Lesson 6 (bonus). Decimal math: add six

After this lesson the adder counts in tens, like a calculator: 38 + 47 = 85 — and the display shows exactly these digits. An optional lesson: the calculator of lesson 5 is complete without it.

6.1. The idea: a digit in four bits

People count in tens, the adder in sixteens. Calculators and cash registers solved it simply: each decimal digit was kept in its own 4 bits. That is binary-coded decimal (BCD): 38 is 0011 (3) and 1000 (8), that is 0011 1000.

An ordinary ADDER4 adds the digits. While the digit sum is at most 9, all is right. If it is more, 4 bits have six codes 1010…1111 that a decimal digit never uses, and we have to jump over them: add 6. A carry into the next digit is born at the same time.

The digit sum is more than 9 when:

• the binary adder gave a carry — that is 16…19, or
• s3 = 1 and (s1 = 1 or s2 = 1) — that is 10…15.

That is how calculators and the first microprocessors counted: the Intel 4004 has the DAA instruction for it, and the 6502 has a decimal mode (SED).

6.2. Why the display is in HEX mode

The display knows nothing about BCD: it takes 8 bits and shows them as a number in the chosen system.

• In HEX one display digit is exactly 4 bits. The low digit is bits 0–3, the high one is 4–7. And in BCD we also put one digit in every 4 bits. So the HEX display shows our digits as they are: 0011 1000 → 38.
• In DEC the display reads the same 8 bits as one binary number: 0011 1000 = 32 + 16 + 8 = 56.

The "+6" correction makes sure a group of four bits never holds 1010…1111, so the display never shows the letters A…F — only digits 0–9. A letter appeared — the correction did not fire.

Real calculators did exactly this: each digit was kept in its own 4 bits and went straight to its own indicator. Converting a binary number to decimal means dividing by 10, and that is far more hardware than the "+6" correction.

6.3. Building

An optional lesson: the calculator of lesson 5 is complete without it.

The same project. Delete the lesson 5 setup (17 switches, two ADDSUB8, lamps, displays); keep Power/Ground.

The idea. People count in tens, the adder in sixteens. Let us write each decimal digit in its own 4 bits (binary-coded decimal, BCD): 38 = 0011 1000. We add the digits with an ordinary ADDER4. While the digit sum is ≤ 9, all is right. Above 9, 4 bits have six codes 1010…1111 that a decimal digit never uses — we have to jump over them: add 6. A carry into the next digit is born at the same time.

The digit sum is more than 9 when:

• the binary adder gave a carry (16…19), or
• s3 = 1 and (s1 = 1 or s2 = 1) — that is 10…15.

That is how calculators and the first microprocessors counted: the Intel 4004 has the DAA instruction for it, the 6502 has the SED mode.

Step 1. BCD1 — one decimal digit

Ctrl+G → name BCD1. Pins: inputs A0–A3, B0–B3, CI; outputs D0–D3, CO.

CO is two things at once: "add 6" (it goes to B1, B2 of the second ADDER4, since 6 = 0110) and the carry into the next digit.

Step 2. Canvas: two digits

The displays are in HEX mode: each hexadecimal digit is exactly 4 bits, so the display shows binary-coded decimal digits as they are: 0011 1000 reads as 38.

How to set the digits

The low digit is switches A0–A3 (B0–B3), the high digit is A4–A7 (B4–B7). Each digit is 0 to 9:

6.4. Check

In the "Switches" columns the top switch (A7) is on the left and A0 on the right; 1 — on, 0 — off. Each group of 4 is one decimal digit: tens on the left, units on the right.

If the answer is right up to 9 but beyond that letters (A…F) appear on the display, the correction does not fire: check OR/AND inside BCD1 and that CO goes to B1 and B2 of the second ADDER4.

Lesson 6 paste (Cmd+I)

Needs AND, OR and ADDER4. Delete the lesson 5 setup, keep the power.

# Lesson 6 (bonus) — decimal math: BCD1 with the +6 correction.chip BCD1 {  portIn(A0)  portIn(A1)  portIn(A2)  portIn(A3)  portIn(B0)  portIn(B1)  portIn(B2)  portIn(B3)  portIn(CI)  portOut(D0)  portOut(D1)  portOut(D2)  portOut(D3)  portOut(CO)  ADDER4(A0=A0, A1=A1, A2=A2, A3=A3, B0=B0, B1=B1, B2=B2, B3=B3, CI=CI, S0=s0, S1=s1, S2=s2, S3=s3, CO=c)  OR(s1, s2, s12)  AND(s3, s12, big)  OR(c, big, CO)  ADDER4(A0=s0, A1=s1, A2=s2, A3=s3, B0=GND, B1=CO, B2=CO, B3=GND, CI=GND, S0=D0, S1=D1, S2=D2, S3=D3)} input(A0)input(A1)input(A2)input(A3)input(A4)input(A5)input(A6)input(A7)input(B0)input(B1)input(B2)input(B3)input(B4)input(B5)input(B6)input(B7)BCD1(A0=A0, A1=A1, A2=A2, A3=A3, B0=B0, B1=B1, B2=B2, B3=B3, CI=GND, D0=D0, D1=D1, D2=D2, D3=D3, CO=c10)BCD1(A0=A4, A1=A5, A2=A6, A3=A7, B0=B4, B1=B5, B2=B6, B3=B7, CI=c10, D0=D4, D1=D5, D2=D6, D3=D7, CO=H)led(H)