Mathematics notes, chapter 2
Number Bases
Textbook Mathematics Form 4 KSSM (DLP), pages 32–53
What this chapter covers
- Represent and explain numbers in various bases in terms of numerals, place values, digit values and number values (2.1.1)
- Convert numbers from one base to another using division by place value, division by the base, and direct conversion between base two and base eight (2.1.2)
- Perform addition and subtraction of numbers in various bases using vertical form or conversion to base ten (2.1.3)
- Solve problems involving number bases (2.1.4)
2.1Number bases
Digits, bases and place values
Number base: a number system that groups quantities in sets of the same size. In base a the only digits are 0 up to a − 1. Base ten (decimal) is the one used in daily life. Computer science uses base 2, base 8, base 10 and base 16.
| Base | Digits that can be used |
|---|---|
| 2 | 0, 1 |
| 5 | 0, 1, 2, 3, 4 |
| 8 | 0, 1, 2, 3, 4, 5, 6, 7 |
| 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
We write the base as a small number at the bottom right: 325 means "the number 32 in base five". Read it "three two, base five". The base sign for base ten may be left out.
A digit must always be smaller than the base. 3415 is possible, but 2915 is not a base-five number because 9 is not a digit of base five.
In base a, the place values from the right are a0, a1, a2, a3, … Each place is worth a times the place on its right.
| Base | a4 | a3 | a2 | a1 | a0 |
|---|---|---|---|---|---|
| 2 | 16 | 8 | 4 | 2 | 1 |
| 5 | 625 | 125 | 25 | 5 | 1 |
| 8 | 4 096 | 512 | 64 | 8 | 1 |
| 10 | 10 000 | 1 000 | 100 | 10 | 1 |
Digit value and number value
Digit value = digit × place value. Number value (the value in base ten) = the sum of all the digit values.
Example 3 (textbook p. 37): digit value
Digit value of the 4 in 44325: the 4 on the left is in the 53 place, so 4 × 53 = 4 × 125 = 500.Example 4 (textbook p. 39): value of 3 4 17 in base ten
(3 × 72) + (4 × 71) + (1 × 70) = 147 + 28 + 1 = 176So 3417 = 17610.
Check on a scientific calculator: press MODE until BASE appears, choose the base of the number (BIN, OCT, HEX or DEC), type the number, then press the key for the base you want.
Find the value of 11012 in base ten.
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(1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 8 + 4 + 0 + 1 = 13
Numbers with the same digits but different bases have different values. 2438 = 163 but 2435 = 73.
Converting a number in base ten to another base
Divide by the new base again and again until the quotient is 0. Write the remainders from the bottom upwards.
Example 5 (textbook p. 41): change 56310 to base five
| Division | Quotient | Remainder |
|---|---|---|
| 563 ÷ 5 | 112 | 3 |
| 112 ÷ 5 | 22 | 2 |
| 22 ÷ 5 | 4 | 2 |
| 4 ÷ 5 | 0 | 4 |
The other method divides by the place values: 563 ÷ 125 = 4 remainder 63, 63 ÷ 25 = 2 remainder 13, 13 ÷ 5 = 2 remainder 3, 3 ÷ 1 = 3. The digits 4, 2, 2, 3 are the same.
Reading the remainders from the top downwards. The first remainder is the last digit of the new number.
Converting between two bases that are not ten
Change to base ten first, then to the base wanted.
Example 6 (textbook p. 42): change 2536 to base nine
Step 1: (2 × 36) + (5 × 6) + (3 × 1) = 72 + 30 + 3 = 10510Step 2: 105 ÷ 9 = 11 remainder 6; 11 ÷ 9 = 1 remainder 2; 1 ÷ 9 = 0 remainder 1
So 2536 = 10510 = 1269.
Base two and base eight directly
One digit in base eight stands for exactly three digits in base two (because 23 = 8).
| Base 8 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Base 2 | 000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 |
- Base 2 to base 8: group the digits in threes from the right; add zeros on the left to complete the last group; change each group to one digit. 11011012 → 1 | 101 | 101 → 1558.
- Base 8 to base 2: change each digit to three binary digits. 5178 → 101 | 001 | 111 → 1010011112.
Grouping from the left. Always start from the right, otherwise the groups are wrong when the number of digits is not a multiple of 3.
Convert 10110102 to base eight, then convert 648 to base two.
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1 | 011 | 010 gives 1, 3, 2, so 1328.
6 = 110 and 4 = 100, so 648 = 1101002.
6 = 110 and 4 = 100, so 648 = 1101002.
2.2Addition and subtraction in various bases
Two methods: (a) work in vertical form in the given base; (b) change both numbers to base ten, calculate, then change the answer back.
Addition in base a: if a column sum reaches a or more, write the remainder and carry the number of groups of a. Subtraction in base a: when you borrow, one unit in the next place is worth a, not 10.
Example 10(a) (textbook p. 46): 1102 + 1112
Right column: 0 + 1 = 1. Middle: 1 + 1 = 2 = 102, write 0 carry 1. Left: 1 + 1 + 1 = 3 = 112, write 11.Answer: 11012.
Check by base ten: 6 + 7 = 13 = 11012.
Example 11(c) (textbook p. 48): 3728 − 778
Units: 2 is smaller than 7, borrow: 8 + 2 − 7 = 3. Eights: 6 (after borrowing) is smaller than 7, borrow: 8 + 6 − 7 = 7. Sixty-fours: 3 − 1 (the borrowed one) = 2.Answer: 2738. Check: 250 − 63 = 187 = 2738.
Borrowing 10 instead of the base. In base eight you add 8 to the digit, not 10.
Calculate 43267 + 16547.
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6 + 4 = 10 = 137, write 3 carry 1. 1 + 2 + 5 = 8 = 117, write 1 carry 1. 1 + 3 + 6 = 10 = 137, write 3 carry 1. 1 + 4 + 1 = 6.
Answer: 63137. Check in base ten: 1 539 + 676 = 2 215 = 63137.
Answer: 63137. Check in base ten: 1 539 + 676 = 2 215 = 63137.
2.3Problems involving number bases
Follow the textbook steps: understand the problem, plan the strategy, carry it out, then conclude (and check). Work in base ten for percentages and money, and change only the final answer to the base required.
Example 12 style (textbook p. 49)
Of 200 pupils, 25% come by bus. Number by bus = 25 ÷ 100 × 200 = 50 pupils. In base four: 50 ÷ 4 = 12 r 2, 12 ÷ 4 = 3 r 0, 3 ÷ 4 = 0 r 3. So the number is 3024.To compare numbers in different bases, change them all to base ten first. Then use >, < or order them.
Summary
| Task | Method |
|---|---|
| Digit value | digit × place value |
| Base a to base ten | add the digit values |
| Base ten to base a | divide by a repeatedly, read remainders upwards |
| Base 2 to base 8 | group in threes from the right |
| Base 8 to base 2 | each digit becomes three digits |
| Add / subtract | vertical form (carry or borrow the base), or via base ten |
Content and examples follow Mathematics Form 4 KSSM (DLP) textbook, Chapter 2, pp. 32–53.