denominator of $\frac{1}{2+\sqrt{3}}$
    • Multiply by $\frac{2-\sqrt{3}}{2-\sqrt{3}}$
    • Result: $\frac{2-\sqrt{3}}{4-3} = 2 - \sqrt{3}$

Laws of Exponents

  • Let $a>0$ be a real number and $n$ be a natural number.
    • $a^m \cdot a^n = a^{m+n}$
    • $(a^m)^n = a^{mn}$
    • $a^m / a^n = a^{m-n}$
  • Extend the laws of exponents to negative exponents:
    • $a^{-m} = 1/a^m$
    • $a^0 = 1$
  • Let $a>0$ be a real number and $m$ and $n$ be integers.
    • $a^{m/n} = \sqrt[n]{a^m}$
    • $(a^m)^{1/n} = \sqrt[n]{a^m}$

Real Numbers

  • A number is rational if it can be expressed in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$
  • A number is irrational if it cannot be expressed in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$
  • The decimal expansion of a rational number is either terminating or non-terminating recurring
  • The decimal expansion of an irrational number is non-terminating non-recurring
  • All rational and irrational numbers make up the collection of real numbers

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Learn about number systems, including the number line and representation of various types of numbers. This chapter covers the basics of number systems, including positive directions and number lines.

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