The number system is a mathematical notation for expressing numbers and performing various arithmetic and algebraic operations. It is a fundamental concept in mathematics and plays a crucial role in various fields such as science, engineering, computer science and more. This chapter includes real numbers and their representation on a number line, the laws of exponents for real numbers, rational and irrational numbers and their properties.
→ Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity and are also used for counting purposes.
It is denoted by N.
Natural Number (N) = 1, 2, 3, 4, 5, 6, 7, 8, 9,...............
→ Natural numbers do not include zero, fractions, decimals or negative numbers.
→ The complete set of natural numbers along with ‘0’ are called whole numbers. It is denoted by W.
Whole Number (W) = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,...............
→ Integers are the collection of whole numbers and negative numbers.
→ Integers are numbers that can be positive, negative or zero but cannot be a fraction. It is denoted by I or Z.
Integers (I or Z) = …..….,−6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6,...........
→ Real numbers are numbers which can be represented on the number line. Every point on the number line represents a unique real number. Some of the real numbers are marked on the number line. It is denoted by R.
→ The real number is defined as the union of a set of rational numbers and the set of irrational numbers. The real numbers include natural numbers, whole numbers, integers, rational and irrational numbers.
Examples of real numbers: Real numbers (R) = 0, 1, −2, ?, −3.4, π, 9.454545, √2, 3√5, etc.
A rational number is a number if it can be written in the form of pq where p and q are integers and q ≠ 0. It is denoted by Q.
Rational number (Q) = 17
It can be represented as 17 by 1 (17/1) where p = 17 and q = 1 which is not equal to 0.
The list of all rational numbers has been listed below:
→ All the natural numbers are rational numbers.
→ All the whole numbers are rational numbers.
→ All the integers are rational numbers.
→ All the fractions are rational numbers.
→ All the terminating decimals are rational numbers.
→ All the non-terminating and repeating decimals are rational numbers.
→ The square roots of all the perfect squares are rational numbers.
→ The cube roots of all the perfect cubes are rational numbers.
An irrational number is any number that cannot be expressed in the form of pq where p and q are integers and q ≠ 0. It is denoted by P.
Irrational number (P) = √3, 1.010024563…….., e, π, etc.
Two rational numbers are equivalent if their lowest form or standard form after reduction is equal. In any rational number, its numerator and denominator are multiplied or divided by the same integer to give the equivalent rational number.
→ To represent the real numbers on the number line we use the process of successive magnification in which we visualise the numbers through a magnifying glass on the number line.
→ The number “0” represents the origin. All the positive numbers are represented on the right side of the origin whereas all the negative numbers are represented on the left side of the origin.
Step 1: The number lies between 3 and 4, so we divide it into 10 equal parts.
Step 2: Now for the first decimal place, we will mark the number between 3.6 and 3.7.
Step 3: Now, we will divide it into 10 equal parts again. The second decimal place will be between 3.66 and 3.67.
Step 4: Hence, mark 3.665 which lies between 3.66 and 3.67.
⇒ Consider a unit square OABC, with each side 1 unit in length.Then by using pythagoras theorem: OB = √(1+1) = √2
Now, transfer this square onto the number line making sure that the vertex O coincides with zero.
With O as centre & OB as radius, draw an arc, meeting OX at P. Then,
OB = OP = radius = √2. Thus, the point P represents √2 on the number line.
⇒ Now draw, BD OB such that BD joins OD such that OD = √3
With O as centre and OC as radius, draw an arc, meeting OX at Q. Then,
OQ = OD = √3. Thus, the point Q represents √3 on the real line.
The following are the properties of irrational numbers:
The addition of an irrational number and a rational number results in an irrational number.
Multiplication of any irrational number with any nonzero rational number results in an irrational number.
The addition or multiplication of two irrational numbers may be rational or irrational number.
Most common Identities for Irrational Numbers:
If p and q are real numbers, then:
→ √pq = √p√q
→ (√p + √q) (√p − √q) = p − q [(√p + √q) (√p − √q) = (√p)^{2} − (√q)^{2} = p − q]
→ (p + √q)(p − √q) = p^{2} − q
→ (√p + √q) (√r + √s) = √pr + √ps + √qr + √qs
→ (√p + √q) (√r − √s) = √pr − √ps + √qr − √qs
→ (√p + √q)^{2} = p + 2√pq + q
→ (√p − √q)^{2} = p − 2√pq + q
Laws of Exponents for Real Numbers
If a, b, m and n are real numbers, then:
→ a^{m} × a^{n} = a^{m+n}
→ (a^{m})^{n} = a^{mn}
→ a^{m} ÷ a^{n} = a^{m−n}
→ a^{n} ÷ a^{m} = a^{n−m}
→ a^{m}b^{m} = (ab)^{m}
Here, a and b are the bases and m and n are exponents.
If ‘a’ is a real number such that a > 0 and ‘p’ and ‘q’ are rational numbers, then:
→ a^{p} × a^{q} = a^{p+q}
→ (a^{p})^{q} = a^{pq}
→ a^{p} ÷ a^{q} = a^{p-q}
→ a^{p}b^{p} = (ab)^{p}
→ a^{p/q} = q√(a^{p})
Properties of Rational and Irrational Numbers
1) The sum of two rational numbers is also a rational number.
Example: ½ + ? = (3+2)/6 = ? which is rational.
2) The product of two rational numbers is also a rational number.
Example: ½ × ? = ? which is rational.
3) The sum of two irrational numbers is not always irrational.
Examples: Sum of √2 and √2 = √2 + √2 = 2√2 which is irrational.
Sum of (2 + 2√ 5) and (−2√ 5) = (2 + 2√ 5) + (−2√ 5) = 2 which is rational.
4) The product of two irrational numbers is not always irrational.
Examples: √2 × √3 = √6 which is irrational.
√2 × √2 = √4 = 2 which is rational.
Decimal representation/expansion of rational numbers is shown as:
→ A terminating decimal is defined as a decimal number that contains a finite number of digits after the decimal point.
Examples: 0.5, −9.25, 9.927, 224.9603, etc. are terminating decimals.
→ All terminating decimals are rational numbers that can be written as reduced fractions with denominators containing no prime number factors other than two or five.
If you can express the denominator of a simplified rational number in the form 2p5q or 2p or 5q, where p, q ∈ N(natural numbers), then the number has a terminating decimal expansion.
Example: 54/400 is a terminating decimal.
As, 54/400 = (27 × 2) (200 × 2) = 27/200 (Simplest form)
Here, 27 and 200 have no common factors.
Further, for denominator = 200 = 2^{3} × 5^{2}
∴ Denominator is of the form 2^{p}5^{q} , where p = 3, q = 2.
Thus, 54/400 is terminating decimals.
→ A number that is not rational is never a terminating decimal number.
→ A non-terminating decimal is defined as a decimal number that contains an infinite number of digits after the decimal point.
→ A repeating decimal or recurring decimal is a decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero.
→ It can be shown that a number is rational if and only if its decimal representation is repeating or terminating.
→ Recurring Decimal numbers are those numbers that keep on repeating the same value after a decimal point.
Examples:
1/3 = 0.33333......................................... (3 repeats forever)
1/7 = 0.142857142857142857................ (142857 repeat forever)
To display a repeating digit in a decimal number, often we put a dot or a line over the repeating digit as shown below:
For Example:
Rationalisation is a process used to eliminate an irrational number from the numerator or denominator of an algebraic fraction. The word rationalise literally means making the fraction reduce into a more effective and simpler form.
If the numerator of a mathematical expression with two terms includes irrational numbers (square root or cube root), then we need to multiply both the numerator and denominator by the conjugate of the numerator.
Example:
If the denominator of a mathematical expression with two terms includes irrational numbers (square root or cube root), then we need to multiply both numerator and denominator by the conjugate of the denominator.
Example:
CREST Olympiads has launched this initiative to provide free reading and practice material. In order to make this content more useful, we solicit your feedback.
Do share improvements at info@crestolympiads.com. Please mention the URL of the page and topic name with improvements needed. You may include screenshots, URLs of other sites, etc. which can help our Subject Experts to understand your suggestions easily.