# Calculating devices (Abacus, Slide Rule, pocket Calculator and the Computer)

**FURTHER MATHEMATICS SS 1 SECOND TERM**

**WEEK 3**

**SUBJECT: FURTHER MATHEMATICS**

**CLASS: SS1**

**TOPIC: CALCULATING AND PROCESSING DEVICES**

**CONTENT**:

- Calculating units (Abacus, Slide Rule, pocket Calculator and the Laptop)
- Decimal, binary, octal, hexadecimal techniques and relationship between them
- Stream charts
- Software of movement – chart

**Sub – Subject: CALCULATING DEVICES**

Calculating units are these guide strategies and machines for dashing up calculations have been devised. One in all such units that makes use of a positional numeration system is the abacus. Others embrace lattice or gallery methodology, slide rule invented by william oughtred, Russian peasant methodology, Egyptian multiplication algorithm, John Napier logarithms, e.t.c. All of the afore-mentioned and plenty of others not talked about have been current and ruling the world of calculation earlier than the invention of pocket, desk and digital calculators, computer systems and the opposite sorted trendy calculating machines. It’s value noting that, all the trendy calculating machines in the present day have been construct up on the present historic calculating and processing units. For instance, from the works of John Von Neumann in 1946, try was made to develop the digital pc, the place a programmer can compose a couple of directions, enter them into the pc reminiscence, after which watch whereas it performs hundreds of thousands of calculations by itself and stories the solutions.

**ABACUS-CALCULATING DEVICE**

Abacus, instrument utilized in performing arithmetic calculations. It consists basically of a pill or body bearing parallel wires or grooves on which counters or beads are moved. A contemporary abacus consists of a picket body with beads on parallel wires, and a crossbar oriented perpendicular to the wires that divides the beads into two teams. Every column—that’s, every wire—represents one place within the decimal system. The column farthest to the proper is those column; the subsequent column to the left is the tens column; and so forth. In every column, there are 5 beads beneath the crossbar, every of which characterize one unit, and two beads above the crossbar, every of which characterize 5 models. For instance, within the tens column, every of the group of 5 beads represents ten, and every of the group of two beads represents fifty. Beads which are to be counted as a part of a quantity are positioned towards the crossbar. Abacus is utilized in locations like China and Japan. Addition and subtraction are carried out by motion of beads back and forth for:

- including or eradicating beads
- Repeated addition is completed with the abacus for multiplication.

**ABACUS**

**CLASS ACTIVITY:**

- Draw abaci to characterize every of the next numbers:
- 4872 (b) 3403
- Stay 5 calculating units of outdated and their inventors

**Sub – matter: DECIMAL AND BINARY SYSTEMS**

**Decimal system:** These are numbers in on a regular basis use. They’re additionally denary numbers or numbers in base 10. Any denary quantity will include any of the ten symbols (i.e. 0,1,2,3,4,5,6,7,8,9).

**Binary system:** The binary system is system of numbers, which has 2 as its base. The digits in base two and 0 and 1. **Laptop is made on the mechanism of a two-way system system**. A two approach system system is a system, which might decide that has two alternate options like:

- True or False assertion
- Sure or No
- Swap ON or OFF in electrical circuits, e.t.c

In an electro – magnetic system as an example, 1 can stand for magnetize whereas 0 stands for demagnetize or in {an electrical} circuits for change off. It is for that reason that the binary techniques dominate different quantity system in pc.

**Addition and Subtraction in Base two, Base ten, Octal and Hexadecimal techniques**

Addition in base two is carried out within the often/frequent approach. Another methodology to traditional methodology of Subtraction in base two is to show it into addition by means of** twos complement** of binary numbers of the **subtrahend**. The twos praise of a quantity is a quantity obtained by including 1 to those praise. The **ones complemen**t is obtained by altering eachzero into one every one into zero.

Octal and hexadecimal numbers may be transformed to binary numbers utilizing **three – bit and 4 – bit equal kinds** respectively, and vice versa

**Example1**: Consider the next in base two utilizing the twos enhances of the subtrahend

11011101 – 10110011

Answer (1)

11011101 minuend

- 10110011 subtrahend

101010

**ALTERNATIVE METHOD **(Use of twos complement of subtrahend)

Twos complement of 10110011 is (01001100) + 1 = 01001101. Including this to the minuend we get

11011101

- 1001101

100101010 the circled 1 is deleted, therefore 11011101 – 10110011=101010

**Instance 2**: Consider the next in base sixteen (hexadecimal) then change your reply to binary quantity utilizing 4 – bit equal kind.

4F6D

- 5BCE

AB3B

to base two offers 1010 1011 0011 1011

**Instance 3: **convert the next binary numbers to octal numbers utilizing three – bit equal kinds:

- 111011011 (b) 11011011101

Answer: (a) grouping 111011011 in threes from proper to left we get 111 011 011

7 3 3

Subsequently,

- 11011011101 in teams of threes (3 – bit equal) 011 011 011 101

3 3 3 5

Subsequently, 11011011101_{2}=3335_{8}

**CLASS ACTIVITY:**

- Consider every of the next in base two utilizing the twos enhances of the subtrahends:
- 11110101 – 10110101 (b) 11101011 – 10111001
- Convert the next binary numbers to: (i) octal and (ii) hexadecimal numbers utilizing three – bit or 4 – bit equal kinds appropriately.
- 1011011011101 (b) 11101101101101

**Sub – Subject: FLOW CHARTS AND ITS APPLICATIONS**

A scheme of labor of particular directions concerned in fixing any mathematical downside is understood algorithm. Whereas when an algorithm is offered in a diagrammatic kind is named movement chart.[mediator_tech]

In different phrases, a movement chart is a diagram which reveals the logical sequence of the operations which should happen. Every field comprises one instruction. The arrows inform us which instruction to observe first, which comes second and so forth. When a course of is repeated in a movement chart, loops are used. Stream chart is principally use for fixing on a regular basis arithmetic issues and different sophisticated operations are diminished to small steps.

The desk beneath reveals the geometrical symbols utilized in movement charting.

Form /image |
Identify |
Perform |

Oval | Terminal image | |

Rectangle | Course of image | |

Parallelogram | Enter/output image | |

Diamond | Resolution image | |

Small circle | Connector image |

**Instance: **

Calculate the imply of the scores of 60 college students in a arithmetic quiz utilizing movement chart.

**Answer:**

Step 1: Receive the rating of every pupil.

Step 2: Sum up the scores.

**START**

**READ SCORE**

**MEAN=TOTALSCORE +60**

**WRITE MEAN SCORE**

**STOP**

Step 3: Divide complete rating by 60.

Step 4: File the imply.

** The movement chart**

**CLASS ACTIVITY**

Draw a movement chart to indicate what occurs while you:

- Go to a library
- Cross a street
- Search for a phrase within the dictionary
- Whenever you bake cake

**PRACTICE QUESTIONS**

- Change the next decimal numbers to : (i) octal numbers (ii) hexadecimal numbers
- 3692 (b) 7386
- Convert every of the next to base ten utilizing the prolonged multiplication:
- 5642
_{8}(b) EF3B_{16} - Convert every of the next binary numbers to hexadecimal numbers utilizing 4 – bit equal kinds:
- 111011011 (b) 101101101011
- Use slide rule to search out the reciprocal of every of the next:
- 2.3 (b) 15.2
- Make an inventory of 5 primary elements of a contemporary pc.

**ASSIGNMENT**

- Decide the flowchart image wherein we will place every of the next directions:
- Learn X,Y
- Y=2*X+3
- IS Y<15
- Write Y
- Provided that draw a easy movement chart to calculate the values of at intervals of 1 given the values of from
- Change every of the next decimal numbers to octal numbers:
- 2462 (b) 9268
- Draw abaci to characterize every of the next numbers:
- 3403 (b) 6575
- Write out the digits in base: two, octal system, and hexadecimal system

**KEY WORDS/PHRASES**

**BINARY****OCTAL****HEXADECIMAL****FLOWCHART****MINUEND****SUBTRAHEND****ONES COMPLEMENT****TWOS COMPLEMENT**