A natural number can be defined as a whole non-negative number. According to KLB book 1, these numbers include 0,1,2,3,4,5,6,7,8 and 9.
Place values of numbers
Each digit in a whole number has a place value, based on its position from the right, as seen in the following place value chart.
Hundred Millions | Ten Millions | Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
100,000,000,000 | 10,000,000,000 | 1,000,000 | 100,000 | 10,000 | 1000 | 100 | 10 | 1 |
The total value of a digit in the number is the digit multiplied by the place value.
The table below shows both the place values and total values of some numbers.
Number | Place Value (of number in bold) |
Total Value (of number in bold) |
2,345 | Ones | 5 |
2,345 | Tens | 40 |
2,345 | Hundreds | 300 |
2,345 | Thousands | 2,000 |
9,765,321 | Ten thousands | 60,000 |
9,765,321 | Hundred thousands | 700,000 |
9,765,321 | Millions | 9,000,000 |
456,789,012 | Ten millions | 50,000,000 |
456,789,012 | Hundred millions | 400,000,000 |
The last number in words can be written as, four hundred and fifty six million, seven hundred and eighty nine thousand, and twelve.
Round off numbers to the nearest tens, hundreds, thousands, millions and billions
Billion
A billion is a thousand millions, written as 1,000,000,000. It has 10 places, ie ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, and billions.
45,827,652,098 can therefore be written as forty five billion, eight hundred and twenty seven million, six hundred and fifty two thousand and ninety eight.
Exercise
Write in words:
- 2,010,951,032
- 8,996,783,154
- 22,917,509,735
- 91,330,000,501
- 609,071,008,209
- 909,999,909,990
Write in numerals:
- Four billion seven hundred twenty-three million one hundred thirty-three thousand four hundred seventy-eight
- Three hundred seventy-four million five hundred twenty-six thousand four hundred thirty-eight
- Ten billion two hundred eighty-eight million three hundred forty-seven thousand eighty-two
- Nineteen billion seven hundred ninety-two million six hundred sixteen thousand three hundred eleven
- Two hundred fifty-nine billion one hundred eighty-six million seven hundred fifty-one thousand three hundred twenty-seven
Rounding Off
Twelve co-workers win Lotto and share the Ksh 11 970 183 first division prize equally amongst themselves. How much does each person win?
If your calculator agrees with mine, they each win Ksh 997 515.25.
Imagine the excitement! Each of the 12 winners would have a long list of sons, daughters, parents, aunts, uncles and friends with whom to share the news. Do you think the winners would phone their friends and family and say, “I’ve just won Ksh 997 515.25 on Lotto!”
Not likely. It would be more likely for them to say, “I’ve just won a million shillings on Lotto!”
Now, the winners were merely doing what we all do many times every day—they were rounding the size of their win in order to get across the important idea, which in this case is the magnitude of the prize.
Numbers are rounded in order to quickly find estimates.
How do we round off numbers?
Consider the digits we have in mathematics. There are only 10 of them.
0 1 2 3 4 | 5 6 7 8 9
There are five numbers that are 4 or lower, and five numbers that are 5 or larger.
If the number is 0, 1, 2, 3 or 4, you round down.
If the number is 5, 6, 7, 8 or 9, you round up.
Examples:
Round off to the nearest number shown in brackets.
- 2,934 (10)
- 453,876 (10,000)
- 98,957 (100)
- 35,872,764,567 (1,000,000,000)
Solutions.
- 2,930
- 450,000
- 99,000
- 36,000,000,000
Operations with Whole Numbers
In this section we will consider the four basic operations with whole numbers: addition, subtraction, multiplication, and division. It is assumed that you are familiar with computations, which are reviewed only briefly here. Our emphasis will be on how the operations are related to each other, as well as their application to real-world situations.
Addition
Addition represents the idea of finding a total count, or summing up, of values.
Example Find the sum: 458 + 375 + 296
Solution Unless you are using a calculator, it is easier to organize this problem vertically:
458
375
+296
______
Starting in the ones place value, 8 + 5 + 6 = 19. Representing this sum as “carrying” into the tens place:
1
458
375
+296
______
9
Now working in the tens place value, 1 + 5 + 7 + 9 = 22. Representing this sum as “carrying” into the hundreds place:
21
458
375
+296
________
29
Finally working in the hundreds place value, 2 + 4 + 3 + 2 = 11. Representing this sum as “carrying” into the thousands place:
21
458
375
+296
_______
1129
Thus 458 + 375 + 296 = 1129.
Subtraction
Subtraction of whole numbers is a natural result of an addition, called the inverse operation of addition. Given the addition statement 6 + 4 = 10, there are two associated subtraction statements:
10 – 6 = 4 and 10 – 4 = 6
Thus subtraction represents the idea of “undoing” addition. In general, if a + b = c, then c – a = b and c – b = a.
Example: Find the difference: 1426 – 548
Solution: As with addition, we align the problem vertically and then “borrow” from the next higher place value, when necessary. Borrowing 1 ten = 10 ones, then subtracting 16 – 8 = 8:
1
1426
-548
________
8
Now borrowing 1 hundred = 10 tens, then subtracting 11 – 4 = 7:
31
1426
-548
_____
78
Finally borrowing 1 thousand = 10 hundreds, then subtracting 13 – 5 = 8:
031
1426
-548
_____
878
Thus 1426 – 548 = 878. Since subtraction is the inverse operation of addition, this answer can be checked by performing the addition 878 + 548 = 1426.
Example After depositing his Ksh 42,800 weekly paycheck, Jose has Ksh 114,500 in his checking account. How much was in his account before the deposit?
Solution Thinking of B as the balance prior to the deposit, the statement B + 42,800 = 114,500 correctly interprets the deposit and final balance. Since subtraction is the inverse operation of addition, then B = 114,500 – 42,800.
Computing the subtraction:
3
1145000
-428000
_______
717000
Thus Ksh 717,000 was in the account before the deposit. Again, note that this answer can be checked by computing the sum 717,000 + 428,000 = 114,500.
Multiplication
Multiplication of whole numbers represents the idea of repeated addition. For example:
4 x 5 = 5 + 5 + 5 + 5 = 20
5 x 4 = 4 + 4 + 4 + 4 + 4 = 20
Example Compute the product: 256 x 47
Solution As with addition, start by writing the product vertically:
256
x47
Starting with the ones place, 6 x 7 = 42. Carrying and continuing the multiplication:
34
256
x47
_____
1792
Now moving to the tens place (remember the 4 represents 40):
122
256
x47
_____
1792
10240
Finally adding the two products:
256
x47
_____
1792
10240
_____
12032
Thus 256 x 47 = 12,032 .
Example The Wheeler family makes monthly mortgage payments of Ksh 846 for a total of 15 years. What is the total amount of their payments?
Solution First we need to find the number of payments they make during the 15 years.
Since they make a payment each month, and there are 12 months in a year, they make a total of
15 x 12 = 180 payments during the 15 year period of time. Since each payment is Ksh 846 and they make 180 payments, they pay a total of
180 x Ksh 846 = Ksh152,280
The total amount of the Wheeler’s payments is Ksh 152,280.
Division
Division of whole numbers represents the idea of repeated subtraction. For example:
36 ÷ 12 = 3 since 36 – 12 – 12 – 12 = 0
15 ÷ 3 = 5 since 15 – 3 – 3 – 3 – 3 – 3 = 0
Division is more commonly thought of as the inverse operation for multiplication. For example:
36 ÷ 12 = 3 since 3 x 12 = 36
36 ÷ 3 = 12 since 12 x 3 = 36
15 ÷ 3 = 5 since 5 x 3 = 15
15 ÷ 5 = 3 since 3 x 5 = 15
In general, if a x b = c, then c ÷ a = b and c ÷ b = a.
Example Compute the quotient: 6120 ÷ 45
Solution Writing the quotient in our more traditional form, and using the “guess and subtract” technique learned in elementary school:
136
45 6120
45
_______
162
135
_______
270
270
_______
0
Thus 6120 ÷ 45 = 136. Note that we can check this quotient with the multiplication 136 x 45 = 6120. We say that the quotient is 136 and the remainder is 0. Often when the remainder is 0 we merely say the quotient is 136 (and the remainder is assumed to be 0).
Example Compute the quotient: 8495 ÷ 27
Solution Using the same approach as in the example above:
314
27 8495
81
_____
39
27
________
125
108
_______
17
Thus 8495 ÷ 27 = 314, with a remainder of 17. This answer can be summarized with the notation 314 R 17. To check this quotient, compute 27 x 314 = 8478, then add on the remainder 8478 + 17 = 8495. That is, 27 x 314 + 17 = 8495.
Example The Omondi's family borrows Ksh 20,880 to purchase a new set of seats at a special 0% interest rate. The seat dealer allows them 5 years to pay back the amount they borrow, and requires equal monthly payments. How much are their monthly payments?
Solution Since there are 12 months in each year, they must make a total of 5 x 12 = 60 payments on the loan. Dividing Ksh20,880 by 60 will result in the monthly payment:
348
60 20880
180
_______
288
240
_______
480
480
________
0
The Omondi's monthly payment will be Ksh 348. As a check 60 x 348 = 20,880.
Exercise Set
Perform the following additions and subtractions.
- 2,456 + 8,946
- 892 + 5,688
- 98 + 1,856
- 9,568 + 5,487
- 17,847 + 6,879
- 6,956 + 65,462
- 534 – 276
- 1,002 – 453
- 2,231 – 859
- 12,458 – 5,674
- 657329 + 400708
- 101,200 – 53,432 12. 102,101 – 57,234
Perform the following multiplications.
- 56 x 35
- 28 x 57
- 154 x 87
- 268 x 67
- 1,859 x 68
- 2,695 x 465
- 10,000 x 64
- 95 x 100,000
- 101 x 1001
- 111 x 1011
- 5,305 x 132
- 6,487 x 328
Perform the following divisions.
- 2,668 ÷ 58
- 7,743 ÷ 89
- 12,549 ÷ 47
- 38,090 ÷ 65
- 8,365 ÷ 27
- 9,740 ÷ 48
- 14,846 ÷ 124
- 33,429 ÷ 132
- 1,450,000 ÷ 1000
- 560,000 ÷ 100
- 105,812 ÷ 1,740
- 867,594 ÷ 2,317
Answer each of the following application questions. Be sure to read the question, interpret the problem mathematically, solve the problem, then answer the question. You should answer the question in the form of a sentence.
- Mwangi has a balance in his checking account of Ksh 8590. He makes a deposit of Ksh 6380, then writes checks for ksh 920, Ksh 3370, and Ksh 2680. What is his new balance?
- Kwamboka has a balance in her checking account of Ksh 1,425. She makes two deposits of Ksh 4350 and Ksh 1690, then writes checks for Ksh 2090, Ksh 1480, and ksh 970. What is her new balance?
- Larry has Ksh 4,600 cash in his wallet. If he just loaned Ksh 3,500 to Moe, how much did he have before he loaned Moe the money?
- Mohammed has Ksh 28,700 in his savings account after he withdrew Ksh 9,700 to buy new roller blades. How much did he have in his savings account before he withdrew the money?
- Wafula buys an RV for which he pays Ksh 24,800 per month for 20 years. What is the total amount he paid for his RV?
- Steve runs 6 miles per day during the weekdays and 15 miles per day on the weekend. How many total miles does he run during one week? If he burns off 124 calories each mile he runs, how many calories does he burn during the week?
- Todd talks 45 minutes per day on the phone during the weekdays and 52 minutes per day on the weekend. How many total minutes does he talk during one week? How many full hours does he talk during the week?
- Wanjiku pays Ksh 3,500 rent each month for five years. How much total rent does she pay during those five years?
- Linda signs a lease to pay Ksh 7,250 rent each month for two years. How much total rent does she owe for the lease?
- Carolyn buys a new sports car at a total loan cost of Ksh 3,240,000. If she makes monthly payments on her loan for five years, how much are her monthly payments?
- Sandra decides to buy a house and signs loan papers which require Ksh 240,000 up-front (down payment), then payments of Ksh 58,600 per month for 30 years. What is the total cost of the loan for Sandra?
- Sandra (from Exercise xi ) is offered another loan with the same down payment, then payments of Ksh 72,200 per month for 15 years. Find the total cost of this loan, and compare it with the loan from Exercise xi. How much money will she save with this new loan?
- After college, you accept a job which pays Ksh 24,000 per month for the first year, then a raise of Ksh 1,750 per month for the second year. What will your total income be for the first two years at your job?
- For college, you rent an apartment for Kah 6,500 per month the first year, with an increase of Ksh 250 per month for each of the second, third, and fourth years. What is the total rent paid for the four years at college?
- You commute to (and from) work 218 times during the year. The distance from your home to work is 24 kilometers. What is the total distance commuting during the year?
- Jerry packs avocados into bags which hold 48 avocados. If he needs to pack 8,900 avocados, how many full bags will he be able to pack?
- Jerry’s avocados trees produce 80 pounds of avocados per tree. He plants 50 trees per acre, and has 86 acres of avocados. Find the total production of avocados from his trees.
Odd numbers
Odd number is a number which has a remainder of 1 upon division by 2, in other words, it is a number that is not divisible by 2. Numbers that end in 1,3,5 and 7 are all odd numbers. eg 567, 43,217, 1,001,001
Even numbers
An even number is a number which has a remainder of 0 upon division by 2, in other words, an even number is a number which is divisible by 2.
Therefore all numbers that end in 0,2,4,6 or 8 are even numbers, eg 23,456, 42, 450,010.
Prime numbers
A prime number can be divided, only by itself and by 1, without a remainder. For example, 17 can be divided only by 17 and by 1.
Here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199