SPM Mathematics Form 4 (Number Bases) Chapter 2 Complete Revision
Summary
TLDRTeacher 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
📘 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.
🔢 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.
🔄 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.
🧮 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.
➕➖ 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.
Mindmap
Keywords
💡Number Bases
💡Digit
💡Place Value
💡Digit Value
💡Number Value
💡Conversion
💡Division Using Place Value
💡Division Using Base Value
💡Calculator Computation
💡Addition and Subtraction
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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