Number Sense

In this chapter, we are going to explore the world of numbers because numbers are so important to mathematics. We’ll be focusing on getting familiar with 7’s and 8’s digits and how to use them through different mathematical operations.

Number

A number is a numerical value that is used to represent a quantity. It is a way of counting, measuring and labelling things.

The smallest seven-digit number is 1,000,000 and the largest seven-digit number is 9,999,999.

The smallest eight-digit number is 10,000,000 and the largest eight-digit number is 99,999,999.

Natural Numbers

Natural numbers are counting numbers. They are all positive numbers that start with 1 and keep going until you get to infinity. You don't have to worry about fractions and decimals - "0" is not even a natural number! To represent natural numbers, we use a symbol called "N".

For instance: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13………………..}

Odd Number: An odd number is a number that is not divisible by 2 leaving a remainder of 1.

Examples of such numbers include 1, 3, 5, 7, 9, 11, 13, 15, etc.

Even Number: An even number is a number that is divisible by 2 without leaving a remainder.

Examples of such numbers include 2, 4, 6, 8, 10, 12, 14, 16, etc.

Whole Numbers

Any natural number that includes zero is a whole number. To represent whole numbers, we use a symbol called "W".

For instance: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13………………..}

Integers

The set of Integers is composed of the set of all counting (natural) numbers, zero and the set of negative counting numbers. To represent integers, we use a symbol called "Z or I".

For instance: Z = {....................., −3,−2, −1, 0, 1, 2, 3,……………..}

Numerals

Numerals are symbols or digits that represent numerical values. They are used to represent numerical values and to facilitate the calculation, measurement and execution of mathematical operations. In our everyday life, we commonly use Arabic numerals which are the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

For example, the numeral ‘7’ represents the number seven.

What is Place Value?

Place value is the relationship between the value of each digit of a number and its position in the number.

For example:

The expanded form of 3,879,524

= 3,000,000 + 800,000 + 70,000 + 9,000 + 500 + 20 + 4

What is Face Value?

The face value of a number is the number itself. It's like taking a picture of a number and selecting the individual numbers without thinking about where they're located.

For example, in the number 13879524, the face value of the digit "9" is just 9 because we are not thinking about where it is in the number, we are only interested in the digit itself.

International System of Numeration

In the international system of numeration, we group digits into sets of three starting from the right side. These groups are called periods and we have the ones period, thousands period and millions period.

In this system, the place values of digits follow a specific sequence: Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousand, Millions, Ten Million and so on.

Here are some key relationships:

1. 1 hundred = 10 tens
2. 1 thousand = 10 hundreds = 100 tens
3. 1 million = 1,000 thousands

The international place value chart is shown as:

Let's take the number 86,345,972 as an example to understand the place values of each digit:

2 is in the Ones.
7 is in the Tens.
9 is in the Hundred.
5 is in the Thousand.
4 is in the Ten Thousand.
3 is in the Hundred Thousand.
6 is in the Million.
8 is in the Ten Million.

Number Names of Seven and Eight-Digit Numbers

Here are the number names for both seven and eight-digit numbers:

Seven-Digit Numbers:

→ 10,000,000 - Ten million
→ 20,000,030 - Twenty million thirty
→ 30,000,047 - Thirty million forty-seven
→ 40,000,123 - Forty million one hundred twenty-three
→ 50,009,000 - Fifty million nine thousand
→ 60,049,050 - Sixty million forty-nine thousand fifty
→ 70,200,000 - Seventy million two hundred thousand
→ 80,301,000 - Eighty million three hundred one thousand
→ 90,500,070 - Ninety million five hundred thousand seventy

Eight-Digit Numbers:

→ 100,000,001 - One hundred million one
→ 200,089,000 - Two hundred million eighty-nine thousand
→ 300,006,000 - Three hundred million six thousand
→ 400,530,000 - Four hundred million five hundred thirty thousand
→ 500,700,000 - Five hundred million seven hundred thousand|
→ 600,013,000 - Six hundred million thirteen thousand
→ 700,002,000 - Seven hundred million two thousand
→ 800,000,100 - Eight hundred million one hundred
→ 900,000,058 - Nine hundred million fifty-eight

These number names are based on the standard naming conventions used. Keep in mind that when dealing with extremely large numbers, you can continue this pattern to name numbers with even more digits.

Successor and Predecessor

In mathematics, the terms successor and predecessor are analogous to the words "succeed" and "precede". 

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

For example, if we take the number 13, its successor is 14 and its predecessor is 12.

Roman Numbers

They are the numerals used in a numerical system derived from the Roman system.

The Roman numerals system is a unique way of numerating numbers which is still used to this day. This ancient numeric system is still employed today to provide a unique representation of numbers. Rather than utilizing standard Arabic numerals like 1, 2, 3, 4, and 5, Roman numerals rely on letters from the Latin alphabet.

Examples:

→ The Roman Numeral I denotes the number 1. 
→ The Roman Numeral V denotes the number 5. 
→ The Roman Numeral X denotes the number 10. 

The combination of these letters can be used to construct larger numerical values. Here is how it is done: 

→ III = 1 + 1 + 1 = 3
→ IV = 5 − 1 = 4
→ VI = 5 + 1 = 6
→ IX = 10 − 1 = 9
→ XI = 10 + 1 = 11
→ XX is 10 + 10 = 20

Let’s look at some of the common Roman numerals.

Factors

Factors are numbers obtained by dividing another number completely without leaving any remainder.

For example:

1 is a factor of 9, 3 is a factor of 9 and 9 is a factor of 9. This means that 9 has 3 factors - 1, 3 and 9.

Multiples

Multiples are numbers obtained by multiplying a particular number by another whole number.

For example:

9 is a multiple of 9, 18 is a multiple of 9, 27 is a multiple of 9, 36 is a multiple of 9 and so on. The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …………..

The multiples and factors are shown as:

Prime Factorization

Prime factorization is a way of breaking down a number into multiples of prime factors. We do this by using prime numbers.

A prime number is a natural number that can only be divisible by a factor of 1 and by the number itself. There are 25 prime numbers between 1 and 100, shown as:

Common Multiples and Common Factors

The concepts of common multiples and common factors are dealt with sets of numbers.

Common Multiples: Common multiples are numbers that are divisible by all the numbers in a given set. Let's consider the numbers 5 and 6 as our set:

→ Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60,...................
→ Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,...............................

Common multiples of 5 and 6 are 30, 60,.............................................

Common Factors: Common factors are numbers that divide evenly into all the numbers in a given set. Let's consider the numbers 10 and 12 as our set:

→ Factors of 10: 1, 2, 5, 10
→ Factors of 12: 1, 2, 3, 4, 6, 12

Common factors of 10 and 12 are: 1, 2

Highest Common Factor (HCF):

It is also known as the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD). The HCF, GCF or GCD of a set of whole numbers is the largest positive number that divides all the given numbers evenly.

In our example of 10 and 12, the HCF is 2 because it is the largest number that divides both 10 and 12 evenly.

The HCF of the given numbers is found by two methods: Division Method and Prime Factorisation Method.

Division Method to find HCF:

Here are the steps for using the division method to find the Highest Common Factor (H.C.F.):

1. Begin by dividing the larger number by the smaller number.
2. If there is a remainder after this division then proceed to divide the first divisor (the smaller number) by the remainder.
3. If the remainder divides the first divisor evenly (without a remainder), then this value is the H.C.F. of the two given numbers.
4. If the remainder does not divide the first divisor evenly, repeat the process by using the remainder as the new divisor and the original first divisor as the new dividend.

Continue these steps until you find a remainder that divides the first divisor evenly. This final dividend is the H.C.F. of the two numbers.

Prime Factorisation Method to find HCF:

Here are the steps for using the prime factorisation method to find the Highest Common Factor (H.C.F.):

1. Find the factors of the given numbers together.
2. Identify the factors that are common to both numbers.
3. Calculate the product of these common factors which is equal to the highest common factor (H.C.F.) of the given numbers.

Example: Find the HCF of 42 and 56.

a) 14
b) 28
c) 42
d) 56

Answer: a) 14

Explanation: The HCF of the given numbers is found by two methods: Division Method and Prime Factorisation Method.

Division Method:

HCF = 14

Prime Factorisation Method:

HCF = 2 × 7 = 14

Lowest or Least Common Multiple (LCM): The LCM of two or more numbers is the smallest positive number that is divisible by all the given numbers.

In our example of 5 and 6, the LCM is 30 because it is the smallest number that can be evenly divided by both 5 and 6.

These concepts of common multiples, common factors, HCF and LCM are essential in various mathematical operations and problem-solving situations.

The LCM of the given numbers is found by Division Method and Prime Factorisation Method.

Division Method to find LCM:

Here are the steps for using the division method to find the Least Common Multiple (LCM):

1. Create a table and place the given numbers horizontally.
2. Continuously divide the numbers by their common factors.
3. Keep dividing until there are no common factors left between the given numbers.
4. In the end, multiply both the common factors and the remaining values to find the L.C.M.

Prime Factorisation Method to find LCM:

Here are the steps for using the prime factorisation method to find the Least Common Multiple (LCM):

1. Find the factors of the given numbers separately.
2. Identify the common prime factors among the numbers and determine the common prime factors and uncommon factors.
3. Calculate the lowest common multiple (L.C.M.) of multiple numbers by multiplying together all the common prime factors once and then including the uncommon factors.

Example: Find the LCM of 14, 42 and 56.

a) 42
b) 56
c) 168
d) 198

Answer: c) 168

Explanation:

The LCM of the given numbers is found by two methods: Division Method and Prime Factorisation Method.

Division Method:

LCM = 2 × 7 × 1 × 3 × 4 = 168

Prime Factorisation method:

LCM = 2 × 7 × 3 × 2 × 2 = 168

Rounding up of Numbers

Rounding up is a method we use to estimate a number to the nearest value that makes sense in a given situation.

How to Round Numbers to the Nearest Tens:

Consider the digit in the one's place (the rightmost digit) and decide whether to round up or down based on it. If the one's place digit is 0 to 4, round down by leaving the tens place digit unchanged and replacing all digits to the right with 0. If the one's place digit is 5 to 9, round up by adding 1 to the tens place digit and setting all subsequent digits to 0.

For example:

How to Round Numbers to the Nearest Hundreds:

Examine the digits in the tens and ones places (the two rightmost digits) to determine whether to round up or down. If these two digits are 0 to 49, round down by maintaining the hundreds place digit as is and setting all following digits to 0. If the two rightmost digits are 50 to 99, round up by increasing the hundreds place digit by 1 and changing all subsequent digits to 0.

For example:

Example: A man invested $2640 for a month. What is his total investment in a year, rounding up to the nearest hundred?

a) $32000
b) $32700
c) $31700
d) $31000

Answer: c) $31700

Explanation: Investment in 1 month = $2640
Total Investment in 12 months (1 year) = $2640 × 12 = $31680
Total Investment in 12 months (1 year), rounding up to the nearest 100 = $31700