Number System and Integers

Number System

International System of Numeration

Comparing Numbers

Word Problems on Number Operation

Rounding of a Numbers

Roman Numerals

Integers

Properties of Integers

Solved Questions on Number System and Integers

Number System

Number Representation: Numbers can be represented in two ways - through figures (numerical symbols) and through words.

Notation: Notation refers to the method of expressing a number using numerical symbols such as digits (0-9). When we write a number using its numerical symbols, it is said to be in notation.

Example: 5000 is a notation.

Face Value and Place Value of a Digit in a Numeral: Face value represents the actual value of a digit and the place value deals with the position of the digit.

The face value and place value of digits in a numeral 79203 are shown:

Numeration: Numeration involves representing a number using words. It is the process of writing a number in its verbal form. When we express a number using words, it is said to be in numeration.

Example: Five thousand is numeration.

International System of Numeration

In the International System of Numeration, numbers are organized into periods. Each contains three place values starting from the right.

1. The first period includes ones, ten and hundred.
2. The next period has thousand, ten thousand and hundred thousand.
3. Moving further to the left, the third period includes million, ten million and hundred million.
4. This pattern continues with each subsequent period with each one consisting of three place values.

This systematic arrangement simplifies the representation of large numbers by breaking them down into manageable groups of three digits, making it easier to read and comprehend numerical values.

Comparing Numbers

When comparing numbers, there are a few fundamental rules to keep in mind.

1. Number of Digits: A number with more digits is always greater than a number with fewer digits.
   
   
2. Equal Number of Digits: When two numbers have the same number of digits, you start comparing the digits from the leftmost place (the highest place value) until you come across unequal digits. The number with the greater digit in that position will be the greater one.

Example: Let's compare 12529 and 12523.

→ These numbers have the same number of digits (five digits each).
→ We start by comparing each digit from the leftmost place.
→ At last 9 is greater than 3 and the number 12529 is greater than 12523.

Word Problems on Number Operation

1. Addition Word Problem:

Example: There was a five-day book fair. The number of books sold on the first, second, third, fourth and final days was 18854, 19812, 29050, 29174 and 30751, respectively. What is the total number of books bought in all five days?

a) 127541
b) 127641
c) 127661
d) 127681

Answer: b) 127641

Explanation: Number of books sold on the first day = 18854
Number of books sold on the second day = 19812
Number of books sold on the third day = 29050
Number of books sold on the fourth day = 29174
Number of books sold on the fifth day = 30751

Total tickets sold (bought) = 18854 + 19812 + 29050 + 29174 + 30751
                                       = 127641

2. Subtraction Word Problem:

Example: Alastair Cook and Joe Root are famous cricket players. Cook has so far scored 12780 runs and Root has scored 17280 runs in the ICC Cricket World Cup. How many more runs did Root score than Cook?

a) 4005 runs
b) 4050 runs
c) 4500 runs
d) 5400 runs

Answer: c)  4500 runs

Explanation: Difference in runs = 17280 − 12780
                                                = 4500 runs

Root scored 4500 runs more than Joe.

3. Multiplication Word Problem:

Example: A packet of Cadbury chocolates holds 30 sachets of Gems. Each sachet has 15 Gems. How many gems can be packed in 35 packets? 

a) 12550
b) 12750
c) 15550
d) 15750

Answer: d) 15750

Explanation: Number of sachet in a packet of Cadbury chocolates = 30
Number of gems in a sachet = 15 
Number of gems in 30 sachets (1 packet) = 15 ×  30
                                                             = 450
Number of gems packed in 35 packets = 450 ×  35
                                                         = 15750

4. Division Word Problem:

Example: Bell is putting 4284 eggs equally in 17 cartons. How many eggs are there in each carton?

a) 152
b) 252
c) 352
d) 452

Answer: b) 252

Explanation: Number of eggs in 17 cartons = 4284

Number of eggs in 1 carton = 4284 ÷ 17
                                         = 252

5. Mixed Operation Word Problem:

Example: Julius sold 50 cinema tickets on Monday and 35 on Tuesday. On Wednesday, he gave 3/5th of his tickets to James which were sold on Monday and Tuesday. How many does he have left?

a) 24
b) 28
c) 32
d) 34

Answer: c) 32

Explanation: Total tickets sold by Julius = Number of tickets sold on Monday and Tuesday
                                                            = (50 + 35) tickets
                                                             = 85 tickets

Number of tickets Julius gave to James on Wednesday = 3/5th × Total number of tickets sold
                                                                                = 3/5 × 85
                                                                                 = 51

Number of tickets left = 85 − 51 = 34

Rounding of a Number to Nearest Ten, Hundred, Thousands and Ten Thousand

Rules for Rounding Numbers: Here is an explanation of the rules for rounding numbers:

a. Rounding to the Nearest Ten:

→ Examine the ones digit of the given number.
→ If the ones digit is less than 5, replace it with 0 and keep the other digits unchanged.
→ If the ones digit is 5 or greater, increase the tens digit by 1 and replace the ones digit with 0.

Examples: 

Rounding 9716534 to the nearest tens results in 9716530.
Rounding 9716537 to the nearest tens results in 9716540.

b. Rounding to the Nearest Hundred:

→ Focus on the tens digit of the given number.
→ If the tens digit is less than 5, replace both the tens and ones digits with 0 and the other digits are to be the same.
→ If the tens digit is 5 or greater, increase the hundreds digit by 1 and replace all digits to the right with 0.

Examples: 

Rounding 9728637 to the nearest hundred results in 9728600.
Rounding 9728657 to the nearest hundred results in 9728700.

c. Rounding to the Nearest Thousand:

→ Examine the hundred digit of the number.
→ If the hundreds digit is less than 5, replace the hundreds, tens and ones digits with 0 while keeping the other digits as they are
→ If the hundreds digit is 5 or greater, increase the thousands digit by 1 and replace all digits to the right with 0.

Examples: 

Rounding 9728157 to the nearest hundred results in 9728000.
Rounding 9728657 to the nearest hundred results in 9729000.

Roman Numerals

Roman numerals are a numeral system that originated in ancient Rome where letters are used to represent numbers. In this system, there are seven fundamental symbols for writing numerals and there is no symbol or character for zero.

Rules for Creating Roman Numerals:

a. Repetition of Symbols for Addition:

→ When a symbol is repeated in a Roman numeral, it signifies addition.
→ The symbols I, X, C, and M can be repeated.
→ However, V, L, and D are never repeated.
→ No symbol in a Roman numeral can be repeated more than three times consecutively.

Examples:

→ II represents (1 + 1) which equals 2.
→ XX represents (10 + 10) which equals 20.
→ XXX represents (10 + 10 + 10) which equals 30.

 b. Placement of Smaller Numerals for Addition:

→ When a smaller numeral is placed to the right of a larger numeral, it is always added to the larger numeral.

Examples:

→ VI represents (5 + 1) which equals 6.
→ XII represents (10 + 2) which equals 12.
→ LX represents (50 + 10) which equals 60.

c. Placement of Smaller Numerals for Subtraction:

→ When a smaller numeral is placed to the left of a larger numeral, it is always subtracted from the larger numeral.

Examples:

→ IV represents (5 − 1) which equals 4.
→ IX represents (10 − 1) which equals 9.
→ XL represents (50 − 10) which equals 40.

d. Use of a Bar for Multiplication:

→If a horizontal bar is placed over a Roman numeral, it signifies that the value of the numeral should be multiplied by 1000 times. It is shown as:

e. Subtraction between Larger Numerals:

→ If a smaller numeral is placed between two larger numerals, it indicates subtraction from the larger numeral immediately following it.

Example:

→ XIV represents 10 + (5 − 1) which equals 14.
→ XLIV represents (50 − 10) + (5 − 1) which equals 44.
→ CXIX represents 100 + 10 + (10 − 1) which equals 119.

These rules provide a structured method for forming Roman numerals and allowing for the representation of numbers using a combination of symbols.

Integers

Integers are a fundamental set of numbers in mathematics that include whole numbers (Zero and positive numbers) and negative numbers.

Integers are used to describe quantities, positions and changes in various real-world situations such as temperature scales, sea levels, etc.

Examples:

a. The average temperatures of hot places are Dubai (+47°C) and Hong Kong (+39°C). The average temperatures of cold places are Kashmir (0°C), Toronto (−2°C) and Colorado (−5°C).

b. 11 km above sea level is +11 km and 11 km below sea level is −11 km.

Representation of Integers on a Number Line

Integers are represented on a number line where zero is the central point. Moving to the right on the number line represents positive integers while moving to the left represents negative integers. A number line is a horizontal number line,

Integers are a set of numbers which includes −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 and so on.

Negative Integers (Negative Numbers) are a set of numbers that includes  −5, −4, −3, −2, −1 and so on.

Zero is neither a positive nor a negative integer. Zero is the number that represents no value in an integer.

Positive Integers (Positive Numbers) are a set of numbers that includes 1, 2, 3, 4, 5 and so on. 

Absolute Value of an Integer

The absolute value of an integer represents its numerical value irrespective of its sign. It always gives positive value. To denote absolute value, we use a pair of vertical lines (| |).

Examples:

→ The absolute value of a is written as |a| which equals a.
The absolute value of 13 is written as |13| which equals 13.

→ The absolute value of −a is written as |−a| which equals a.
The absolute value of −13 is written as |−13| which equals 13.

→ The absolute value of 0 is written as |0| which equals 0.

Additive Inverse of an Integer

The additive inverse of an integer is another integer with the same absolute value but the opposite sign.

Examples:

→ The additive inverse of 12 is −12.
→ The additive inverse of −13 is 13.

When you add an integer to its additive inverse, you always get zero. In mathematical terms, if we consider an integer 'a' and its additive inverse is '−a', then this relationship can be expressed as: 

a + (−a) = 0

This property shows that the sum of an integer and its additive inverse is always equal to zero.

Multiplicative Inverse of an Integers

The multiplicative inverse of a number is a value that when multiplied by the original number equals 1.

For any integer, "a" the multiplicative inverse is represented as "1/a".

When you multiply "a" by its multiplicative inverse, it results in 1. This relationship is expressed as:

Fundamental Operations on Integers

There are four basic operations that can be performed with integers: addition, subtraction, multiplication and division.

Let's explore the rules for each operation:

Rules for Addition of Integers

a. When you add two integers with the same sign (both positive or both negative), you add their values together and give the sum the common sign.

Examples:

(+19) + (+10) = 19 + 10 = +29                                  
(−19) + (−10) = −19 − 10 = −29

b. To add a positive integer and a negative integer, you subtract the absolute value of the smaller number from the absolute value of the larger number and keep the sign of the larger number.

Examples:

(−19) + (+10) = −19 + 10 = −9
(+19) + (−10) = +19 − 10 = 9

c. To add the numbers, move right on a number line.

In the number line given below, 2 is added to 3 as shown in the figure.

In the number line given below, −2 is added to 3 as shown in the figure.

Rules for Subtraction of Integers

a. Subtracting one integer from another is the same as adding the additive inverse of the second integer to the first integer. Afterward, follow the rules for the addition of integers.

Examples:

(19) − (10) = 19 − 10 = 9 
(−19) −  (+10) = −19 − 10 = −29
(−19) − (−10) = −19 + 10 = −9 
(+19) − (− 10) = 19 + 10 = 29

b. To subtract the numbers, move left on a number line.

In the number line given below, 4 is subtracted from 2 as shown in the figure.

Rules for Multiplication of Integers

a. When you multiply two integers with opposite signs, you find the product of their values and give it a minus sign.

Examples:

19 × −10 = −190
−19 × 10 =  −190

b. When you multiply two integers with the same sign, you find the product of their values and give it a plus sign.

Examples: 

19 × 10 = 190
−19 × −10 = 190

Rules for Division of Integers

a. To divide one integer by another when they have the same sign, you divide their values and give the quotient a plus sign.

Examples:

190 ÷ 10 = 19
(−190) ÷ (−10) = 19

b. To divide one integer by another when they have opposite signs, you divide their values and give the quotient a minus sign.

Examples:

(−190) ÷ 10 = −19
(−190) ÷ (−10) = 19

Successor and Predecessor

A successor is a number that comes after a given number whereas a predecessor is a number that comes before a given number.

Successor of a given number = Given Number + 1

Predecessor of a given number = Given Number − 1

For example, if we consider the number 19, its successor is 20 and its predecessor is 18.

Properties of Integers

a. Closure Property of Addition, Subtraction and Multiplication: If you have two integers a and b then when you add (a + b), subtract (a − b) or multiply (a × b) them, the result is also an integer. However, it is important to note that division (a ÷ b) does not necessarily give an integer.

For example, If 9 and 2 are integers, then

Addition: 9 + 2 = 11 which is an integer. 
Subtraction: − 9 + 2 = − 7 which is an integer.
Multiplication: (− 9) × (2) = − 18 which is an integer.
Division: 9 ÷ 2 = 4.5 which is not an integer. It is a decimal number.

b. Commutative Property of Addition and Multiplication: When dealing with the addition and multiplication of integers a and b, changing the order of the numbers being added or multiplied doesn't affect the result. The commutative property is shown as:

For example, If 9 and 2 are integers, then

Addition: 9 + 2 = 2 + 9 which simplifies to 11. 
Subtraction: 9 − 2 ≠ 2 − 9 which does not have the same simplification.
Multiplication: 9 × 2 =  2 × 9 which simplifies to 18.
Division: 9 ÷ 2 ≠ 2 ÷ 9 which does not have the same simplification.

c. Associative Property of Addition and Multiplication: When you group integers a, b and c in either addition or multiplication, the order of grouping doesn't change the result.

The associative property is shown as:

For example, If 9, 2 and 5 are integers, then

Addition: (9 + 2) + 5 = 9 + (2 + 5) which simplifies to 16.
Multiplication: (9 × 2) × 5 = 9 × (2 × 5) which simplifies to 90.

d. Distributive Property of Multiplication: This property states that when you add or subtract two numbers and then multiply the result by a factor is equivalent to multiplying each number by that factor and then adding or subtracting the products.

The distributive property is shown as

For example,

If 9, 2 and 5 are integers, then

Multiplication:

9 × (2 + 5) = 9 × 2 + 9 × 5 which simplifies to 63.
9 × (2 − 5) = 9 × 2 − 9 × 5 which simplifies to −27.

These properties are fundamental in understanding how integers behave under various mathematical operations.