1.3 names and addition

Names of Numbers

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million (six 0’s)1,000 = one thousand (three 0’s)

1,000,000,000 = one billion1,000,000,000,000 = one trillion

(nine 0’s) (twelve 0’s)

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million (six 0’s)1,000 = one thousand (three 0’s)

1,000,000,000 = one billion1,000,000,000,000 = one trillion

(nine 0’s) (twelve 0’s)

Hence 3,054,208 is

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million1,000,000,000 = one billion1,000,000,000,000 = one trillion

(six 0’s) (nine 0’s) (twelve 0’s)

1,000 = one thousand

Hence 3,054,208 is “three million

(three 0’s)

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million1,000,000,000 = one billion1,000,000,000,000 = one trillion

(six 0’s) (nine 0’s) (twelve 0’s)

1,000 = one thousand

Hence 3,054,208 is “three million fifty four thousand

(three 0’s)

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million1,000,000,000 = one billion1,000,000,000,000 = one trillion

(six 0’s) (nine 0’s) (twelve 0’s)

1,000 = one thousand

Hence 3,054,208 is “three million fifty four thousand two hundred and eight.”

(three 0’s)

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million1,000,000,000 = one billion1,000,000,000,000 = one trillion

(six 0’s) (nine 0’s) (twelve 0’s)

1,000 = one thousand

Hence 3,054,208 is “three million fifty four thousand two hundred and eight.”

(three 0’s)

The number 40 is 10 times as much as 4 since the 0 shifted the 4 to a higher value slot.

For conveniences, we name some of the larger units in our decimal (base 10) system as shown.

Names of Numbers

1’s10’s100’s1,000’s10,000’s

ones tens hundredthousandten–thousand

100,000’s1,000,000’s

hundred–thousandmillion

1,000,000 = one million1,000,000,000 = one billion1,000,000,000,000 = one trillion

(six 0’s) (nine 0’s) (twelve 0’s)

1,000 = one thousand

Hence 3,054,208 is “three million fifty four thousand two hundred and eight.”

(three 0’s)

The number 40 is 10 times as much as 4 since the 0 shifted the 4 to a higher value slot. But the number 04 is the same as 4 since the 0 to the left indicating an empty slot so it’s valueless.

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

5

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

6

1

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

6

1

9,So the sum is 9,635.

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

vs

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

vs

The Mayan symbols visually show us that

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

vs

The Mayan symbols visually show us that so it’s easier to “see” and memorize adding digits with Mayan symbols:

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

vs

The Mayan symbols visually show us that

5 + 5 = 10

+ =

so it’s easier to “see” and memorize adding digits with Mayan symbols:

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

5 + 6 = 5 + 5 + 1= 11

+ =

vs

The Mayan symbols visually show us that so it’s easier to “see” and memorize adding digits with Mayan symbols:

5 + 5 = 10

+ =

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

5 + 6 = 5 + 5 + 1= 11

+ = + =

vs

The Mayan symbols visually show us that

5 + 7 = 5 + 5 + 2= 12

so it’s easier to “see” and memorize adding digits with Mayan symbols:

5 + 5 = 10

+ =

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

5 + 6 = 5 + 5 + 1= 11

5 + 8 = 5 + 5 + 3= 13

+ = + = + = + =

vs

The Mayan symbols visually show us that

5 + 7 = 5 + 5 + 2= 12

5 + 9 = 5 + 5 + 4= 14

so it’s easier to “see” and memorize adding digits with Mayan symbols:

5 + 5 = 10

+ =

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

5 + 6 = 5 + 5 + 1= 11

5 + 8 = 5 + 5 + 3= 13

+ = + = + = + =

vs

The Mayan symbols visually show us that

5 + 7 = 5 + 5 + 2= 12

5 + 9 = 5 + 5 + 4= 14

6 + 6 = 5 + 5 + 2= 12

6 + 8 = 5 + 5 + 4= 14

+ = + = + = + =6 + 7 = 5 + 5 + 3= 13

6 + 9 = 5 + 5 + 5= 15

so it’s easier to “see” and memorize adding digits with Mayan symbols:

5 + 5 = 10

+ =

Visually, the Mayan numerals reveal many basic addition relations that Arabic numerals do not.

Addition

6=5+1, 7=5+2, 8=5+3 and 9=5+4

5 + 6 = 5 + 5 + 1= 11

5 + 8 = 5 + 5 + 3= 13

+ = + = + = + =

vs

The Mayan symbols visually show us that

5 + 7 = 5 + 5 + 2= 12

5 + 9 = 5 + 5 + 4= 14

6 + 6 = 5 + 5 + 2= 12

6 + 8 = 5 + 5 + 4= 14

+ = + = + = + =6 + 7 = 5 + 5 + 3= 13

6 + 9 = 5 + 5 + 5= 15

Tables for the addition of 7, 8 and 9 are on the next slide.

so it’s easier to “see” and memorize adding digits with Mayan symbols:

5 + 5 = 10

+ =

Addition

7 + 7 = 5 + 5 + 2 + 2= 14

7 + 9 = 5 + 5 + 2 + 4= 16

+ = + = + =

7 + 8 = 5 + 5 + 2 + 3= 15

8 + 8 = 5 + 5 + 3 + 3= 16

+ += =

8 + 9 = 5 + 5 + 3 + 4= 17

+ =

9 + 9 = 5 + 5 + 4 + 4= 18

(The Mayan addition table of 7,8 and 9)

Addition

7 + 7 = 5 + 5 + 2 + 2= 14

7 + 9 = 5 + 5 + 2 + 4= 16

+ = + = + =

7 + 8 = 5 + 5 + 2 + 3= 15

8 + 8 = 5 + 5 + 3 + 3= 16

(The Mayan addition table of 7,8 and 9)

+ += =

8 + 9 = 5 + 5 + 3 + 4= 17

+ =

9 + 9 = 5 + 5 + 4 + 4= 18

Note the pattern of adding 9 with another nonzero digit. To get the answer, we reduce the digit by 1 and carry.

9 + 1 = 10

9 + 2 = 11

9 + 3 = 12

9 + 4 = 13

9 + 5 = 14

9 + 6 = 15

9 + 7 = 16

9 + 8 = 17

9 + 9 = 18

Addition

+

Addition

+

Addition

+ +

Addition

+=

+

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

+ +

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

+ +

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

+ + ++

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

+ +

=

++

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

If we are gathering three piles of apples, it does not matter which two piles we group together first,

+ +

=

++

i.e. (A + B) + C = A + (B + C) where the “( )” means “do first.”

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

If we are gathering three piles of apples, it does not matter which two piles we group together first,

+ +

We say that “addition is associative.”

=

++

i.e. (A + B) + C = A + (B + C) where the “( )” means “do first.”

In general, if A and B are two numbers, then A + B = B + A. and we say that “addition is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

If we are gathering three piles of apples, it does not matter which two piles we group together first,

+ +

We say that “addition is associative.”

=

++

i.e. (A + B) + C = A + (B + C) where the “( )” means “do first.”

The addition operation being commutative and associative allows us to add multiple numbers in any order and we can take advantage of that.

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

then sum the rest of digits.

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

then sum the rest of digits.

8

3

7

2

4

+

Example A. a. Calculate.

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

61 2

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

1 + 4 = 2 + 3 = 5

then sum the rest of digits.

8

3

7

2

4

+

Example A. a. Calculate.

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

61 2

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10 then sum the rest of digits.

8

3

7

2

4

+

Example A. a. Calculate.

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

61 2

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

10 10

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

10 10

24

b. Calculate.

3 8

+

2 3

51 7

3 2

1 1

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

10 10

or24

b. Calculate.

3 8

+

2 3

51 7

3 2

1 1

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

10

or24

b. Calculate.

3 8

+

2 3

51 7

3 2

1 1

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

20

10

24

or24

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

20

10

24

or24

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

10

10

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

20

10

24

or24

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

10

10

6

2

total 26, carry the 2

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

20

10

24

or24

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

10

10

2

5

5

total 26, carry the 26

To add multiple numbers, we may first collect those digits that sum to 5, 10 or 15 as shown here:

Addition

8

3

7

2

4

2 + 3 + 4 + 7 + 8 =

+

Example A. a. Calculate.

1 + 4 = 2 + 3 = 5 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10

6 + 9 = 7 + 8 = 15

then sum the rest of digits.

5

10

15

20

10

24

or24

3 8

+

2 3

51 7

3 2

1 1

b. Calculate.

10

10

total 26, carry the 26

2

5

5

1 2

AdditionGiven a number, the sum of all its digits is called the “digit sum” of that number.

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

Given a number, the sum of all its digits is called the “digit sum” of that number.

35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.

10

35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.

Hence we must have 10 + X + X = 16.10

35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.

Hence we must have 10 + X + X = 16.10

Therefore X + X = 6 or X = 3 and the number must be 3,352,003

35,328,792 is

For example, the digit sum of

Addition

3 + 5 + 3 + 2 + 8 + 7 + 9 + 2

10 1010

Example B. a. The digit sum of the number 3,X52,00X is 16 where X is an unknown digit, what is the number?

We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

Given a number, the sum of all its digits is called the “digit sum” of that number.

= 39.

Hence we must have 10 + X + X = 16.10

Therefore X + X = 6 or X = 3 and the number must be 3,352,003

35,328,792 is

Qn: Is it possible for the digit sum of the number 3,X52,00X to be 15? Explain.

For example, the digit sum of

Additionb. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

Additionb. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Addition

15

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Addition

Hence we must have X + Y = 1 and X and Y must consist of a “0” and a “1”

15

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Addition

Hence we must have X + Y = 1 and X and Y must consist of a “0” and a “1”

15

So the number must be 20,581 or 21,580.

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Addition

Hence we must have X + Y = 1 and X and Y must consist of a “0” and a “1”

15

So the number must be 20,581 or 21,580.

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Digit sums is one of the basic procedures used in a computer program designed for checking to see if the transmitted data is corrupted, i.e. transmitted incorrectly. We will use this sum later on when we address the division operation.

Addition

Hence we must have X + Y = 1 and X and Y must consist of a “0” and a “1”

15

So the number must be 20,581 or 21,580.

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Digit sums is one of the basic procedures used in a computer program designed for checking to see if the transmitted data is corrupted, i.e. transmitted incorrectly. We will use this sum later on when we address the division operation.

We call “0” the additive identity because x + 0 = 0 + x = x. i.e. when 0 is added with another value x, we get back x.

Addition

Hence we must have X + Y = 1 and X and Y must consist of a “0” and a “1”

15

So the number must be 20,581 or 21,580.

b. The digit sum of the number 2X,58Y is 16 where X and Y are two possibly different digits, what could the number be?

We have that 2 + X + 5 + 8 + Y = 16.

Digit sums is one of the basic procedures used in a computer program designed for checking to see if the transmitted data is corrupted, i.e. transmitted incorrectly. We will use this sum later on when we address the division operation.

We call “0” the additive identity because x + 0 = 0 + x = x. i.e. when 0 is added with another value x, we get back x. We end this section with some step by step exercises for mental mathematics.

AdditionIdeally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 1. Memorize the sums of two digits.

AdditionIdeally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93Step 3. Practice the sum of a two-digit number with a multiple of 10;

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 = for example, do mentally,

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 = 53 + 20 + 8 for example, do mentally,

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 =

= 73 + 8

53 + 20 + 8 for example, do mentally,

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 =

= 73 + 8

53 + 20 + 8 for example, do mentally,

= 81

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 =

= 73 + 8

97 + 55 = 53 + 20 + 8 for example, do mentally,

= 81

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 =

= 73 + 8

97 + 55 = 53 + 20 + 8 97 + 50 + 5 for example, do mentally,

= 81

Step 1. Memorize the sums of two digits.

Addition

for example, do mentally,

Ideally, one should be comfortable with mental addition of two two-digit numbers such as totaling $28 and $45. Below are a series of mental exercises that will help one to do that.

Step 2. Practice the sum of a two-digit number with a digit;

33 + 4 = 37 57 + 8 = 65 9 + 84 = 93

for example, do mentally,

Step 3. Practice the sum of a two-digit number with a multiple of 10;

13 + 40 = 53 57 + 40 = 97 29 + 90 = 119

Step 4. Do the sum of a two-digit number in two steps by adding the 10’s first, then add the unit-digits;

53 + 28 =

= 73 + 8

97 + 55 = = 147 + 5 = 152

53 + 20 + 8 97 + 50 + 5 for example, do mentally,

= 81

AdditionYour Turn: Do the following mentally in two steps.26 + 27 = 26 + 20 + 7 = 44 + 39 = 44 + 30 + 9 =87 + 48 = 87 + 40 + 8 =

HW A. Fill in the values.

Number of nickels

Sum in ¢’s

2 4 6 8 10 12 14 16 18 20

Addition

Number of nickels

Sum in ¢’s

1 3 5 7 9 11 13 15 17 19

2. Arrange the following values from the largest to the smallest then add them.

10431 9776

10429

513869

513896

a.1045 93036

504

639869

837372

b.

Exercise B. Do the following problems two ways. * Add the following by summing the multiples of 10 first. * Add by adding in the order.to find the correct answer. 1. 3 + 5 + 7 2. 8 + 6 + 2 3. 1 + 8 + 9 4. 3 + 5 + 15 5. 9 + 14 + 6 6. 22 + 5 + 8 7. 16 + 5 + 4 + 3 8. 4 + 13 + 5 + 79. 19 + 7 + 1 + 3 10. 4 + 5 + 17 + 311. 23 + 5 + 17 + 3 12. 22 + 5 + 13 + 2813. 35 + 6 + 15 + 7 + 14 14. 42 + 5 + 18 + 1215. 21 + 16 + 19 + 7 + 44 16. 53 + 5 + 18 + 27 + 2217. 155 + 16 + 25 + 7 + 344 18. 428 + 3 + 32 + 227 + 22

Addition