SPM Mathematics Form 4 (Number Bases) Chapter 2 Complete Revision

Summary

Teacher Daisy's video script offers an educational exploration into number bases, starting from binary to decimal. It explains the concept of digit representation in various bases and demonstrates counting balls in base 2, 3, and 10. The script delves into place values and digit values, using examples to clarify calculations. It also covers methods for converting numbers between bases, including division by place value and base value, with examples for converting to base 5 and base 8. The educational content is complemented by practical exercises on digit value calculation and number conversion, making the lesson interactive and informative.

Takeaways

• 📚 The chapter introduces various number bases, explaining that each base uses a specific number of digits (e.g., base 2 uses 0 and 1, base 10 uses 0 to 9).
• 🧮 It demonstrates counting balls in base 2, base 3, and base 10 to illustrate how different bases represent quantities.
• 🔢 The concept of place values in number bases is explained, showing how they are calculated as repeated multiplications of the base raised to the position number.
• 📈 An example is provided to calculate the place value of digits in numbers represented in base 8 and base 2.
• 🔑 The script explains digit value as the product of a digit and its place value within a number.
• 💡 Digit values for numbers in base 8 and base 2 are calculated to show how to find the value of each digit in a number.
• 🔄 The script outlines how to determine the numerical value of a number in various bases by summing the digit values.
• 🔀 Two methods for converting numbers from one base to another are discussed: division using place value and division using base value.
• 🗂️ An example shows how to convert the decimal number 563 to base 5 and base 8 using both methods, confirming they yield the same result.
• 🔄 It explains how to convert a number from any base to base 10 and then to another base, using base 6 to base 9 as an example.
• 💻 The use of a calculator for base conversions is briefly mentioned, including how to set the calculator to different base modes.
• ➕➖ The script covers addition and subtraction in different number bases, using both vertical form and conversion to base 10 as methods.

Q & A

What are the digits used in base two?

Base two uses two digits which are zero and one.

How many digits does base ten have?

Base ten uses 10 digits which are 0 to 9.

What is the place value of each digit in the number 6231 in base eight?

The place values from right to left are eight to the power of zero, eight to the power of one, eight to the power of two, and eight to the power of three.

How do you calculate the digit value of a particular digit in a number?

The digit value is calculated by multiplying the digit by its place value.

What is the number value of the number 6231 in base eight?

The number value is calculated by summing the digit values: 6 * 512 + 2 * 64 + 3 * 8 + 1 * 1 which equals 3220.

How do you convert a base 10 number to a different base using division by place value?

You write down the place values of the target base, starting from the rightmost digit. Then you determine how many times each place value fits into the number, writing down the quotient and carrying over the remainder to the next place value.

What is the base 5 representation of the number 563 in base 10?

The base 5 representation of 563 in base 10 is 4223.

How can you convert a number from base 6 to base 10?

You multiply each digit by its corresponding place value and sum up all the products to get the base 10 number.

What is the process of converting a number from base 8 to base 2?

Each digit in base 8 is equivalent to three digits in base 2. You convert each digit into three binary digits using the values 4, 2, and 1, and then combine them.

How do you perform addition in different number bases?

You can either write the numbers vertically and add them digit by digit, carrying over as necessary, or convert the numbers to base 10, perform the addition, and then convert the result back to the original base.

Can you provide an example of subtracting numbers in base 6?

Yes, to subtract numbers in base 6, you can either perform the operation directly in base 6 by borrowing as necessary, or convert both numbers to base 10, perform the subtraction, and then convert the result back to base 6.

Outlines

00:00

📘 Introduction to Number Bases

Teacher Daisy introduces the concept of number bases in Form 4 Chapter 2. She explains that different bases use varying digits, such as base 2 with 0 and 1, base 3 with 0 to 2, and so on up to base 10 with 0 to 9. The script provides an example of counting balls in base 2, base 3, and base 10, emphasizing the use of only two digits in base 2 (0 and 1). It also explains the concept of place values, which are determined by the base raised to the power of the position number. The script concludes with an example of calculating place values for numbers in base 8 and base 2.

05:03

🔢 Understanding Digit Values

This section delves into the concept of digit values, which are calculated by multiplying a digit by its place value. The script provides examples of determining the value of an underlined digit in numbers given in base 8 and base 9. It then explains how to calculate the number value of a given number in various bases by summing the digit values. Examples are given for numbers in base 8 and base 2, with the process of converting these to base 10 values and then back to their original base.

10:03

🔄 Converting Numbers Between Bases

The script teaches two methods for converting numbers from one base to another: division using place value and division using base value. It demonstrates the conversion of the decimal number 563 to base 5 and base 8 using both methods. The process involves dividing the number by increasing powers of the target base and recording the remainders to construct the number in the new base. The script shows that both methods yield the same result.

15:05

🧮 Advanced Base Conversions and Calculations

This part of the script covers the conversion of numbers from base 6 to base 9 and from base 2 to base 8, as well as the reverse, from base 8 to base 2. It explains the process of grouping digits in threes for base 2 to base 8 conversions and vice versa, using the equivalence of three binary digits to one octal digit. The script also mentions the use of calculators for base conversions, suggesting the use of specific functions for different bases.

20:07

➕➖ Addition and Subtraction in Number Bases

The final paragraph discusses methods for performing addition and subtraction in different number bases. It introduces the use of vertical form for direct calculations and the conversion of numbers to base 10 for easier computation. Examples are provided for adding and subtracting numbers in base 2 and base 6, showing how to convert to base 10 for calculation and then back to the original base. The script concludes with a concept map for the chapter and an invitation for feedback and engagement from viewers.

Keywords

💡Number Bases

Number bases refer to the system of representing numbers using a set of digits. In the video, various bases such as binary (base 2), ternary (base 3), and decimal (base 10) are discussed. Each base uses a different number of digits, for example, binary uses 0 and 1, while decimal uses 0 to 9. The concept is central to the video's theme of teaching number systems, as it lays the foundation for understanding how numbers are represented in different bases.

💡Digit

A digit is a single symbol that represents a value in a positional numeral system. In the context of the video, digits are the building blocks of numbers in any base. For instance, binary digits are '0' and '1', and decimal digits range from '0' to '9'. Understanding digits is crucial for grasping how numbers are constructed and interpreted in different bases.

💡Place Value

Place value denotes the value of a digit based on its position within a number. The video explains that in any base, the place value is calculated by multiplying the base raised to the power of the digit's position. For example, in base 8, the rightmost digit has a place value of 8^0, the next has 8^1, and so on. This concept is essential for calculating the value of numbers in different bases.

💡Digit Value

Digit value is the numerical value of a particular digit in a number, which is found by multiplying the digit by its place value. The video uses examples such as calculating the digit values in base 8 and base 2 numbers to demonstrate how to find the value of each digit within a number. This is a key step in determining the overall value of a number in any given base.

💡Number Value

Number value is the total value of a number in a given base, calculated by summing the digit values. The script provides examples of finding the number value of base 8 and base 2 numbers by adding up the digit values. This concept is fundamental to understanding the magnitude of numbers across different bases.

💡Conversion

Conversion in the video refers to the process of changing a number from one base to another. The script describes methods for converting numbers between bases like base 2, base 5, and base 8. Conversion is a central theme as it allows understanding and comparison of numbers across different numeral systems.

💡Division Using Place Value

This method of conversion involves dividing the number by the new base and recording the remainders to construct the number in the new base. The video uses this method to convert base 10 numbers to base 5 and base 8, demonstrating how to align the number's value with the new base's place value system.

💡Division Using Base Value

Division using base value is another method for converting numbers between bases, where the number is repeatedly divided by the new base, and the remainders are used to construct the new number. The video shows this method in action when converting base 10 to base 5 and base 8, emphasizing a different approach to base conversion.

💡Calculator Computation

Calculator computation in the video refers to using a calculator's base mode to perform conversions between number bases. The script provides examples of converting binary numbers to octal and octal to binary using a calculator, illustrating a practical tool for base conversion and reinforcing the concepts taught.

💡Addition and Subtraction

The video covers how to perform addition and subtraction across different number bases. It introduces two methods: using vertical form and converting to base 10. These operations are essential for understanding arithmetic in various bases and are demonstrated with examples in binary and base 6 to show practical applications.

Highlights

Introduction to number bases and their digits

Explanation of base-2 using only zero and one

Counting balls in base-2, base-3, and base-10

Understanding place values in different bases

Calculating digit values by multiplying digit with its place value

Determining number value by summing digit values

Conversion of numbers from one base to another

Method 1 for base conversion using place value

Method 2 for base conversion using base value

Conversion of base-6 to base-10 and then to base-9

Conversion of base-2 to base-8 by grouping digits

Conversion of base-8 to base-2 by expanding each digit

Using a calculator for base conversion

Performing addition in different number bases

Performing subtraction in different number bases

Concept map for form 4 chapter 2 on number bases

Invitation to like, share, and subscribe for more educational content

Transcripts

hi i am teacher daisy

now let's learn form 4 chapter 2 number

basis

in this chapter you will learn 2.1

number bases

2.1 number bases

now let us look at the number bases and

the digit of the number base

base two will use two digits which are

zero and one

base three will use three digits which

are zero one and two

base four will use four digits which are

zero to three base 5 will use 5 digits

which are 0 to 4. base 6 will use 6

digits which are 0 to 5

followed by base 7

base 8 base 9

and base 10 will use 10 digits which are

0 to 9.

for instance how many balls are there in

the following

please give your answer in base 2 base 3

and base 10

base two only use two digits which are

zero and one

let us count together

first one then cannot have two so become

one zero

after that one one one zero zero one

zero one one one zero one one one and

lastly one zero zero zero

base three use three digits which are

zero one and two

let us count together

first one

followed by two one zero one one one two

two zero two one and lastly two two

base ten use ten digits which are digits

from zero to nine

first one

then two three four five six seven and

lastly eight

therefore the number of ball is one zero

zero zero in base two

two two in base three

and eight in base ten

the place values of a base are the

repeated multiplication of that base

raised to the power of the position

number

example

state the place value of each digit in

the numbers

a six two three one in base eight b

one one one one zero one and base two

solution a

write down the digit in base eight

six two three one

after that write down the place value

from right to left

eight to the power of zero eight to the

power of one a to the power of two and

eight to the power of three

b

write down the digit in base 2

1 1 1 1 0 1

after that write down the place value

from right to left

2 to the power of zero two to the power

of one two to the power of two two to

the power of three

two to the power of four and two to the

power of five

digit value

the value of a particular digit in the

number is the multiplication of a digit

and the place value that represents the

digit

example

state the digit value of each digit in

the numbers a six two three one in base

eight

b

one one one one zero one in base two

solution

a first write down the digit in base 8.

after that write down the place value

from right to left

next in order to find out the digit

value digit in base 8 times place value

one times one equals one

three times eight equals twenty-four

two times sixty-four equals one hundred

twenty-eight and

six times five hundred twelve equals

three thousand seventy-two

b write down the digit in base 2 then

write down the place value from right to

left

next in order to find out the digit

value digit in base 2 times place value

one times one equals one

zero times two equals zero

one times four equals four

one times eight equals eight

one times sixteen equals sixteen and

one times thirty-two equals thirty-two

example

state the value of the underlying digit

in each of the following numbers

a three four one in base eight

b five zero three seven in base nine

solution

a

write down the digit in base eight

after that write down the place value

then use the underline digit times the

place value

three times eight to the power of two

equals one hundred ninety two

therefore the value of the underlying

digit is 192.

b

write down the digit in base nine

after that write down the place value

then use the underline digit times the

place value

5 times 729

equals 3645

therefore the value of the underlined

digit is 3645.

number value

the numerical value of a number in

various bases can be determined by

calculating the sum of digit values of

the number

state the number value of each digit in

the numbers

a 6231 in base eight

b

one one one one zero one and base two

solution

a first write down the digit in base 8.

after that write down the place value

from right to left

next in order to find out the digit

value digit in base 8 times place value

then the number value is sum of all the

digit value

3072

plus 128 plus 24 plus 1 equals 3220

b first write down the digit in base 2.

after that write down the place value

from right to left

next in order to find out the digit

value digit in base 2 times place value

then the number value is sum up all the

digit value

32 plus 16 plus 8 plus 4 plus 0 plus 1

equals 61.

after learnt the number bases

now let us learn how to convert numbers

from one base to another base

there are two methods will be discussed

here

one division using place value and 2

division using base value

example

rajon river is

563 kilometers

convert

563 in base 10 to a number in

a base 5 b base 8

solution

a base 5

method 1 division using place value

first write down the place value

five to the power of zero equals one

five to the power of one equals five

five to the power of two equals

twenty-five

five to the power of 3 equals 125

and 5 to the power of 4 equals 625

after that let us check is 563

greater or equal to 625

no

thus write 0 in base 5

then is 563 greater or equal to 125

yes thus

563 divide by 125

4 times 125 equals 500

563 minus 500 will get 63.

thus write down 4 in base 5 and carry 63

to the next process

is 63 greater or equal to 25

yes

thus 63 divide by 25

2 times 25 equals 50.

63 minus 50 will get 13.

thus write down 2 in base 13 and carry

13 to the next process

is 13 greater or equal to 5

yes

thus 13 divided by 5 2 times 5 equals 10

13 minus ten will get three

thus write down two in base three

and carry three to the next process

is three greater or equal to one

yes

thus three divide by one

three times one equals three

three minus three will get zero

thus write down three in base three

therefore

563 in base 10 is equal to 4 2 2 3 and

base 5.

now let us move on to method 2 division

using base value

563 divide by 5 will get 112 with the

remainder 3.

112 divide by 5 will get 22 with the

remainder 2 22 further divide by 5 will

get 4 with the remainder 2

4 divide by 5 will get 0 with the

remainder 4.

after that give the value of the

remainder from bottom to top

therefore

563 and base 10 is equal to 4 2 2 3 and

base 5.

you can see both method will give you

the same answer

b base 8

method 1 division using place value

first write down the place value

a to the power of zero equals one a to

the power of one equals eight eight to

the power of two equals sixty-four

8 to the power of 3 equals 512

8 to the power of 4 equals 4096

after that let us check is

563 greater or equal to 4096

no

thus write 0 in base 8

then is 563 greater or equal to 512

yes

thus

563 divide by 512

1 times 512 equals 512.

563 minus 512 will get 51.

thus write down 1 in base 8

and carry 51 to the next process

is 51 greater or equal to 64

no

thus write 0 in base 8

since we never divide by anything

so carry 51 to the next process

is 51 greater or equal to 8

yes

thus 51 divide by eight

six times eight equals forty-eight

fifty-one minus forty-eight will get

three

thus write down six in base eight

and carry three to the next process

is three greater or equal to one

yes

thus three divide by one

three times one equals three

three minus three will get zero

thus write down 3 in base 8.

therefore

563 in base 10 is equal to

1063 in base 8.

now we move on to method 2 division

using base value

563 divided by 8 we'll get 70 with the

remainder 3

70 divide by eight we'll get eight with

a remainder six

eight further divide by eight we'll get

one with a remainder zero

one divide by eight will get zero with

the remainder one

after that give the value of the

remainder from bottom to top

therefore

563 in base 10 is equal to 1 0 6 3 and

base 8.

you can see both method will give you

the same answer

convert a number in a certain base to

base 10 and then to another base

a number in base p can be converted to

base 10 and then the base q

example

convert 2 5 3 and base 6 to a number in

base 9.

solution

step 1

convert base 6 to base 10

first write down the digit in base 6.

after that write down the place value

and then count the digit value and

lastly add up all the digit value and

get the number value

thus 2 5 3 and base 6 equals 105 in base

10

[Music]

step 2 convert base 10 to base 9

105 divide by 9 we'll get 11 with the

remainder 6

11 divide by nine we'll get one with the

remainder two

one divide by nine we'll get zero with

the remainder one

after that get the value of the

remainder from bottom to top therefore

105 in base 10 is equal to 1 2 6 in base

9.

convert a number in base 2 to base 8.

each digit in base 8 is equivalent to

three digits and base two

steps

one separate each of the three digits of

a number in base two from the right to

the left

arrange the digits in group of three

two determine the sum of the digit

values for the combined three digits in

base two

three combine the number in base eight

example

convert one one zero one 1 1 in base 2

to base 8

solution write down the digit in base 2.

after that write down the place value 3

in one group

then write down the digit value

add up the digit value in every group

for base eight

thus the one one zero one one one in

base two is six seven in base

eight the alternative method ways to

solve whis problem is by listing down

the number in base 2 in base 8 and then

compare the values

1 1 0 is 6

1 1 1 is 7 thus the one one zero one one

one in base two is six seven in base

eight

convert a number in base eight to base

two

each digit in base eight is equivalent

to three digits in base two

to convert a number in base eight to a

number in base two

convert each digit into three digits in

base two by using four two and one

example

convert the numbers in base eight to

numbers in base two

a five one seven in base eight b seven

two five in base eight

solution

a write down the digit in base eight

five

one seven

five is four plus zero plus one

one is zero plus zero plus one

seven is four plus two plus one

after that write down the place value

two to the power of zero equals one

two to the power of one equals two

two to the power of two is four

then repeat the place value for every

group

next write down the digit in base two

five is four plus one so below the of

the place value of four and one right

one and the rest right zero one is one

so below the of the place value of one

right one and the rest right zero

seven is four plus two plus one so below

the of the place value of four two and 1

right one

therefore 5 1 7 in base a equals 1 0 1 0

0 1 1 1 1 in base 2

b

write down the digit in base eight

seven two five

seven is four plus two plus one

two is zero plus two plus zero

five is four plus zero plus one after

that write down the place value

two to the power of zero equals one two

to the power of one equals two two to

the power of two is four then repeat the

place value for every group

next write down the digit in base two

seven is four plus two plus one so below

the of the place value of four two and

one right one and the rest right zero

two is two so below the of the place

value of two right one and the rest

right zero

five is four plus one so below the of

the place value of four and one right

one

therefore

seven two five in base eight equals one

one one zero one zero one zero one in

base two

now let us learn how to use calculator

to convert number bases

calculator computation

1.

set the calculator to the base mode by

pressing

2.

set the calculator to the desired number

system by pressing

bin for base 2 binary

deck for base 10 decimal

oct for base 8 octal

example

a convert 1 0 0 1 1 1 in base 2 into

base 8.

therefore 1 0 0 1 1 1 in base 2 equals 4

7 in base 8.

b convert one five seven in base eight

into base two

therefore one five seven in base eight

equals one one zero one one one one in

base two

addition and subtraction and number

bases using two methods

a using vertical form

write the numbers vertically when

performing addition and subtraction

b conversion of base

convert the numbers in certain base to

base 10

[Music]

addition and number bases using two

methods

example

calculate each of the following

a 1 1 0 in base 2 plus 1 1 1 in base 2

solution

a e using method one using vertical form

one one zero plus one one one

zero plus one is one

one plus one is one zero

one plus one is one zero

one zero plus one is one one

therefore the answer is one one zero one

in base two

using method two conversion of base

convert to base ten

one one zero when base two is six in

base 10

1 1 1 in base 2 is 7 in base 10

6 plus 7 equals 13 in base 10

after add up in base 10 we need to

convert back to base two

thirteen divide by two we'll get six

with the remainder one six divide by two

we'll get three with the remainder zero

three divide by two will get one with

the remainder one one divide by two

we'll get zero with the remainder one

then get the answer from bottom to top

one one zero plus one one one equals 1 1

0 1 in base 2.

you can see both answers are also the

same

subtraction and number bases using two

methods example

calculate each of the following a four

zero zero five in base six minus three

two five in base six

solution

a using method one using vertical form

four zero zero five in base six minus

three two five in base six

five minus five is zero

zero cannot minus two so need to borrow

from the left side

so zero need to borrow from four four

minus one equals three zero get six

and zero borrow from six

six minus one equals five

and six minus two equals four

five minus three equals two and write

down three

therefore the answer is three two four

zero

[Music]

using method two conversion of base

convert to base 10

four zero zero five in base six is eight

six nine in base ten

three two five in base six is one two

five in base 10

869 minus 125 in base 10 equals 744

after add up in base 10 we need to

convert back to base 6.

744 divide by 6 will get 124

with the remainder zero

124 divide by six we'll get 20 with the

remainder four 20 divide by six we'll

get three with the remainder two

three divide by six will get zero with

the remainder three

then get the answer from bottom to top

four zero zero five minus three two five

equals three two four zero in base 6

you can see both answers are the same

the concept map for form 4 chapter 2

number bases is as follow

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